@sriprasadjoshi3036 some say 0 is even, some say it is neither even nor odd. Recent advances in set theory strongly suggest the ancient mathematicians made an error and 0 is actually a natural number. There is an ongoing debate at the moment.
Yeah easily D, the rest are like good debates that are just worn out. D is just faulty written. Barely anyone even uses the division symbol, the just write it fractionally depending on what they want to portray
I really don't think D is that bad of a notation. To me, D is very clear and has a clear answer. In terms of bad notation, tan^-2(x) should take the top spot. : )
Right. I don't like D not so much for mathematical reasons, but rather because its one of the low-effort comment engagement posts that bots post on social media. And I hate that.
@@blackpenredpen probably not to someone who really knows their math, but 12/3(4) would be very different to (12/3)4, give two answers because they are different. And pemdas is not taught well, there’s always confusion on multiplication or division first. The way it’s written is meant to confuse people into thinking it’s 12/(3 * 4) so I would say it’s bad notation, since most people use fraction to mistake division now
I cannot believe the amount of people who don't understand D. Like, if you have 5/4+4*5 do they seriously think that means 5/(4+4*5)? In what world does it make sense to take everything after a division sign and throw it together in parentheses when there are no parentheses... Perhaps PEMDAS needs to be taught a few more times in school.
@@gudadada Actually, both sides of the debate are correct. This expression can be solved either way, as both interpretations of the expression are commonly accepted. When you have a number adjacent to a parenthesis, it's called a juxtaposition, and is solved before other operations. Outside the US, some countries instead learn BEJMA (Brackets, Exponents, _Juxtaposition_ , Multiplication, Addition). It's very handy for factoring. Imagine the expression [(2x² + 4x)], which can be rewritten as [2x(x + 2)] So technically, both forms are correct. Just make sure to use extra brackets when inputting into a calculator (I have two different calculators that solve it differently), or when sharing your problems with others, to make sure everybody is on the same page!
No, you are misinterpreting what juxtaposition is. Juxtaposition refers to a sign being implied, but it does NOT change grouping. For example, 20÷3x,x=5 is NOT the same as 20÷(3x),x=5. The former is 100/3, the latter is 4/3. This is a rule agreed upon by all mathematicians and functionally by all calculators. Of course, this "issue" is usually mitigated by using fractions which have much better visual clarity (everything in the numerator and denominator is contained), but there is really no debate, only one answer is correct. The reason your factored example works is because of left-right order. (2x)(x+2) is the same as (2)(x)(x+2). Really, it's incorrect to view it as (2x)(x+2), because that's one step into expanding. If you instead gave an example with division, say 2/x(x+2),x=5, now you'd have an issue. 2/(x(x+2)) is NOT the same as 2/x(x+2). The lone x does not get attached to the (x+2) without parentheses - that is plain wrong. Try plugging these examples in yourself to an algebra calculator if you don't trust the people who do math for a career.@@brickbot2.038
This is partly because B and C already have solved answers. There’s not anything to argue about there. “A” *also* has a solved answer, but the answer is “both, but it depends on the context”. There are plenty of formulas where 0^0=1 is required because it still outputs correct answers, but there are other cases where the output would be undefined. You could kinda argue there, but you’d still reach the final answer pretty quickly. D gets argued about because it’s an argument about the understanding of syntax. It’s not arguing about a mathematical concept itself outside of parsing syntax correctly. In reality, whoever writes that should use clearer syntax regardless of - even if there is only one way to parse it, it’s still an awkward way of writing an expression.
@gudadada It's not that the first is wrong, it's just that absolutely no one would write it that way outside of inciting a debate. The second you see all the time halfway through a problem. Not everyone is so careful with parens when solving z=20/y; y=3x; x=5.
The division symbol isn't the problem. It's the juxtaposition of terms that people assume takes priority over the division symbol, that is the problem. We'd have the same issue if it were a simple slash. It would be a much more efficient order of operations, if juxtaposition DID take priority over division, because it would allow you to write your denominators without snaring them, and professional scientists and mathematicians use this order of operations all the time. It's middle school teachers who don't want to deal with this problem, and the curriculum they follow that created this problem, who tell you that multiplication and division have the same priority regardless of notation.
@@AlejandroMéndez-j6j scientist here, I see 1/2π all the time, nobody ever writes 1/(2π). Makes sense too, because if 1/2π did mean the same thing as (1/2)π, then you'd just write π/2 to begin with, why write a complex form when it can be simple. By having it this way you have short expressions that are unambiguos
@@Alvin853: You can also write 1/2/π. Actually, you can just write /2/π, although a lot of people won't understand that; but once you get used to it, it's very convenient.
@@carultch Professional mathematicians wouldn't write it like that because it is just intentionally vague. If you want to do it properly without parentheses you use one horisontal dash and put the 12 on top and the 3 on the bottom
@@omerdvir1709 != Means not equals in programming so 0 != 1 means 0 does not equal to 1. In maths 0! Means 0 factorial which should be interpreted as (0!)=1 which is also true
@@ghotifish1838you’re right. I’m actually doing programming on school so o should have got that reference but I thought he was referring to the factorial and it would make more sense to put the exclamation at the end
Side note: != as a substitute for ≠ is common (mostly via C) but not universal; Haskell uses /= and is another common inequality operator (e.g. BASIC, SQL, Excel). In C 0==!1 too, but 1=0! fails to parse (! is prefix not in C, postfix factorial in common maths).
B really seems almost more like a communication issue than a math issue. There is no question that 1 and -1 are both square roots of one. it's more that there is confusion over the fact that the square root function is only looking for the principle positive square root instead of all square roots.
You said it yourself, "the square root function is looking at positive values". It only uses positive values because the principal square root is a function, whereas the +- version is not
@@radupopescu9977um actually complex square root is a different function from the principal square root. just like complex logarithm is multivalued, so is complex square root. But normal square roots and logarithms have one value. No one would say that ln(2) is a bunch of numbers, in fact they would write the answer to the complex logarithm IN TERMS OF normal one! Like Ln(2) = ln(2) + 2n*pi*i or something. Just like the actual answer to square equation is written in terms of principal root with additional symbols, like Sqrt(2) = ± sqrt(2)
Especially since the given answer in the video is wrong. The right answer is that its contextual; right-most inner-most is the standard evaluation order since left-descent parsers are problematic. So in most cases, the solution really is 1. But some mathematicians who don't care about formal languages use the order given in this video. The binary division operator is non-standard to begin with.
Yeah, so many people always answer that "you forgot the +/-" in the comments of a maths RUclips video, when it's completely inappropriate to the solution development
I definitely would say (d) is the worst because (a), (b), and (c) are all about defining different math concepts and people who don't understand the concepts, while (d) is about clarity and people being taught different things in school
@@DatBoi_TheGudBIAS No, the sum was established before the current convention of convergence, and shouldn't be interpreted purely analytically. The "identity" depends on an expanded version of algebraic manipulations that are consistent with analytic continuation.
@@radupopescu9977 Yeah, I should have addressed this, since this is the main criticism, and it would definitely be a big deal if it would be possible to get any other real number using manipulations like this. Mathematics of course depends on any two independent parties being able to get the same result. Although I can't proof that there are no such manipulations, I can at least point out some applications of the "identity". First of all, ζ(-1)=-1/12, but this is not the big application everyone talks about. The real application is in algebra, as are the manipulations. You have this formula (Weyl denominator formula) that requires the halfsum of positive roots of a root system (important concept in Lie algebra), and the integers form a root system of a generalised Kac-Moody algebra (kinda like an infinite Lie algebra). Well, the proper constant for that formula is -1/24, which could be interpreted as half of 1+2+3+4+... You can apply the same logic to why the modular form (these are important complex functions) of weight 1/2 looks the way it looks, but this happens to follow from my "main" application.
I think D is the worst because writing it in fraction form would clear any debate so I think it's more of a communication/notation problem than a debate really. I can't think of a single situation where I would rather write ÷ instead of just expressing division as a fraction.
Not laying out division and multiplication in an intuitive order or using brackets unambiguously should be considered as invalid as not having a closing parenthesis... To paraphrase The Big Lebowski on PEMDAS: "You're not wrong Walter, you're just an asshole."
"I think D is the worst because writing it in fraction form would clear any debate so I think it's more of a communication/notation problem than a debate really." EXACTLY THIS IS WHAT I WAS SAYING. The debate is stupid because it revolves around an ambiguity that should not be their in the first place.
D is the least interesting as it's mostly a question on syntax. The people who say the answer is 1, generally do so because they view 3(4) as implied multiplication, which has been taught (by some) to have higher precedence than standard multiplication (using the "x" or "÷" symbols). I wonder how the responses may change if we did some alterations to the question: 12/3(4)=? or evaluate 12/3x where x = 4? or what about: 12 ÷ 3π=? would you evaluate that as (12 ÷ 3) x π or 12 ÷ (3 x π) I'm not arguing for one or the other, it's just that I can see how people would find it ambiguous and I can see an argument for both sides. But all in all, it's just not an interesting problem.
D is not about syntax, it's about associativity of operators of same order. It appears to be "left associative" in USA, and "right associative" where I'm from
@@dfhwze The order is exactly what's in question. Division has been used as lower than multiplication, or specially lower only to the right (meaning the division itself could imply the answer). And implicit multiplication has been higher than explicit. All else was added to fuel the flames. E.g. the parenthesis are there to allow the numerals, which are there to get more people to weigh in without grasping the question.
So people really teach that that would have a higher precedence? That is just weird. That is just a sign people get this wrong because the school system sucks.
@@thomasdewierdo9325 I wouldn’t say that. I would say the fault lies in the person who wrote the question. And of course it was written that way purposely for the controversy.
I think D. The ISO 80000-2 standard for mathematical notation recommends only the solidus / or "fraction bar" for division, or the "colon" : for ratios; it says that the ÷ sign "should not be used" for division.
@iF3lixDE The ISO is just one more nail in the coffin that leads to the issue. If you write it as a fraction with 12 on top (numerator) and 3(4) on the bottom (denominator), you get 1.
@iF3lixDE I did choose D because, besides the parentheses, Im with the ISO on this one. So this is bad notation through and through. Also, the world is a big place, and not everyone is in school.
It's (D). The others at least require some mathematical thought. (D) is just dumb and is only an issue because people hate that particular division symbol and assume it means something that it does not.
@leaDR356 basically. Some people, especially old people learned to multiply the parentheses first so 12 / 4(3). Tbh thats how I learned it and managed to get a math minor... so it matter little when you are calculating on your own cause your not going to write it that way past 6th grade anyway
@@leaDR356 Not necessarily. It is perfectly valid to consider juxtaposed multiplication as higher priority than explicit multiplication or division. In which case the answer absolutely is 1.
D is not a math problem it's a notation interpretation issue So Why when it's 12/3X we interpret 3X as a number and not as 3 * X, Prioritising implicit multiplication is more consistent
there is no 'we interpret' such and such, this notation is ambiguous and I would personally ask for clarity if this was given to me, the use of '/' is something that is typically only seen online and therefore the correct interpretation is undetermined however if you were to say 12÷3x this notation is not ambiguous and clearly implies 12÷3*x
Exactly why notation with ''/'' is limited and not typically taught in math classes. Regardless, no it is not more consistent at all. 'Left to right' is more consistent than 'left to right but implicit multiplication before division'...
@@aMyst_1 It can also be equal to x/4. If you use multiplication by juxtaposition. The problem is not that one answer is the correct answer, the problem is that there's no universally agreed upon way to interpret the problem
"Multiplication denoted by juxtaposition (also known as implied multiplication) creates a visual unit and has higher precedence than most other operations. In academic literature, when inline fractions are combined with implied multiplication without explicit parentheses, the multiplication is conventionally interpreted as having higher precedence than division, so that e.g. 1 / 2n is interpreted to mean 1 / (2 · n) rather than (1 / 2) · n."
What are you quoting? And what is said to be ''conventional'' in certain literature (kind of suspect this is not mathematic literature but rather scientific or engineering literature) does not change the actual rules of mathematics.
@@FrenkieWest32 That's not how that works, you just made that up. There is no rules of mathematics saying what should the mathematical notation look like. 12/3x = 4/x for people that say that implied multiplication is done before division, and 12/3x = 4x for people that say that implied multiplication is done with the same priority as division. There's nothing mathematics tells us about how it should be interpreted, math notation is made by humans and could be completely different
@@F_A_F123 What exactly did I make up? Your comment is dubious, ironic considering the topic. 'Rules' are made by people. I would not refer to the intrinsic nature of reality with 'rules'. Orders of operation in mathematical communication are 'rules' made by humans. Just how one can say it is a rule to use 'x' for your first variable, even though this is completely manmade. With all that said: yes, rules are not set in stone completely. And apparently there is more controverse around this than I thought.
@@FrenkieWest32They're quoting the Wikipedia page for the Order of Operations, but conveniently left out the part right after the quote saying that some academic authors advise against the form a/bn (the form in their comment) and say that you should use the much less ambiguous forms of (a/b)n and a/(bn). In the case of the video, that would give both (12/3)*4 and 12/(3*4).
I think the issue with D is that there's disagreement about whether or not implicit multiplication takes priority over explicit division. I remember The How and Why of Mathematics made a couple videos about this debate, that I thought were really good.
What really bugs me is how people pick a position and start making stuff up about all other positions; in fact, this video is guilty of it. It presents it as "do the parenthesis first", but the parenthesis is only there to distinguish 3x where x=4 from 34. In a conversation, this could be a simple misunderstanding and be resolved. But when it moves to lecturing like this video, it's a straw man, misrepresenting the other position(s). I agree that The How and Why of Mathematics presented this well, including actual research.
Nah the issue is the garbage ÷ sign which no respectable professor in math1 / math2 or any of my engineering classes in college ever uses. I feel like if they ever saw someone write equations in ambiguous notations as in d) they'd fail them on the spot. If I ever have to use such "cramped" notations (matlab programs for example) I'd just write (12/3)*4 or as shown in the video 12 * (1/3) * 4
@@pointyheadYTit is garbage but using / doesn’t resolve the ambiguity, at least not for layman. In scientific academia, multiplication by juxtaposition is the norm. So you if saw a/bc, the standard interpretation is a / (b*c) and not a/b * c.
@@mordret103 you’d be surprised how common it is. For example, the amount of substance n in the ideal gas law is commonly written n = pV/RT You’re expected to interpret this as pV/(RT) and not pV/R * T
I think the problem with D is that even with the same operation, it's usually implied that when the operation doesn't appear, it should be done first. e.g. when you write 12÷4a, you kinda want to do "4a" first. Obviously the whole thing with order of operations is just a convention. As a programmer, when I occasionally write math operations in the code, I often add parentheses which are technically redundant, just to make it clearer what is going on. e.g. I write var1+(var2*var3) instead of var1+var2*var3. They're technically the same, but the first is much easier to understand from a quick glance, and unlike what some people think, the point of writing things in math (and code) is to make it *easier* for other people to understand what they're reading. As for the debates, personally I think the worst debate is C. Debates A, B and D are just about conventions. You can define these things however you want, it's just for the sake of convenience, there's no hidden meaning there. Like, you could define square root to be a function that returns pairs of values. It would be less convenient to work with, but nothing would break. C is the only debate that is actually about the *meaning* of something, that actually shows a fundamental misunderstanding of what real numbers are and of how series work.
I really like your point about notation. I always teach that notation in maths is like good punctuation in English. Your main objective is to communicate your intention to your readers not to be technically right but misleading - that's no good for anyone. Things like D only exist for the sake of it. The others are all real things to define or discuss.
@@yann8765 why pick the positive root and not the negative root as the single answer? It's just a convention, like rounding 7.5 up to 8 rather than down to 7. I mean it is sensible but not mathematically forced on us?
@@72kyle Rounding 7.5 to 8 isn't a convention either ; if it was rounded to 7, rounding would create a bias toward lower values (0,1,2,3,4,5 (so 6 digits) rounded below, but 6,7,8,9 (only 4 digits) rounded above. EDIT : I take that back, actually it seems to me that it is secondary to another convention, which is that when splitting a continuous interval, we tend to do it as [low, up) rather than (low, up]
Because for D multiplication by juxtaposition is often done first. Even in other fields you will catch scholarly papers dividing by stuff and not putting parentheses around their denominators.
@@adamwalker8777 The parentheses are incidental - they're only there to allow for juxtaposition between numerical values (à la “3a”). The contents of the parentheses are trivial: (4) = 4 So that has no bearing on the controversy. What's controversial is whether juxtaposition takes higher precedence than other multiplication and division; i.e. whether “xy” represents “(x ⋅ y)” or just “x ⋅ y”. Its simply a matter of notational convention - either way would work entirely consistently - but people's intuitions seem to differ, so it's best to make your intentions clear using brackets or fraction bars.
@@TheJamesM When I talk about parentheses, I don't mean an expression like this (4). putting one number in parentheses is stupid and makes no sense. when we have a complex expression in parentheses and we are reducing it, then we must remove the parentheses after all the calculations inside the parentheses. That's what I mean.
D is on the verge of not being a math problem, it’s a problem of semantics not math. It’s closer to being a freaking English problem. I truly feel sorry to those that find problem D mathematically interesting / provocative because there are endless ACTUALLY interesting ideas to explore in mathematics.
All math guides from professionals state that adjacency notation for multiplication is an implied parenthesis in the order of operations. For example: 12 ÷ 3n, when n = 4. 3n is pre-grouped single operand. 3(4) is the same as 3n, when n = 4. Similarly, fraction notation (which is division) will be performed prior to left-to-right order as an implied parenthesis.
@@forbidden-cyrillic-handle what style guides from professionals do you think I'm referencing? The ISO standard (one of) is the style guides from professionals. Multiplication (the operator) in the ISO standard is a cross (×) or a half-height dot (⋅). Adjacency is a product in the standard, but is a single expression rather than being 2 operands separated by an operator. It takes precedence over spaced operators.
3n takes priority sure; but 3(4) is not equivalent to 3n and n = 4. That would be (3*4). If you substitute n with 4 and write 3n as 3(4) then that is just the wrong procedure
Okay, but let’s say you set n = 5 + 6. See how you would have to add an extra parentheses for each little thing you tack on to it? Honestly, I think it’s just easier to just assume that 5n; n = 5 + 4 can just be expressed as 5(5+4), rather than having to rewrite it as (5(5 + 4)).
This video 'I hate internet math' by doorwaydude is what I consider to be a 'sequel' to this vid. It attempts to explain why these debates exist in the first place (and why some of them are pointless). I really think people in this comment section should have a look if they are still unsure about things like 0.999...=1
no (for me) because this means its the exact next number. think of 1 quantum of numbers . its 0.00..1 so this is the smallest value .. can you have 1.5 cents? no because 1 cent is the quantum of euros , thats why theres no "cent" between 1 cent and 2 cents
A. Undefined unless convenient. B. Principal square root if we're applying the function (and we nearly always do); plus or minus if we're trying to find all solutions. C. Boring debate due to people not understanding how infinite sums work ("but they can't be equal because there's still a difference of .0000000000...001!') or who just outright deny the use of infinity for philosophical reasons (finitism) in the first place. D. Old clickbait tactic used to drive engagement via making people think they're arguing about mathematics when they're really arguing semantics. From my experience D is the least interesting because it's a semantic debate and almost always clickbait but people get really heated about C and, as annoying as it is, people never tire going at it about it because everyone is really convinced of the sense of their argument and the nonsense of the other side. Never really seen people argue about A or B. I think you should have put that approximation of pi meme there instead if you know what I'm talking about.
"C. Boring debate due to people not understanding how infinite sums work ("but they can't be equal because there's still a difference of .0000000000...001!') or who just outright deny the use of infinity for philosophical reasons (finitism) in the first place." - That's a strawman argument. Those of us who object to this equality claim that you have never proved that 0.999... is a real number. If it is a real number, then, yes, it is most definitely equal to 1. But simply assuming that it's real is circular logic. One could just as easily assume that 0.9 recurring is how we write an infinitesimal (a hyperreal or surreal number that is infinitely close to 1, but less than 1, while at the same time being greater than every real-number-less-than-1).
@brendanward2991 and what level of precision requires an infinite number of 0's in front of it to be accurate? At some point all math is rounded to the number of significant digits.
C. Also relies on people not understanding that decimal notation is not unique, just like rational notation. 1 can be written as the 2/2, 3/3, etc.etc. 1 can also be written as 0.999... or 1.00...01
Definitely D. Terrible notation and not ISO compliant (ISO-80000-1 and ISO-80000-2 not followed). It's simply ambiguous notation. A trick. Academically, multiplication by juxtaposition implies grouping but the more programming/literal interpretation does not. Wolfram Alpha's Solidus article mentions the a/bc ambiguity and modern international standards like ISO-80000-1 mention about division on one line with multiplication or division directly after and that brackets are required to remove ambiguity. Even over in America where the programming interpretation is more popular, the American Mathematical Society stated it was ambiguous notation too. Multiple professors and mathematicians have said so also like: Prof. Steven Strogatz, Dr. Trevor Bazett, Dr. Jared Antrobus, Prof. Keith Devlin, Prof. Anita O'Mellan (an award winning mathematics professor no less), Prof. Jordan Ellenberg, David Darling, Matt Parker, David Linkletter, Eddie Woo etc. Even scientific calculators don't agree on one interpretation or the other. Calculator manufacturers like CASIO have said they took expertise from the educational community in choosing how to implement multiplication by juxtaposition and mostly use the academic interpretation (1) Just like Sharp does. TI who said implicit multiplication has higher priority to allow users to enter expressions in the same manner as they would be written (TI knowledge base 11773) so also used the academic interpretation (1). TI later changed to the programming interpretation (16) but when I asked them were unable to find the reason why. A recent example from another commenter: Intermediate Algebra, 4th edition (Roland Larson and Robert Hostetler) c. 2005 that while giving the order of operations, includes a sidebar study tip saying the order of operations applies when multiplication is indicated by × or • When the multiplication is implied by parenthesis it has a higher priority than the Left-to-Right rule. It then gives the example 8 ÷ 4(2) = 8 ÷ 8 = 1 but 8 ÷ 4 • 2 = 2 • 2 = 4 A,B,C I 100% agree with here, but D, no, 16 is not the corrext answer according to the evidence. 16 is 'a' correct answer, along with 1. The expression is wrong. That is the correct answer.
The *internet argument* expression is wrong, because it's intentionally lacking context. The notation makes sense in most articles it is used, but could be clarified, which is what engineering standards like ISO 80000 directs. Casio did state that the reason they made regional models that don't prioritize juxtaposition over division (but not exponents, it isn't parenthesis) is that teachers of lower level maths insisted on it. Meanwhile, in higher level maths it's common to define new notation within an article.
@@0LoneTech In higher level maths you will NEVER SEE ambiguity. The only time you'd EVER see something like a/bc in real maths is if the journal explicitly states that that is the convention that will be used. And even then, you are very unlikely to see it unless it's involving something like 2pi, where it's a commonly used number, usually a multiple of an irrational number. Otherwise, pretty much no one is going to write a/bc in a real publication. Not if they want to be taken seriously, or if the journal is too broke to format for a fraction bar or to print brackets (which isn't going to happen).
@@Aestareth_it's not simple and it's not obvious. It's deliberately constructed in a confusing way by omitted * sign before parenthesis. This makes a lot people think that this operation (multiplication without operator) has higher priority, because it's often used with variables symbols. For example, -b/2a for a parabola vertex.
Saying D) isn't the worst debate out of those is the same as saying that "I saw a man with a telescope" is worth discussing whether it's as seeing a man through a telescope or as seeing a man holding a telescope. It's just ambiguity and the flaw is found on the very phrase
Obviously D is stupid, but in this problem it’s still clear enough where there is a commonly accepted answer. There are some that are like 5/2a, which would be a better debate, but this one is at least somewhat clear. B, in the other hand, is stupid because literally both answers are correct. Like the square root is one is both 1 and -1 because both of them square to 1, but at the same time, it’s commonly accepted that a singular number inside the radical is looking for a single number when solving. Sure, +- 1 is technically correct, but that doesn’t matter. If somebody wants to know the square root of 49, it doesn’t help that it could also be -7 because negatives aren’t as useful in basic math, so we ignore that solution. Either is completely correct, so it’s a complete and utter waste of time arguing. At least arguing over syntax has a reason, because if an accepted solution is found, it will solve arguments in the future. Arguing over solutions to a square root just don’t matter. Like who cares whether or not cereal is a soup, it can be both. Colloquially, we define it separately so it’s not the same thing, but going by definition, it would be. All these stupid arguments just end up creating more confusion than they solve, so I think B is the worst, though D is a close second.
D is the worst because even BPRP is wrong. The answer is that there is no consensus. Different publications disagree on the relative precedence of inline division vs implied multiplication.
Exactly. The answer is either "both" or "need more context" (or so forth). The fact that both "12 / 3 * 4" can show one answer while "f(x) = 12/3x" can show another is the whole issue. They're the same equation, but will result in different answers. If someone doesn't recognize that and instead consider it to be a single answer (either exclusively 1 _OR_ 16), that is what is incorrect.
@@ThomasTheThermonuclearBomb It's still an issue if you use / for division, which some publication style guides recommend if it makes a nested fraction easier to read.
D is the worst because order of operations is an arbitrary linguistic layer applied over the top of mathematics and not itself pure mathematics, so all arguing about it in a mathematic context is inherently insipid.
Suppose you're throwing a dart at the real interval [3,4) , and that the outcome X (= the number where your dart lands) has a uniform probability density across the interval. Then Pr(X ≠ 5) = 1 (as 5 lies outside the interval, hence it's impossible to be the outcome), but Pr(X ≠ π) = 0.99999.... (as it's possible, though highly improbable, that the dart lands on the number π = 3.14159... ; the chance that the first decimal of X and π don't match is 0.9 , the chance that at least one of the first two decimals of X and π don't match is 0.99 , the chance that at least one of the first three decimals of X and π don't match is 0.999 , etcetera.) Since the probability Pr(X=π) must "clearly" be greater than the probability Pr(X=5) (as the event X=π is possible while the event X=5 is not), the probability Pr(X ≠ π) must "clearly" be less than the probability Pr(X ≠ 5) . Therefore, I think it can be argued that 0.99999... does not equal 1 .
@@yurenchu 0.9 repeating is just another form of 3/3, due to base 10 numbers not being able to resolve 1/3. If 1/3 = 0.333... and 1/3 * 3 = 3/3, and 0.333... * 3 = 0.999... then 1 = 3/3 = 0.999... There's no odds or probabilities involved. Equations overcomplicate the whole thing. Think of it like a display glitch in our number system. It's not that it's "close enough" or being rounded or anything like that. It's literally equivalent, just displayed differently.
@@yurenchu Actually the probability of X ≠ π IS 1. Read about the concept of "almost surely". The probability that the dart will land exactly on pi is 0, although the set of points it could land for that result to be true is not empty. It's just weirdness that comes from probabilities with infinite sets, but mathematically, you would say that P(X ≠ π) = 1. Of course, you are always encouraged to note that that 1 is representing an "almost sure" probability and not a certain one. But your argument actually ends up helping the case for 0.999... = 1, because when you know about the concept of almost surely, you know that P(X ≠ π) = 1, and your creative way to calculate the probability by each decimal, also gives the 0.999..., so they have to be equivalent.
@@ric6611 You guys are missing my point. I'm not saying that it cannot be argued that 0.9999... equals 1. I _know_ that 0.9999... equals 1 within the common mathematical framework that we learn in school. And I went to school too, so I too learned this and am fully aware of this. My point is that arguably it could make sense to devise an alternative mathematical framework in which 0.9999... does not equal 1. Such a mathematical framework.would allow us to distinguish between different scenarios such as the ones that I sketched above. Therefore, I can see that there would exist (non-trivial) debate regarding option C.
0^0 = 1 since 0^0 is the set of all maps from the empty set to the empty set, where there is exactly one such map. It's also the IEEE standard. The often-made flaw is to assume that x^y requires to be continuous, and then argue with lim.
Could you point me to a resource that explains that definition please? I remember being told during my studies that 0^0 is 1 "for a reason you'll learn about later", but I never pursued far enough to encounter it, and I've always been curious. This seems very interesting and I'd love to learn more
@@procop314 Maybe one of the more intuitive approaches is starting with really understanding what a power set 2^S of a set S is, then more generally a function space in set theory, and finally an exponential object in category theory. Thereafter, you'll have a very solid basis for knowing that S^0 must be 1 for any set S, including finite ordinal numbers S. Once you understand the set theoretical necessity, understand the natural numbers on the basis of a von Neumann definition of ordinals. Then get an understanding of how natural numbers can be considered as being embedded into rational and real numbers. Persue this path step by step. Your understanding will get more and more solid.
The order of operations for (D) has nothing to do with giving priority to parenthesis. 3(4) is implied multiplication, not explicit multiplication, and does take priority in this expression so the answer is in fact 1. The confusion with this problem is the result of calculators designed for the United States vs the rest of the world. Calculators designed for the US market will allow you to enter an expression using implied multiplication, but will auto correct and add the multiplication sign making it explicit multiplication when you hit "=". Calculators designed for the rest of the world will not add the multiplication sign and give priority to implied multiplication over division and explicit multiplication (As stated in the order-of-operations section of their respective instruction manuals). If you want proof, enter the expression exactly as it is written into any calculator designed for the global market and you will get the answer of 1. Enter the expression into a calculator designed for the US market, and if the calculator will display what you entered as well as the answer, you will see that it adds the multiplication sign and returns the answer of 16. This is not because we do not give priority to implied multiplication in the US. We in fact do just like the rest of the world. The mindset is that you should not (and usually can not) use implied multiplication when programming, which you are essentially doing on a calculator that allows you to enter the entire expression before hitting =. Matlab for example will not let you use implied multiplication. In any text book written for the US, implied multiplication does take priority and you see this all the time with coefficients. For example; 1 / 2y would never be interpreted as 2 / y. 2 / y would be 1 / 2 * y which uses explicit multiplication and has the same priority as division but the order is left to right.
the most annoying math debate i had was with someone who would go “aha! just a theorem!” to everything, yet never had anything to add. the worst part is we weren’t debating math, i was just trying to explain some math for a foundation of my argument.
How would you evaluate: 12÷3x It feels very wrong to say this is (12÷3)x now that it's a variable, juxtaposition to me seems like "treat this as one entity" Edit: So the debate basically reduces to whether 3(4) is implied multiplication like 3x or normal multiplication like 3*x. Implied seems like a better convention to me 🤷
3X is implied multiplication, 3(X) is not, 3(X) is just normal multiplication, following the syntax of multiplication (left to right) it would be ((12/3)*4). Some people might argue that 12/12=1, 12/(6+6)=1, 12/3(2+2)=1, 12/3(4)=1 but that's not how it works. (6+6) Is after a division sign meaning the value of (6+6) is inverted into 1/12, by pulling (3(4)) out of the parentheses you would have to invert the equation into 12/3/4 and using the division syntax (left to right) you get ((12/3)/4) which is 1, it would be written as 12/12 = 12/(3(4)) = 12/3/4 = 1
@@arno_grnfld455What is "implied" vs "normal" multiplication? It's juxtaposition in both cases. The reason we add parentheses to get 3(4) instead of just writing 34 is because 34 is a different number.
@@TheUnlocked no, 3(4) is normal explicit multiplication, you can shift the 3 around and multiply it for example, 7(x+y)*6(-2x+y) = 42(x+y)(-2x+y). It is not tied down to a variable like how 3x is (3*x) (implicit multiplication) Implicit or implied multiplication is like 3X where 3 cannot be seperated or shifted around, e.g. 3x/4 ≠ 3/4x, the 3 (or 4) is glued to the variable like: (3*X)/4, 3/(4*X), if 3(4) is implicit multiplication, you'd write it as (3*(4)) not 3(4) which is just normal multiplication. In this case 12/3(4) would follow the normal Syntex of left to right, (12/3)*(4), if 3(4) is implicit multiplication, 12/3X, X=4, it would work like 12/(3*(4)) instead
I found out there is no agreement for D. Much of the world, especially academia and Europe, follow the juxtaposition of multiplication which gives 1 because multiplication is implied. North America, especially teachers in the U.S., follow strict PEMDAS, which gives 16. Even calculators don’t agree between the two and some have been known to document the order they use and some companies have been known to change between the two orders over time (and back). In short, don’t write notation like this. It’s confusing and I swear it’s used just to start flame wars. And if order is confusing, add parentheses for clarity.
@@TheMassacreOfTheBanuQurayzahQu Are you American? In my experience, it seems to be Americans that don't want 0 to be a natural number. Besides, 0 is the sum of a series of 1s. It's just the series with 0 terms. :)
@@omp199 Hmmm. I suppose it would be better to say Positive Integers for 1,2,3,... And use Nonnegative Integers for 0,1,2,... At least that's how Prealgebra, the Art of Problem Solving book gets past the natural vs whole debate.
For D I recently learned that there are some conventions where multiplication by jusxtaposition (when you have a parenthesis adjacent to a number, with no explicit multiplication sign) comes before both regular multiplication and division. Some calculators even have this as a part of their order of operations. Under this order of operations, PEJMDAS, the correct answer would be 1, but most people don't use this order of operations.
A is a matter of what branch of math you're working in B is a matter of following a definition of a radical function C is a matter of understanding this in term of limit question D is a matter of confusing notation I'm thorn between B and D as being the worst of those 4
The notation on D is not ambiguous. Some people are confused by it because they either cannot or refuse to understand basic order of operations, or because they incorporate misguided assumptions into the calculation.
@@anewman513 okay, the notation is technically not ambiguous, but it's for sure impractical. there are better ways to write down the same arithmetic operation without causing so much confusion about something that hardly has anything with the math itself
@@anewman513 It is ambiguous because some countries basically teach PEJMDAS by telling students to only remove operators if things are "grouped" So someone who learned that would see 12/3(4) as 12/(3(4)) because omitting the multiplication is an implied grouping. For variables most people do this most of the time. Usually when I see someone write 1/2x they don't mean (1/2)x.
@@drdynanite - Yes, I agree it could be written in a way that even the most easily confused persons would not be confused by it. Though, the problem is with the persons who are easily confused, not with the notation itself
D is the worst debate because it just a notational thing that boils down to how we want to define it, there are no deep secrets hiding within this. B I have never heard of and is also just notation.....
if anyone wrote D on an exam or in a paper, I would consider that reason to flunk them. (They have misunderstood the central requirement in math to explain yourself clearly, not just be technically right.) I'd also like to offer this counterpoint: 12 / 3x in this case it's fairly obvious that the x should go in the denominator, but it's actually the same rule as in (D), an implicit multiplication.
C drives me crazy the most, my friend asked me why it's equal to 1, did all the same stuff you did, and he just said, nah, they aren't equal cause this says one and this says 0.999... it made me want to rip my hair out
Can you please consider making a wordless definition of the limit shirt? Meaning only quantifiers and other math notation exclusively. I would purchase it so fast, by far my favorite calculus topic!
A isn't even a debate, it's just context-dependent. For limits it's an indeterminate form, for power series it's 1, otherwise just define your terms and run with it.
A is worth nothing. When you are in multivariable calculus and you have 1 limit from the x-axis and one from the y-axis do you say "let's debate"? NO. It's undefined. If sometimes it's convenient to DEFINE it (VERY LOCALLY FOR THIS PROBLEM) to get continuity, fine. Otherwise, UNDEFINED.
Honestly, I don’t think it’s a question of what comes first. Rather, it’s a question of how that division symbol should even function. Obviously the fraction brackets are self explanatory, they take the everything at the top and divide it by everything on the bottom. I personally think that the division symbol should take everything on the left and put it over everything on the right, so it ends up working similar to a fraction, thus the answer being 1 for me. Now I am fully aware that isn’t currently the convention, and there are multiple reasons why it might be a bad idea to do it that way, but that’s just my opinion.
"I personally think that the division symbol should take everything on the left and put it over everything on the right" - it hasn't mean that for well over 100 years - it was changed no later than the late 19th Century. It's the term on the left divided by the term on the right, same as any other operator works (which didn't use to be the case). "so it ends up working similar to a fraction" - it already DOES work similar to a fraction! 3+1÷2+6=3+½+6=9½. Under the old meaning that would've been 3+1÷2+6=(3+1)÷(2+6)=4÷8=½ "thus the answer being 1 for me" - yes, correct, but for the modern reason, not the old reason (the answer to this particular expression in fact has ALWAYS been 1, since there's only 2 terms in it anyway - 12 divided by 3(4)). "Now I am fully aware that isn’t currently the convention" - yes it is. Under ALL rules, both old AND new, this expression has ALWAYS been equal to 1, as we were only dividing by 1 term anyway! "there are multiple reasons why it might be a bad idea to do it that way" - no, there are multiple GOOD reasons for doing it this way, which is why it was changed more than 100 years ago, but as I said that change doesn't affect THIS expression anyway - the answer to this has ALWAYS been 1.
A: maybe also add 0 to the options B*: ³√(-1) (principal root vs real root) B**: arcsin(2) D*: 2x / 2x (options: 1 or x²) to see who picks (2x/2)x instead of (2x)/(2x) by that "order of operations" E: f(x)ⁿ for f that can be written without brackets (e.g. sin(x)² ln(x)³) E*: f⁻ⁿ(x) for f which fⁿ(x) is used as (f(x))ⁿ and f⁻¹(x) being an inverse function F: any notation which limit exists but written without limit (e.g. x³/(5x-x²) at x=0) (whether it equals a value when it gets only one possible value) F*: sum of divergent series (e.g. 1+2+3+...) F**: step function at 0 G: non-integer factorials without using gamma function (e.g. i! , 3.5!) G*: analytic functions / continuations H: Division as inverse (e.g. matrix, modular arithmetic) I: Symbols that can act as both prefix and suffix operators (e.g. 3³3, 3!3) I*: Symbols that can also used multiple times at a row (e.g. 3!!3) J: Iterated binary operations without ending mark, unlike integration (e.g. Σx+1, Π2x) J*: Operators which are not associative, but the operation orders or associativity directions are not well known (e.g. P → Q → R)
Follow-up to B*: do irrational equations such as ³√x = −2 or ³√(x−6) = x have solutions over the field of complex numbers, and if so, what definition of the cube root should we use.
For (B) I think its important to point out that the ± is evaulated separately from the radical. We can see its OUTSIDE the radical, so it is its own thing, done after you get the radical's output. The radical only gives one output because it is a function. That output is defined as the principal root. See: ±√2 means plus and minus the (principal) square root of 2. √2 = ±... would be wrong.
not really math "debates" its usually just people who struggled to pass high school maths thinking they suddenly understand anything about the subject and get really loud about it
i feel like the problem with C is that a lot of people dont grasp that 0.999... is non-terminating. any proof stating that 1-0.999...=0.00...1 doesnt get that their "proof" contradicts the nature of the statement, because to imply that you're left with a fractional part after taking the difference is to imply that 0.999... terminates, which is a contradiction to how its defined
It may depend on a tool you are using to evaluate the expression, and conventions may vary across different tools, so, yeah, it sort of depends on context (of conventions).
Some advanced calculators (e.g. CalcES aka Scientific calculator plus 991on Android) simply let you choose how you want them to treat implied multiplication: as *1/2π = 1/2∗π* or as *1/2π = 1/(2∗π).* And letting users choose the way they want their implied multiplication being evaluated is a wise decision - it sort of solves the problem by providing both available options.
In a similar fashion, the value of =2^2^2^2 or =−2^2 also depends on conventions being used, and different tools may give different results, And some tools will simply refuse to return a value for the second expression, stating it is ambiguous and "parentheses must be used to disambiguate operator precedence" - exactly the case with the expression from the video.
It's the correct response though. It's simply ambiguous notation. A trick. Academically, multiplication by juxtaposition implies grouping but the programming/literal interpretation does not. That's the issue. You can't prove either answer since it comes from notation conventions, not any rules of maths. Wolfram Alpha's Solidus article mentions the a/bc ambiguity and modern international standards like ISO-80000-1 mention about division on one line with multiplication or division directly after and that brackets are required to remove ambiguity. Even over in America where the programming interpretation is more popular, the American Mathematical Society stated it was ambiguous notation too. Multiple professors and mathematicians have said so also like: Prof. Steven Strogatz, Dr. Trevor Bazett, Dr. Jared Antrobus, Prof. Keith Devlin, Prof. Anita O'Mellan (an award winning mathematics professor no less), Prof. Jordan Ellenberg, David Darling, Matt Parker, David Linkletter, Eddie Woo etc. Even scientific calculators don't agree on one interpretation or the other. Calculator manufacturers like CASIO have said they took expertise from the educational community in choosing how to implement multiplication by juxtaposition and mostly use the academic interpretation. Just like Sharp does. TI who said implicit multiplication has higher priority to allow users to enter expressions in the same manner as they would be written (TI knowledge base 11773) so also used the academic interpretation. TI later changed to the programming interpretation but when I asked them were unable to find the reason why. A recent example from another commenter: Intermediate Algebra, 4th edition (Roland Larson and Robert Hostetler) c. 2005 that while giving the order of operations, includes a sidebar study tip saying the order of operations applies when multiplication is indicated by × or • When the multiplication is implied by parenthesis it has a higher priority than the Left-to-Right rule. It then gives the example 8 ÷ 4(2) = 8 ÷ 8 = 1 but 8 ÷ 4 • 2 = 2 • 2 = 4
With part b, its important to note that sqrt(x^2) is abs(x) BY DEFINITION. Thats where the plus or minus comes from. From "cancelling" the absolute value. Too many people believe it's just voodoo, which you kind of lend credence to by saying "oh, its when we solve this equation." NO! Sqrt(x^2) = abs(x). That is where the plus or minus comes from. Please bprp, I rely on you to note this kind of nuance since you are an authority, so i can point people to your videos when people get real resistant to being told they are wrong.
The definition is actually not a mere abs, the y-th root of x is the z with minimum principal argument that solves z ^ y = x. It just happens to be abs when dealing with non-negative reals. But, for example, cbrt(-8) is not -2, unless you are restricted to real numbers.
Every problem needs to say x ∈ ℍ or x ∈ ℝ or whatever. If it doesnt then whoever wrote the question wrote it for a classroom, not for youtube or for the real world.
Maybe you should do a video about the Monty Hall Problem. Pretty sure that might come in pretty high on your list of "most debated math topics" if you kept getting comments about it :P
4:27 The response I hear most often for that is along the lines of "A number between 0.999… and 1 is 0.9999… And a number between _that_ and 1 is 0.99999… You can always add another 9, you just keep getting more specific." Which, of course, comes from a fundamental misunderstanding. The intuition that these people listen to says that 0.999… is not 1, but (1 - 0.0…1), but that's not really a valid expression, is it?
The problem with D is that it when you deal with variables or symbols (𝜋 etc), implied multiplication does take priority. Take this question from a recent GCSE maths paper for example: simplify 12x⁷y³ ÷ 6x³y. The correct answer is 2x⁴y² not 2x¹⁰y⁴. And nobody would see 1/2𝜋 and think it means 𝜋/2 instead 1/(2×𝜋)
Yes, because I have learnt algebra, I have learnt about implied multiplication. Because I have learnt implied multiplication I will use it when solving D even though there are no variables in the equation.
I don't think D is actually 16. Like A, there is no agreement. Also, there is no doubt that 5 / 3a is 5/(3*a), so why should it not work for 5 / 3(4)? Some calculators explicitally ask you which meaning do you want to give it. So I think there is no answer agreed upon, just like 0^0.
More fundamentally, the reason to read is to understand. It's not useful to look at an article and go "Ha! They wrote it wrong!" in preference over "so that's what they mean".
@@strongbrain3128 You have only asserted your conclusion, not addressed the question. Your reasoning would be "I refuse to distinguish implicit and explicit multiplications." This is one possible consistent position, but quite rare; it conflicts with established use.
It’s definitely D. Only because people are so adamant about debating about what is the correct answer and what is the correct order of operations. A lot of them don’t even realize that all of the order of operations are just an agreed upon convention on how to solve mathematical problems. They are arbitrary. Some people just added another arbitrary rule and that is to solve juxtaposed multiplication before explicit multiplication and division. This is what makes the question somewhat ambiguous. Since there is a rule (juxtaposed multiplication) that is not agreed upon by everyone. But instead of just agreeing that it’s ambiguous, people will fight tooth and nail for their way of solving it.
I fight tooth and nail for better notation. But imo worst is still c because there are multiple proofs and there isn't any argument other than "nuh uh", but people still fight it.
@@michaelsorensen7567 Suppose you're throwing a dart at the real interval [3,4) , and that the outcome X (= the number where your dart lands) has a uniform probability density across the interval. Then Pr(X ≠ 5) = 1 (as 5 lies outside the interval, hence it's impossible to be the outcome), but Pr(X ≠ π) = 0.99999.... (as it's possible, though highly improbable, that the dart lands on the number π = 3.14159... ; the chance that the first digit of X and π don't match is 0.9 , the chance that at least one of the first two digits of X and π don't match is 0.99 , the chance that at least one of the first three digits of X and π don't match is 0.999 , etcetera.) Since the probability Pr(X=π) must "clearly" be greater than the probability Pr(X=5) (as the event X=π is possible while the event X=5 is not), the probability Pr(X ≠ π) must "clearly" be less than the probability Pr(X ≠ 5) . Therefore, I think it can be argued that 0.99999... does not equal 1 .
@@yurenchu if your dart is magnetically confined to whole numbers, the odds you'll hit 3 instead of 4 (in a random throwing) is 1 in 2. If your magnetic confinement is expanded to include tenths, the odds of hitting 3.1 is 1 in 11. That's a significant decrease in odds, right? If you expand to hundredths, the odds of hitting 3.14 is 1 in 101. We're already at a less than one percent chance to hit the approximation of pi that fits into our magnetically bounded random throw. However, if you abandon the constraint and get infinitely small dart tips, the odds of hitting pi exactly are zero. You WILL get off by a little bit one way or another, EVENTUALLY.
@@michaelsorensen7567 The 4 is not included in this (half-open) interval, and instead of "odds" you actually mean "probability" (or "chance"); but okay, those points are not really important. I'd say that the probability of hitting pi is not exactly zero -- or at least not the same zero as the probability of, say, hitting 5 . To put it another way: suppose I throw the dart, and it lands on (pi + 1/7). The _a priori_ chance of hitting (pi + 1/7) equals the (also _a priori_ ) chance of hitting pi. But since the dart did hit (pi + 1/7), it proves to be actually possible; so that chance cannot be equal to zero. Or at least not be equal to the same zero as the chance of hitting 5 (which is physically impossible). Sure, the chance of hitting (pi +1/7) is infinitesimally small, but it's (arguably) _not_ 0 .
4:19 i didnt realize you could prove it like that, i always thought it was equal to 1 just because is supposed to be 3/3 = 0.9999..., but when the denominator and numerator of a fraction are the same then it becomes 1
I think you're wrong on D. There's a very significant difference between two objects concatenated and those same objects separated by a multiplication sign. Suppose the (4) were a variable like x. Then you would have 12÷3x, which is clearly 12 / (3x) not (12/3)x. This is the same situation - just replace the x with (4).
As the other person said. Monomials are one term and implicit multiplication is two terms. So the are not the same. People keep bring up variables as though "Ha! Found this example that proves you completely wrong!" When really those two things are not comparable.
D) you've been miscommunicated. We don't multiply the parenthesis first because we "misheard" doing the inside first, afaik that's just your strawman. The actual reason is due to juxtaposition which is considered of higher order than a multiplication dot •. Example: 12÷3x vs 12÷3•x where x = 4 The first statement has a juxtaposition of 3 and x, wheras the 2nd statement has a multiplicative dot between 3 and x. Thus the first statement is 12/12 = 1 and the second is 12/3*4=4^2=16 Now do this with a parenthesis instead as you can juxtaposition those as well 12÷3(4) = 1 ≠ 12÷3•4 = 16
The thing with b is that it depends on the context. Without any, it should be assumed that you're taking the principal root. In another context, it can be a multivalued function.
Exactly. For a cube root *³√−̅8̅* Wolfram Alpha gives: • a principal complex cube root: *³√−̅8̅ = 1+𝒊√3̅* • a principal real-valued cube root: *³√−̅8̅ = −2* • a set of all complex cube roots: *³√−̅8̅ = {1+𝒊√3̅; −2; 1−𝒊√3̅}* All three possible definitions of a cube root.
C) I think people have this notion that a real number only has ONE decimal representation. And I find that very understandable. Pretty sure that's the whole crux why people argue about this at all. The idea of 1 = 0.999... would break that notion.
It's also that they give symbols different meanings in their heads instead of referring to clear definition. Hence we get stuff like "it's infinite 9s", as if "infinite 9s" was somehow a number.
I have to disagree on D, 3(4) should be done first like 3x because usually juxtaposition takes higher precedence than multiplication and devision, it should be 1
@@thewhat2 And yet read some academic journals' publication style guidelines and you'll see that in academic papers, it would be interpreted the other way and YOU'd be wrong. But the keyword is "some". It's just a matter of convention, and different places have different conventions. The trick is to realize that it is a dumb argument because you're literally arguing for which definition is better, and the fact there's a disagreement at all shows that neither is good and you should adopt a better way to notate it (which already exists).
D is less of a math issue and more of an issue with how some people interpret the syntax. It’s not that bad. The first one is also an easy hit or miss because like you said, there’s no agreement. I can’t decide whether I hate B or C more. I’ll say C only because that not equal 1 is a mistake I can’t understand how they’d make (if they are a beginner the +-1 on B is an easy mistake) but 0.99999… not equaling 1 is not justifiable in my eyes (at least not comparatively) so it’s the one I dislike the most. B is a close second however.
As a tutor, A is my least favorite. If only for the reason that I don't like saying anything "wrong." So when I bring up "Any number to the power 0 is 1" I always have to throw in "except 0, there's actually some disagreement on what that should be."
You misinterpreted the argument for why D should be equal to 1. The actual argument is that putting 3 and 4 next to each other like that makes it a multiplication by juxtaposition, which is usually considered to have a higher priority than division. Many calculators can't agree on that one.
12/3(4) is not the same as 12/3*4. We all agree the latter is 16 but the former uses juxtaposition. The order to use for a juxtaposed operation depends entirely on the context. If you are a college level North American student you are expected to ignore the juxtaposition and treat it as a regular multiplication because that's what they decided to teach, as a convenient simplification which unfortunately introduce this confusion. If you are writing a post graduate scientific math paper for peer review, on the other hand, you are expected to use PEJMDAS, because that's the international standard for math publications in science. Also, many countries outside North America introduce PEJMDAS already at college level and thus scientific calculators sold in those regions are programmed to take juxtaposition into account and perform it first. There is no right or wrong, it depends on the context, because PEMDAS and PEJMDAS are just different conventions that have different applications, in general the former is more scholastic, the latter is more scientific. That's also why it's a popular debate: different people have been taught different and there is no truth.
The problem with the root, is that people don't understand that it's not the root itself that gives us two options x²=1 √(x²)=√1 |x|=1 x=±1 you either use absolute value x²=1 x²-1²=0 (x-1)(x+1)=0 x=±1 or the difference of squares
I wouldn't say it's a big debate, though. While it's certainly confusing, there aren't a lot of actual arguments about it, mostly just people going "huh?" It's also been thoroughly debunked multiple times in super long, very detailed videos.
@@ivansmashem The reason it's a thing is because of the zeta (ζ) function. The analytic continuation of it is very useful, but also gives weird answers like that. There's not much to argue about.
@motobike3904 The Riemann zeta function doesn't give an answer of -1/12 for the analytic continuation of positive integers, though. That requires setting the parameter to -1, which is not valid, as it doesn't keep the analytic continuation. It's officially gobbledygook and nothing other than someone saying, "Hey, what happens if we start plugging in invalid numbers?" The analytic continuation of the function is indeed useful, but not when the parameter is -1. In that case, the result may be interesting, but it is pretty meaningless.
No, they are the exact same thing. Both have only positive values (on positive values) (ok, unless I'm wrong as well). If x^(1/2) would have 2 answers, you would be unable to say (x^(1/2))^2 = x for example.
I don't understand the last one, why is 3(4) not prioritary if something like 3x would be calculated first? It seems that just changing the 4 for an x will totally change the order. Maybe I have been tought wrong but I would solve 12÷3x as if 3x was in parentheses because I have been thought, even if that may be wrong, that implied multiplication by parentheses are prioritary
If I have a items that must be divided into b lots of c people, then we are looking for x per c to get how many items each person gets... a ÷ (b × c) = x/c. If I have to figure out how many items I need to give c people so that for every lot of a, each person would get b, then the required items are x... (a ÷ b) × c = x. When given 12÷3(4), there is an assumed "= x." The second solution is therefore correct, since we simply assign the values of a, b, and c to the equation as shown. Since we are looking for x and not x per c, then we get 16 instead of 1.
@Draconic404 Basically, we have a known quantity in 4, so it acts differently to an unknown quantity in a variable like x. Because we are solving for = x, we can do the order of operations left to right without implied multiplication. The ambiguity of the equation is solved by the word problems since known quantities can be seen as tangible things like items, lots, and people.
@MrCmon113 Put it this way. When you read it how, do you say it? Twelve (12) divided by three (3) times four (4)? Or twelve divided by the quantity three times four? If the former, the answer is 16. If the latter, the answer is 1; however, you would need to demonstrate a reason to call 3(4) "the quantity 3 times 4" instead of just "3 times 4." There isn't a feasible way to do so without adding more symbols such as parentheses to the equation, so this answer is incorrect. It is easiest if you write the problem out left to right as you would say it to make it more obvious. 12 divided by 3 (=4) times 4 (=16), for example. On the other hand, 12 divided by the quantity 3 times 4 would look like this--12÷(3×4)-- when written as spoken. Clearly different notation, yes? Does that help?
I believe your answer for D deserves an "Incomplete" mark. It's correct, inasmuch as PEMDAS is the final word on order of operations. But PEMDAS is *not* the final word on order of operations everywhere. In some contexts "multiplication by juxtaposition" (eg,, signifying multiplication by just putting two objects next to each other) is given a higher precedence than regular multiplication. Don't believe me? Spend some time going through the manuals for many different calculators, preferably of different brands and sold in different countries. You'll find that each manual (usually) has a whole section on "order of operations" and that some do assign higher precedence to "multiplication by juxtaposition" and some do not. Also note that in the submission guidelines for many scientific/mathematical journals, authors are instructed to observe the convention that "multiplication by juxtaposition" takes precedence over regular multiplication. I would argue that D should also get a "no agreement" answer on this basis.
@@oc-steve It's remarkable, considering BPRP himself published a video on how PEMDAS is incomplete, and as usually presented it doesn't even specify what the value of -1+1 is.
3(4) is not implicit multiplication, you cannot force it to be that's not how the math works. If 3X, X=4 then it is implicit multiplication, you write that as (3(4)) not 3(4) because 3(4) is just normal multiplication
@@arno_grnfld455 You keep using that word. I do not think you know what it means. Implicit means unstated, as in unwritten. There's no multiplication symbol in 3(4) so if it involves multiplication, it is implicit. This is a plain fact of the notation only. If I see 1-(2+3), I do not read "parenthesis without multiplication". It doesn't matter how many pairs of parenthesis you put around an expression, it's still the same expression; the point of the parenthesis is to keep it so. Just like how the order of operations in common algebraic notation is implicit unless made explicit by parenthesis. You can always include parenthesis when doing variable substitution, e.g. x=4 in 12÷3x. Juxtaposition is another clear fact. 3 and (4) in 3(4) are juxtaposed as there's nothing written between them. Again, this is notation, not arithmetic operation. 1 and 2 are also juxtaposed in 12, and in that case it is part of decimal notation. BPRP presented one incorrect reasoning for arriving at 12÷3(4)=1. Using that to assert 1 is incorrect is the fallacy of denying the antecedent. He then implied that 12÷3(4)=12÷3×4, which is a begging the question fallacy. The question is: does implicit multiplication read exactly as explicit multiplication? The form is therefore a÷bc, and the numerals and parenthesis are distraction.
@@0LoneTech I agree with all the points in your comment, but I think you have missed one important point. Someone who doesn't know what "implicit" means is, with almost complete certainty, not going to understand what you mean by "the fallacy of denying the antecedent" or "a begging the question fallacy". You have to write to the level of your reader; otherwise you are not going to get through to them.
@@omp199 Good point. I am not particularly good at rhetoric (the art of speaking convincingly) but more writing to let my thoughts out sometimes. I should also have included the observation that the parenthesis in 3(4) were there to preserve the 4 rather than make 34. They could equally be around either (3)4 or both (3)(4). I had written the reasoning motivating that use (that juxtaposition can mean other things than implied multiplication).
Hi very clear, 2 comments: (1) I think it depends on the type of 0. If we are talking about finite math: combinatorics, graphs etc, and dealing with integers or natural numbers, then 0^0 will always be 1. But if we are talking about reals / complex numbers, then 0^0 usually is defined to be the limit of something, and the result may be undefined. It's not that there is no agreement on the math: instead the generally agreed meanings vary depending upon the context. (4) If I write: 12 - 8 + 8 - 3 - 9, then there is no confusion. We just evaluate from left to right to reach the answer 0. It can be just the same for multiplication/division: 12 / 3 * 4 / 2 / 2 = 4. There's nothing different structurally going on. This also means that people (including YOU! heh) should feel free to write 27 / 3 / 3 = 3. It's certainly very convenient, but for some reason there is a taboo on this that most people aren't even aware of.
At my grade school, when we learned order of operations, we were told to do PEMDAS in exact lettering order, so instead of going from left to right, we would always multiply any terms before dividing. Pretty weird in hindsight.
When a number precedences a parenthesis with no symbol between they’re one term. I will fight you over that. No one would read or write 1/2x and think .5x
"No one would... _write_ 1/2x..." Exactly. That is simply bad notation. So is 1÷2x. It should be written as a fraction as either: 1 ---- 2x Or ½x These are two different expressions. Let x = 5. The first expression then equals ⅒, and the second 2 ½. This is because the numerator and denominator are evaluated independently. Using either the / or the ÷ in this context creates ambiguity.
@@JustifiedNonetheless "That is simply bad notation" - it's NOT bad notation. It's a Term. ab=(axb) BY DEFINITION. "So is 1÷2x. It should be written as a fraction" - fractions and division aren't the same thing. 1÷2x is 2 terms (separated by an operator), ½x is 1 term. Terms are separated by operators and joined by grouping symbols (brackets, vinculum, exponents).
Regarding the 0.999... == 1, I was thinking that if it's not equal to 1, then there must be some value x>0 such that 1-x > 0.999.... One could somewhat easily prove that x does not exist, since there will always be some extension of the geometric series that disprove an arbitrary x. Yeah, I spent way too many years grading for discrete mathematics.
0 < ε = 1/ω Neither ε nor ω are Real numbers (though they're exactly as real as the badly named so-called "Real" numbers), nor is this particular use of 0.999.... If you insist that 0.999... is the Real number associated with the given geometric series, then the only option is 1.
@@kadanseward3022 And you can have multiple things with the same notation in different contexts. In the context of the ℝeal numbers, 0.999... and 1 refer to the same number. In the context of the hyperreals, 0.999... _might_ be its own thing that is legitimately less than 1 by an infinitesimal amount. In the context of the integers, 0.999... is nonsense.
That makes C the best though - it's a debate where one side is right and the other side is wrong, so it's an opportunity for the wrong side to learn something (if they have an open mind). The others, especially D, are pointless debates that can go on forever because they're just arguments about conventions. Nobody can ever "win" the debate for A, B or D because they aren't about a mathematical truth, they're just about humans disagreeing about arbitrary choices.
@@forbidden-cyrillic-handle the way ur describing it makes it seem like the most ok one but remember even tho it is an agreement issue it is still the order of operations we r talking abt sonething u should know from private school
D: The reason for me is that terms can be brought out from within the paranteses. (2*2) = 2(2) = 4(1) = 1(4) = 4. Something infront of paranteses is part of the the paranteses. x(y) = (x*y). This is where the confusion comes from I think. Noone would write 1/2(4) if they ment 1/2 * 4, simular noone would write 1/2(4) if they ment 1/(2*4). As a programmer I would always write it as (1/2) * 4 or 1/(1*4) to make it clear. (4) = 2(2) = 4 same as (2x) = 2(x) = 2x
surprisingly, his solution for D isn't correct. admittedly, it's not really recommended to write it down like this, but it looks like that the international physics literature overwhelmingly agrees upon that, say, "2x" refers to an "implied multiplication" which cannot spilt up any further and thus has predominance over anything else of seemingly equal (!) order. or in order words, "2 × x" is treated differently from the other notation. no one has to believe me, but that's also how most scientific calculators work (which isn't by mistake but design.) but what I actually tried to say is that in those rare instances where it actually says, "x / 2y," the expected operation to be performed is, "x / (2 × y)." at the end of the day, it's just a convention and doesn't break actual math.
also, I wouldn't even call it a math debate as unlike all the other examples, it's not even about math at all, but sheer syntax, which belongs somewhere else, as far as I'm concerned. one way to tell them apart is: math is the part which works for aliens as well, while syntax is invented by humans.
@@TheSketchGuy672 that is were we disagree, 12/3*(4) and 12/3(4) is not the same if you use PEJMDAS. 12/3*(4) is equal to 16, 12/3(4) is equal to 1. This is what the J in PEJMDAS means. It consider juxtapositions (multiplications without written sign) to be of higher importance than explicitly written operations.
Option D is not so clear. The RUclips video "PEMDAS is wrong" by "The How and Why of Mathematics" tells some examples why multiplication by juxtaposition should be made before division (at least slashing fractions). It is indicated like that in an article by the AMS, another Physical Review Style and Notation Guide, and is usually used when using x, like 1/2x, which is interpreted as 1/(2x) and not x/2.
D is a matter of multiplication by juxtaposition where 3(4) takes precedence over 3x4. It used to be taught that way 100 years ago, and it is coming back. Some calculators are programmed now to do #(#) before doing x / left to right. My calculator can be set to do it either way.
Another reason for defining 0^0 to be 1 is that in general for non-negative integers m and n, m^n is the number of functions from a set with n elements to a set with m elements, and there is 1 function from the empty set to the empty set.
The problem with D is that academic textbooks above a certain level almost unanimously disagree with him. It might be 16 in a grade school, but in any setting that matters at the higher level, the answer will almost always be 1. No one at that level is even attempting to work out a grade schooler's mnemonic device because it isn't relevant to people's understand of math at that level. (The division sign used doesn't particularly matter, so I won't address the difference in division signs.) Let's say you have 12 / 3b. Hard liner PEMDAS users will say that it's equal to (12 / 3) * b, which is equal to 12b/3. So hard liner PEMDAS people believe that 12/3b = 12b/3. That is prima facie absurd. The only reason to have written 12b/3 as 12/3b is if (a) you're a hard liner PEMDAS addict who wants to intentionally confuse people for some reason, or (b) you understand that 12b/3 is not the same as 12/3b. The fact is that anyone in an academic context will easily understand what 12/3b means without having to apply a children's mnemonic device that was created to dumb down math enough to teach kids the basics. Kids are also taught that they can't take the square root of a negative number, but we all know that's not actually true. Why is it so hard for some people to understand that the "rules" they teach kids are actually just helpful guidelines to ease lower level learning? For those who think that 3b where b = 4 isn't the same as 3(4), consider how we typically substitute b with 4 in 3b. Mathematicians at a higher level tend to be lazy and avoid writing unnecessary notation as much as possible. It is simply understood by convention. Some fields aren't as lazy and demand that it be made more explicit, but the general understanding is that 3(4) and 3b where b=4 are the exact same thing. Only people who cling to PEMDAS as some golden rule even as they enter spaces where PEMDAS isn't even a consideration would keep prattling on about how 1 is never correct. 16 might be correct in lower level education, but it's not useful in the slightest to cling to children's mnemonic devices at higher levels.
Dear anyone who disagrees with him on D. As someone who studied in advanced mathematics. You are exactly wrong. As for pemdas...I haven't thought about it in years. It's just built in to our calculators. It's built into the axioms of math. I mean what you are implying is nonsense. The reason why 12/3(4) = 16 is because it follows naturally from the definition of mathematical operations. A. Multiplication acts on two adjacent numbers. (key words for adjacent and two) B. Multiplication is associative. C. Division is multiplication of the inverse of a number. D. A number multiplied by its inverse is the identity and every number excluding 0 has an inverse. etc. 12/3(4) = (12/3)4, by necessity of the axioms of multiplication.
@@insainsinwell that's just not true, calculators themselves are split on what to use for their orders of operations. with some only changing to pemdas over pejmdas due to lower level education requesting it. It's not a settled issue either way
D is interesting because we're talking about the parse tree and which needs to be evaluated first. If I was writing that in C, if you looked at 12 / 3 * 4, you'd have to know how it'll be parsed and evaluated which isn't obvious at first. I can never remember. In the interests of clarity for my future self and my coworkers, I'd write it as (12 / 3) * 4. I wonder if teaching parsers would help people understand PEDMAS. :)
These math debates are so much fun to dive into-each one has its own twists and turns! Personally, I think this video really breaks down the logic behind each debate brilliantly. I’ve been exploring similar tricky concepts, and SolutionInn has been super helpful for clarifying the details.
Yes, in countries that learn multiplication by juxtaposition There is no universal way to interpret x÷y(z), so their are multiple valid solutions. The problem is poorly written
When you stop using the division sign and use fractions, you are putting brackets around the numerator and denominator. This is not using fractions however, just the division sign so there is no brackets.
@@forbidden-cyrillic-handle Considering new Casio calculators are still being made and sold, all of which using implied multiplication, I'm gonna have to disagree with you on that. It's what my algebra and my geometry teachers in highschool taught me, and it's the calculator I own. I got an A in every math related class, so I certainly wasn't napping through them. It's my favorite subject after all.
@@ultimatedude5686 I've never seen a coherent argument for why it should be zero, in any context. Most of the time the argument is "BuT zErO!!1" which doesn't really cut it. It's 1.
@@cooperisedyea, “Nothing to the power of nothing equals something” makes so much sense in comparison.. right. The right answer is 0, unless you’d like to prove me wrong in a video, by holding a 0 object, putting it to the power of a 0 object, and somehow creating an object. Go ahead pls. You can’t multiply nothing by nothing and create something lol. If that was the case, then why can’t we solve world hunger by simply creating food from no resources? The only slight chance 0^0 has a chance of being 1 is in some BS philosophical debate where you say crap like “Oh but the existence of nothing is counted as a state of existence, making it 1” which is also wrong
@@ultimatedude5686 But doesn't it require dividing by 0? As far as I understand: x³ = x × x × x x² = x × x x¹ = x x⁰ = x/x x ^(1) = x/x² Etc. Wouldn't that mean that 0⁰ =0/0?
@@donedonefree6828 I don't think that's true. Why are you representing x^0 as x/x and x^(-1) as x/x^2? I would represent x^0 as simply 1, and x^(-1) as 1/x. The issue here is that you're defining x^1 = x as the base case. If we instead said the base case is x^2 = x*x, then by your logic we could say that 0^1 is undefined as it would be represented as x^1 = x*x/x. If we instead define our base case as x^0 = 1, then everything works out. This is a valuable definition because it comes up in practice. For example, the Taylor series for e^x is written as x^0/0! + x^1/1! + x^2/2! + x^3/3! + ... This converges for all real x, but unless we define 0^0 as 1, it does not work when x=0. This kind of thing happens all the time with power series.
I think that 4½ should instead be written as 4 + ½, or 9/2, or 4.5. In fact, I have not seen mixed fractions since elementary school, they are simply a bad way to represent fractions
We simply used the / symbol throughout college. Literally never seen anyone remotely knowledgeable about math use ÷. I vaguely remember seeing it used in kindergarten and elementary school.
No that's not the argument for 1 AT ALL for the D, the argument is that this implicit multiplication can be seen as prioritary over division, replacing putting many parenthesis. ex : 1/2x instead of 1/(2*x), which is much longer and uglier. Is it just a notation and is purely SUBJECTIVE, so no, the answer is not simply 16, it depends on the convention you are using.
@@Termenz1 e is a function, the exp function, and "raising e to a power" is just putting that number through the function. It looks like how normal powers work, which is why the notation is to raise e to a power, but that only makes sense for real numbers; you can also raise e by complex numbers and matrices
@@robertveith6383 causing confusion is ambiguity, which that problem does. It's a 'bad' question. Parenthesis, parentheses, idc I wasn't bright at grammar.
It's much better to explicitly write (2a)÷(3b) to avoid any ambiguity (as with Wolfram Alpha, Maple, Mathcad or some online, mobile or handheld calculators).
Can you solve x^ln(4)+x^ln(10)=x^ln(25)?
ruclips.net/video/xBpZRWCGw30/видео.html
B is hotly debated, true. Is zero odd/even? Is zero a natural number?
@@deltalima6703 Zero is even and non-natural number
@@deltalima6703 What? I don't think that anyone has ever argued that zero is odd.
@sriprasadjoshi3036 some say 0 is even, some say it is neither even nor odd. Recent advances in set theory strongly suggest the ancient mathematicians made an error and 0 is actually a natural number. There is an ongoing debate at the moment.
@@deltalima6703 how come 0 is so bullied all the time? Everyone tells him "you're nothing" and he's always excluded from set parties.
D is just intentionally bad notation. The others at least have some interesting mathematics behind them. So D is indeed the worst debate.
Yeah easily D, the rest are like good debates that are just worn out. D is just faulty written. Barely anyone even uses the division symbol, the just write it fractionally depending on what they want to portray
I really don't think D is that bad of a notation. To me, D is very clear and has a clear answer.
In terms of bad notation, tan^-2(x) should take the top spot. : )
Right. I don't like D not so much for mathematical reasons, but rather because its one of the low-effort comment engagement posts that bots post on social media. And I hate that.
@@blackpenredpen probably not to someone who really knows their math, but 12/3(4) would be very different to (12/3)4, give two answers because they are different. And pemdas is not taught well, there’s always confusion on multiplication or division first. The way it’s written is meant to confuse people into thinking it’s 12/(3 * 4) so I would say it’s bad notation, since most people use fraction to mistake division now
I see what you mean, but at that point the debate is not about really about mathematics, it is about syntax.@@blackpenredpen
Half the comments are saying D is the worst debate, the other half are arguing about how it's really solved. ABC are just about forgotten
I cannot believe the amount of people who don't understand D. Like, if you have 5/4+4*5 do they seriously think that means 5/(4+4*5)? In what world does it make sense to take everything after a division sign and throw it together in parentheses when there are no parentheses... Perhaps PEMDAS needs to be taught a few more times in school.
@@gudadada Actually, both sides of the debate are correct. This expression can be solved either way, as both interpretations of the expression are commonly accepted.
When you have a number adjacent to a parenthesis, it's called a juxtaposition, and is solved before other operations. Outside the US, some countries instead learn BEJMA (Brackets, Exponents, _Juxtaposition_ , Multiplication, Addition).
It's very handy for factoring. Imagine the expression [(2x² + 4x)], which can be rewritten as [2x(x + 2)]
So technically, both forms are correct. Just make sure to use extra brackets when inputting into a calculator (I have two different calculators that solve it differently), or when sharing your problems with others, to make sure everybody is on the same page!
No, you are misinterpreting what juxtaposition is. Juxtaposition refers to a sign being implied, but it does NOT change grouping. For example, 20÷3x,x=5 is NOT the same as 20÷(3x),x=5. The former is 100/3, the latter is 4/3. This is a rule agreed upon by all mathematicians and functionally by all calculators. Of course, this "issue" is usually mitigated by using fractions which have much better visual clarity (everything in the numerator and denominator is contained), but there is really no debate, only one answer is correct. The reason your factored example works is because of left-right order. (2x)(x+2) is the same as (2)(x)(x+2). Really, it's incorrect to view it as (2x)(x+2), because that's one step into expanding. If you instead gave an example with division, say 2/x(x+2),x=5, now you'd have an issue. 2/(x(x+2)) is NOT the same as 2/x(x+2). The lone x does not get attached to the (x+2) without parentheses - that is plain wrong. Try plugging these examples in yourself to an algebra calculator if you don't trust the people who do math for a career.@@brickbot2.038
This is partly because B and C already have solved answers. There’s not anything to argue about there. “A” *also* has a solved answer, but the answer is “both, but it depends on the context”. There are plenty of formulas where 0^0=1 is required because it still outputs correct answers, but there are other cases where the output would be undefined. You could kinda argue there, but you’d still reach the final answer pretty quickly.
D gets argued about because it’s an argument about the understanding of syntax. It’s not arguing about a mathematical concept itself outside of parsing syntax correctly. In reality, whoever writes that should use clearer syntax regardless of - even if there is only one way to parse it, it’s still an awkward way of writing an expression.
@gudadada It's not that the first is wrong, it's just that absolutely no one would write it that way outside of inciting a debate. The second you see all the time halfway through a problem. Not everyone is so careful with parens when solving z=20/y; y=3x; x=5.
D is the reason we don't use ÷ after elementary school
The division symbol isn't the problem. It's the juxtaposition of terms that people assume takes priority over the division symbol, that is the problem. We'd have the same issue if it were a simple slash.
It would be a much more efficient order of operations, if juxtaposition DID take priority over division, because it would allow you to write your denominators without snaring them, and professional scientists and mathematicians use this order of operations all the time. It's middle school teachers who don't want to deal with this problem, and the curriculum they follow that created this problem, who tell you that multiplication and division have the same priority regardless of notation.
@@carultch here mathematician, I'd tell you to write a parentheses or for me that's undefined.
@@AlejandroMéndez-j6j scientist here, I see 1/2π all the time, nobody ever writes 1/(2π). Makes sense too, because if 1/2π did mean the same thing as (1/2)π, then you'd just write π/2 to begin with, why write a complex form when it can be simple. By having it this way you have short expressions that are unambiguos
@@Alvin853: You can also write 1/2/π. Actually, you can just write /2/π, although a lot of people won't understand that; but once you get used to it, it's very convenient.
@@carultch Professional mathematicians wouldn't write it like that because it is just intentionally vague. If you want to do it properly without parentheses you use one horisontal dash and put the 12 on top and the 3 on the bottom
Both Mathematicians and computer scientists agree that 0!=1
Ah damn I commented a similar thing
Say it the other way around 1=0! Looks more confusing
@@omerdvir1709 != Means not equals in programming so 0 != 1 means 0 does not equal to 1. In maths 0! Means 0 factorial which should be interpreted as (0!)=1 which is also true
@@ghotifish1838you’re right. I’m actually doing programming on school so o should have got that reference but I thought he was referring to the factorial and it would make more sense to put the exclamation at the end
Side note: != as a substitute for ≠ is common (mostly via C) but not universal; Haskell uses /= and is another common inequality operator (e.g. BASIC, SQL, Excel). In C 0==!1 too, but 1=0! fails to parse (! is prefix not in C, postfix factorial in common maths).
230 - 220 × (1 ÷ 2)
You might not believe me, but the answer is actually 5!
yeah.
I can't believe it's 5!
yeah many people don't get that it's actually 5!
Such a good meme
@@shauryamathbasics You ruined it. You took the funny away by explaining it. I wanted to see people confused.
B really seems almost more like a communication issue than a math issue. There is no question that 1 and -1 are both square roots of one. it's more that there is confusion over the fact that the square root function is only looking for the principle positive square root instead of all square roots.
Yes, it's a matter of notation: knowing what the radical sign means.
You said it yourself, "the square root function is looking at positive values". It only uses positive values because the principal square root is a function, whereas the +- version is not
Same with D. People are doing the operations correctly, just in the wrong order. Poor communication around the necessary order of operations
@@radupopescu9977um actually complex square root is a different function from the principal square root. just like complex logarithm is multivalued, so is complex square root. But normal square roots and logarithms have one value. No one would say that ln(2) is a bunch of numbers, in fact they would write the answer to the complex logarithm IN TERMS OF normal one! Like Ln(2) = ln(2) + 2n*pi*i or something. Just like the actual answer to square equation is written in terms of principal root with additional symbols, like Sqrt(2) = ± sqrt(2)
@@gabrielfonseca1642 thank you for repeating exactly what I said😅
B is the one I meet most often but D is just stupid
agreed
Especially since the given answer in the video is wrong. The right answer is that its contextual; right-most inner-most is the standard evaluation order since left-descent parsers are problematic. So in most cases, the solution really is 1. But some mathematicians who don't care about formal languages use the order given in this video. The binary division operator is non-standard to begin with.
@@ontoversecorrect.
6÷6 =1
but 6 ÷ 2(3) 6÷3(2) lol. I'm not sure why they ignore juxtaposition or implied multiplication lol
Yeah, so many people always answer that "you forgot the +/-" in the comments of a maths RUclips video, when it's completely inappropriate to the solution development
@@ontoverseWhat? There's no case where it's 1. Only people who don't know how to do arithmetic gets the answer 1.
I definitely would say (d) is the worst because (a), (b), and (c) are all about defining different math concepts and people who don't understand the concepts, while (d) is about clarity and people being taught different things in school
You missed 1+2+3... = -1/12
Dat result comes from the foolish assumption of the sum converging in the first place
@@DatBoi_TheGudBIAS No, the sum was established before the current convention of convergence, and shouldn't be interpreted purely analytically.
The "identity" depends on an expanded version of algebraic manipulations that are consistent with analytic continuation.
@@radupopescu9977 Yeah, I should have addressed this, since this is the main criticism, and it would definitely be a big deal if it would be possible to get any other real number using manipulations like this.
Mathematics of course depends on any two independent parties being able to get the same result.
Although I can't proof that there are no such manipulations, I can at least point out some applications of the "identity".
First of all, ζ(-1)=-1/12, but this is not the big application everyone talks about. The real application is in algebra, as are the manipulations. You have this formula (Weyl denominator formula) that requires the halfsum of positive roots of a root system (important concept in Lie algebra), and the integers form a root system of a generalised Kac-Moody algebra (kinda like an infinite Lie algebra).
Well, the proper constant for that formula is -1/24, which could be interpreted as half of 1+2+3+4+...
You can apply the same logic to why the modular form (these are important complex functions) of weight 1/2 looks the way it looks, but this happens to follow from my "main" application.
@@radupopescu9977 I'd recommend reading about analytic continuation before talking about this.
That is really a battleground should really have been in the list..
I think D is the worst because writing it in fraction form would clear any debate so I think it's more of a communication/notation problem than a debate really. I can't think of a single situation where I would rather write ÷ instead of just expressing division as a fraction.
t was used in grade 3 then forgotten about until it shows up as a button on a calculator.
I do the division sign when I'm dividing a fraction by another fraction, there just isn't enough space to write 4 "layers"
Not laying out division and multiplication in an intuitive order or using brackets unambiguously should be considered as invalid as not having a closing parenthesis...
To paraphrase The Big Lebowski on PEMDAS: "You're not wrong Walter, you're just an asshole."
@@Firefly256id probably multiply by the inverse of the fraction
"I think D is the worst because writing it in fraction form would clear any debate so I think it's more of a communication/notation problem than a debate really."
EXACTLY THIS IS WHAT I WAS SAYING. The debate is stupid because it revolves around an ambiguity that should not be their in the first place.
D is the least interesting as it's mostly a question on syntax. The people who say the answer is 1, generally do so because they view 3(4) as implied multiplication, which has been taught (by some) to have higher precedence than standard multiplication (using the "x" or "÷" symbols).
I wonder how the responses may change if we did some alterations to the question:
12/3(4)=?
or evaluate 12/3x where x = 4?
or what about:
12 ÷ 3π=? would you evaluate that as (12 ÷ 3) x π or 12 ÷ (3 x π)
I'm not arguing for one or the other, it's just that I can see how people would find it ambiguous and I can see an argument for both sides. But all in all, it's just not an interesting problem.
D is not about syntax, it's about associativity of operators of same order. It appears to be "left associative" in USA, and "right associative" where I'm from
Completely agree
@@dfhwze The order is exactly what's in question. Division has been used as lower than multiplication, or specially lower only to the right (meaning the division itself could imply the answer). And implicit multiplication has been higher than explicit.
All else was added to fuel the flames. E.g. the parenthesis are there to allow the numerals, which are there to get more people to weigh in without grasping the question.
So people really teach that that would have a higher precedence? That is just weird. That is just a sign people get this wrong because the school system sucks.
@@thomasdewierdo9325 I wouldn’t say that. I would say the fault lies in the person who wrote the question. And of course it was written that way purposely for the controversy.
I think D. The ISO 80000-2 standard for mathematical notation recommends only the solidus / or "fraction bar" for division, or the "colon" : for ratios; it says that the ÷ sign "should not be used" for division.
The division sign isnt the problem - so why would it matter that it does not follow ISO?
@iF3lixDE The ISO is just one more nail in the coffin that leads to the issue. If you write it as a fraction with 12 on top (numerator) and 3(4) on the bottom (denominator), you get 1.
@@p4z9m You would. But there is no fraction.
If you wrote a fraction with 12 on top and 3 on the bottom youd get 16
@iF3lixDE I did choose D because, besides the parentheses, Im with the ISO on this one. So this is bad notation through and through. Also, the world is a big place, and not everyone is in school.
It's (D). The others at least require some mathematical thought. (D) is just dumb and is only an issue because people hate that particular division symbol and assume it means something that it does not.
no I'm pretty sure it's caused by multiplication by juxtaposition being weird on some calculators (i.e. PEMDAS vs PEJMDAS)
@@NOT_A_ROBOT Correct. Multiplication by Juxtaposition is still very common, so that's why we're getting different answers.
So it is 12/3 × (4), right? This sign creates confusion cus it is just next to a paranthesis, right?
@leaDR356 basically. Some people, especially old people learned to multiply the parentheses first so 12 / 4(3). Tbh thats how I learned it and managed to get a math minor... so it matter little when you are calculating on your own cause your not going to write it that way past 6th grade anyway
@@leaDR356 Not necessarily. It is perfectly valid to consider juxtaposed multiplication as higher priority than explicit multiplication or division. In which case the answer absolutely is 1.
D is not a math problem it's a notation interpretation issue
So Why when it's 12/3X we interpret 3X as a number and not as 3 * X,
Prioritising implicit multiplication is more consistent
there is no 'we interpret' such and such, this notation is ambiguous and I would personally ask for clarity if this was given to me, the use of '/' is something that is typically only seen online and therefore the correct interpretation is undetermined however if you were to say 12÷3x this notation is not ambiguous and clearly implies 12÷3*x
Yeah people need to realize that pemdas is just a convention and not a mathematical truth
12/3x=4x right?
Exactly why notation with ''/'' is limited and not typically taught in math classes. Regardless, no it is not more consistent at all. 'Left to right' is more consistent than 'left to right but implicit multiplication before division'...
@@aMyst_1 It can also be equal to x/4. If you use multiplication by juxtaposition. The problem is not that one answer is the correct answer, the problem is that there's no universally agreed upon way to interpret the problem
"Multiplication denoted by juxtaposition (also known as implied multiplication) creates a visual unit and has higher precedence than most other operations. In academic literature, when inline fractions are combined with implied multiplication without explicit parentheses, the multiplication is conventionally interpreted as having higher precedence than division, so that e.g. 1 / 2n is interpreted to mean 1 / (2 · n) rather than (1 / 2) · n."
What are you quoting? And what is said to be ''conventional'' in certain literature (kind of suspect this is not mathematic literature but rather scientific or engineering literature) does not change the actual rules of mathematics.
@@FrenkieWest32 That's not how that works, you just made that up. There is no rules of mathematics saying what should the mathematical notation look like. 12/3x = 4/x for people that say that implied multiplication is done before division, and 12/3x = 4x for people that say that implied multiplication is done with the same priority as division.
There's nothing mathematics tells us about how it should be interpreted, math notation is made by humans and could be completely different
@@F_A_F123 What exactly did I make up? Your comment is dubious, ironic considering the topic.
'Rules' are made by people. I would not refer to the intrinsic nature of reality with 'rules'. Orders of operation in mathematical communication are 'rules' made by humans. Just how one can say it is a rule to use 'x' for your first variable, even though this is completely manmade.
With all that said: yes, rules are not set in stone completely. And apparently there is more controverse around this than I thought.
@@FrenkieWest32They're quoting the Wikipedia page for the Order of Operations, but conveniently left out the part right after the quote saying that some academic authors advise against the form a/bn (the form in their comment) and say that you should use the much less ambiguous forms of (a/b)n and a/(bn). In the case of the video, that would give both (12/3)*4 and 12/(3*4).
@@zaleshomeowner3493 You made that shit up, I didn't look at wiki.
I think the issue with D is that there's disagreement about whether or not implicit multiplication takes priority over explicit division. I remember The How and Why of Mathematics made a couple videos about this debate, that I thought were really good.
What really bugs me is how people pick a position and start making stuff up about all other positions; in fact, this video is guilty of it. It presents it as "do the parenthesis first", but the parenthesis is only there to distinguish 3x where x=4 from 34. In a conversation, this could be a simple misunderstanding and be resolved. But when it moves to lecturing like this video, it's a straw man, misrepresenting the other position(s).
I agree that The How and Why of Mathematics presented this well, including actual research.
Nah the issue is the garbage ÷ sign which no respectable professor in math1 / math2 or any of my engineering classes in college ever uses. I feel like if they ever saw someone write equations in ambiguous notations as in d) they'd fail them on the spot. If I ever have to use such "cramped" notations (matlab programs for example) I'd just write (12/3)*4 or as shown in the video 12 * (1/3) * 4
@@pointyheadYTit is garbage but using / doesn’t resolve the ambiguity, at least not for layman.
In scientific academia, multiplication by juxtaposition is the norm. So you if saw a/bc, the standard interpretation is a / (b*c) and not a/b * c.
@@rubicon24One wouldn't use a / but proper fractions
@@mordret103 you’d be surprised how common it is. For example, the amount of substance n in the ideal gas law is commonly written n = pV/RT
You’re expected to interpret this as pV/(RT) and not pV/R * T
I think the problem with D is that even with the same operation, it's usually implied that when the operation doesn't appear, it should be done first. e.g. when you write 12÷4a, you kinda want to do "4a" first. Obviously the whole thing with order of operations is just a convention. As a programmer, when I occasionally write math operations in the code, I often add parentheses which are technically redundant, just to make it clearer what is going on. e.g. I write var1+(var2*var3) instead of var1+var2*var3. They're technically the same, but the first is much easier to understand from a quick glance, and unlike what some people think, the point of writing things in math (and code) is to make it *easier* for other people to understand what they're reading.
As for the debates, personally I think the worst debate is C. Debates A, B and D are just about conventions. You can define these things however you want, it's just for the sake of convenience, there's no hidden meaning there. Like, you could define square root to be a function that returns pairs of values. It would be less convenient to work with, but nothing would break. C is the only debate that is actually about the *meaning* of something, that actually shows a fundamental misunderstanding of what real numbers are and of how series work.
How is B about convention ?
I really like your point about notation. I always teach that notation in maths is like good punctuation in English. Your main objective is to communicate your intention to your readers not to be technically right but misleading - that's no good for anyone. Things like D only exist for the sake of it. The others are all real things to define or discuss.
@@yann8765 why pick the positive root and not the negative root as the single answer? It's just a convention, like rounding 7.5 up to 8 rather than down to 7. I mean it is sensible but not mathematically forced on us?
@@72kyle
None is "picked", the result is ±1 ?
@@72kyle
Rounding 7.5 to 8 isn't a convention either ; if it was rounded to 7, rounding would create a bias toward lower values (0,1,2,3,4,5 (so 6 digits) rounded below, but 6,7,8,9 (only 4 digits) rounded above.
EDIT : I take that back, actually it seems to me that it is secondary to another convention, which is that when splitting a continuous interval, we tend to do it as [low, up) rather than (low, up]
Because for D multiplication by juxtaposition is often done first. Even in other fields you will catch scholarly papers dividing by stuff and not putting parentheses around their denominators.
PEJMDAS vs PEMDAS moment
It's literally the more natural way without calculators or computers to look at it. It's why it's common in older education
and it's a big mistake to ignore parentheses
@@adamwalker8777 The parentheses are incidental - they're only there to allow for juxtaposition between numerical values (à la “3a”). The contents of the parentheses are trivial:
(4) = 4
So that has no bearing on the controversy. What's controversial is whether juxtaposition takes higher precedence than other multiplication and division; i.e. whether “xy” represents “(x ⋅ y)” or just “x ⋅ y”. Its simply a matter of notational convention - either way would work entirely consistently - but people's intuitions seem to differ, so it's best to make your intentions clear using brackets or fraction bars.
@@TheJamesM When I talk about parentheses, I don't mean an expression like this (4). putting one number in parentheses is stupid and makes no sense. when we have a complex expression in parentheses and we are reducing it, then we must remove the parentheses after all the calculations inside the parentheses. That's what I mean.
D is on the verge of not being a math problem, it’s a problem of semantics not math. It’s closer to being a freaking English problem. I truly feel sorry to those that find problem D mathematically interesting / provocative because there are endless ACTUALLY interesting ideas to explore in mathematics.
100%
All math guides from professionals state that adjacency notation for multiplication is an implied parenthesis in the order of operations. For example: 12 ÷ 3n, when n = 4. 3n is pre-grouped single operand. 3(4) is the same as 3n, when n = 4. Similarly, fraction notation (which is division) will be performed prior to left-to-right order as an implied parenthesis.
@@forbidden-cyrillic-handle what style guides from professionals do you think I'm referencing? The ISO standard (one of) is the style guides from professionals. Multiplication (the operator) in the ISO standard is a cross (×) or a half-height dot (⋅). Adjacency is a product in the standard, but is a single expression rather than being 2 operands separated by an operator. It takes precedence over spaced operators.
3n takes priority sure; but 3(4) is not equivalent to 3n and n = 4. That would be (3*4). If you substitute n with 4 and write 3n as 3(4) then that is just the wrong procedure
Okay, but let’s say you set n = 5 + 6. See how you would have to add an extra parentheses for each little thing you tack on to it? Honestly, I think it’s just easier to just assume that 5n; n = 5 + 4 can just be expressed as 5(5+4), rather than having to rewrite it as (5(5 + 4)).
3n is NOT the same as 3(anynumber)
B is the worst debate, because it shouldn't be a debate to begin with.
This video 'I hate internet math' by doorwaydude is what I consider to be a 'sequel' to this vid. It attempts to explain why these debates exist in the first place (and why some of them are pointless). I really think people in this comment section should have a look if they are still unsure about things like 0.999...=1
Thanks, dude! 👍
No problem with dude, I hope?
".999 repeated is 1 becuase you cant find a number between them" is a really cool observation
And IF .999... is not equal to one, then there must be an infinitude of numbers between .999... and one.
no (for me) because this means its the exact next number. think of 1 quantum of numbers . its 0.00..1 so this is the smallest value .. can you have 1.5 cents? no because 1 cent is the quantum of euros , thats why theres no "cent" between 1 cent and 2 cents
It doesnt make any sense for me because if you look at integer Numbers 2 and 3 you cant find any number between them and they arent equal
@@fab3f2.5
@@namespaced4437 "integer Numbers"
A. Undefined unless convenient.
B. Principal square root if we're applying the function (and we nearly always do); plus or minus if we're trying to find all solutions.
C. Boring debate due to people not understanding how infinite sums work ("but they can't be equal because there's still a difference of .0000000000...001!') or who just outright deny the use of infinity for philosophical reasons (finitism) in the first place.
D. Old clickbait tactic used to drive engagement via making people think they're arguing about mathematics when they're really arguing semantics.
From my experience D is the least interesting because it's a semantic debate and almost always clickbait but people get really heated about C and, as annoying as it is, people never tire going at it about it because everyone is really convinced of the sense of their argument and the nonsense of the other side. Never really seen people argue about A or B. I think you should have put that approximation of pi meme there instead if you know what I'm talking about.
Tell that to any teacher I've ever had for B. If I didn't write down -1 as well I'd get points off.
Yep, D I'd say, write parentheses or it's your fault.
"C. Boring debate due to people not understanding how infinite sums work ("but they can't be equal because there's still a difference of .0000000000...001!') or who just outright deny the use of infinity for philosophical reasons (finitism) in the first place." - That's a strawman argument. Those of us who object to this equality claim that you have never proved that 0.999... is a real number. If it is a real number, then, yes, it is most definitely equal to 1. But simply assuming that it's real is circular logic. One could just as easily assume that 0.9 recurring is how we write an infinitesimal (a hyperreal or surreal number that is infinitely close to 1, but less than 1, while at the same time being greater than every real-number-less-than-1).
@brendanward2991 and what level of precision requires an infinite number of 0's in front of it to be accurate? At some point all math is rounded to the number of significant digits.
C. Also relies on people not understanding that decimal notation is not unique, just like rational notation. 1 can be written as the 2/2, 3/3, etc.etc. 1 can also be written as 0.999... or 1.00...01
Definitely D. Terrible notation and not ISO compliant (ISO-80000-1 and ISO-80000-2 not followed).
It's simply ambiguous notation. A trick.
Academically, multiplication by juxtaposition implies grouping but the more programming/literal interpretation does not.
Wolfram Alpha's Solidus article mentions the a/bc ambiguity and modern international standards like ISO-80000-1 mention about division on one line with multiplication or division directly after and that brackets are required to remove ambiguity.
Even over in America where the programming interpretation is more popular, the American Mathematical Society stated it was ambiguous notation too.
Multiple professors and mathematicians have said so also like:
Prof. Steven Strogatz, Dr. Trevor Bazett, Dr. Jared Antrobus, Prof. Keith Devlin, Prof. Anita O'Mellan (an award winning mathematics professor no less), Prof. Jordan Ellenberg, David Darling, Matt Parker, David Linkletter, Eddie Woo etc.
Even scientific calculators don't agree on one interpretation or the other.
Calculator manufacturers like CASIO have said they took expertise from the educational community in choosing how to implement multiplication by juxtaposition and mostly use the academic interpretation (1) Just like Sharp does. TI who said implicit multiplication has higher priority to allow users to enter expressions in the same manner as they would be written (TI knowledge base 11773) so also used the academic interpretation (1). TI later changed to the programming interpretation (16) but when I asked them were unable to find the reason why.
A recent example from another commenter:
Intermediate Algebra, 4th edition (Roland Larson and Robert Hostetler) c. 2005 that while giving the order of operations, includes a sidebar study tip saying the order of operations applies when multiplication is indicated by × or • When the multiplication is implied by parenthesis it has a higher priority than the Left-to-Right rule. It then gives the example
8 ÷ 4(2) = 8 ÷ 8 = 1
but 8 ÷ 4 • 2 = 2 • 2 = 4
A,B,C I 100% agree with here, but D, no, 16 is not the corrext answer according to the evidence. 16 is 'a' correct answer, along with 1.
The expression is wrong.
That is the correct answer.
thank you, I'm sure if bprp was given: f(x) = 12 ÷ 3x then he would agree that f(4) = 1
The *internet argument* expression is wrong, because it's intentionally lacking context. The notation makes sense in most articles it is used, but could be clarified, which is what engineering standards like ISO 80000 directs. Casio did state that the reason they made regional models that don't prioritize juxtaposition over division (but not exponents, it isn't parenthesis) is that teachers of lower level maths insisted on it.
Meanwhile, in higher level maths it's common to define new notation within an article.
Thank you! I'm honestly disappointed bprp did not say this. His answer is just perpetuating the issue.
OK fine thanks for the thesis 🤓
@@0LoneTech In higher level maths you will NEVER SEE ambiguity. The only time you'd EVER see something like a/bc in real maths is if the journal explicitly states that that is the convention that will be used. And even then, you are very unlikely to see it unless it's involving something like 2pi, where it's a commonly used number, usually a multiple of an irrational number. Otherwise, pretty much no one is going to write a/bc in a real publication. Not if they want to be taken seriously, or if the journal is too broke to format for a fraction bar or to print brackets (which isn't going to happen).
D is the worst cause it’s never ending debate where you will not convince anyone ever.
Such deliberately ambiguous expressions should be officially banned by the UN.
It's not possible to "convince" anyone (ever), and it immediately follows from the Bayes' theorem (when a prior probability is exactly zero).
One may consider viewing The Bayesian Trap by Veritasium (for more details) if one wishes.
even though it's so damn simple and obvious. it really is the worst debate because it makes you lose faith in people
@@Aestareth_it's not simple and it's not obvious. It's deliberately constructed in a confusing way by omitted * sign before parenthesis. This makes a lot people think that this operation (multiplication without operator) has higher priority, because it's often used with variables symbols. For example, -b/2a for a parabola vertex.
Saying D) isn't the worst debate out of those is the same as saying that "I saw a man with a telescope" is worth discussing whether it's as seeing a man through a telescope or as seeing a man holding a telescope. It's just ambiguity and the flaw is found on the very phrase
Obviously D is stupid, but in this problem it’s still clear enough where there is a commonly accepted answer. There are some that are like 5/2a, which would be a better debate, but this one is at least somewhat clear. B, in the other hand, is stupid because literally both answers are correct. Like the square root is one is both 1 and -1 because both of them square to 1, but at the same time, it’s commonly accepted that a singular number inside the radical is looking for a single number when solving. Sure, +- 1 is technically correct, but that doesn’t matter. If somebody wants to know the square root of 49, it doesn’t help that it could also be -7 because negatives aren’t as useful in basic math, so we ignore that solution. Either is completely correct, so it’s a complete and utter waste of time arguing. At least arguing over syntax has a reason, because if an accepted solution is found, it will solve arguments in the future. Arguing over solutions to a square root just don’t matter. Like who cares whether or not cereal is a soup, it can be both. Colloquially, we define it separately so it’s not the same thing, but going by definition, it would be. All these stupid arguments just end up creating more confusion than they solve, so I think B is the worst, though D is a close second.
@@Chris_M5you could of just said the answer was 1..
@@icecube-n2d True, but I could’ve also said the answer was +- 1, which is the problem with the debate
Obviously it means a telescope and you visited a man together. :p
D is the worst because even BPRP is wrong. The answer is that there is no consensus. Different publications disagree on the relative precedence of inline division vs implied multiplication.
Hence why only idiots ever even write that kind of stuff unless the entire reason is the generate clicks.
Exactly. The answer is either "both" or "need more context" (or so forth). The fact that both "12 / 3 * 4" can show one answer while "f(x) = 12/3x" can show another is the whole issue. They're the same equation, but will result in different answers. If someone doesn't recognize that and instead consider it to be a single answer (either exclusively 1 _OR_ 16), that is what is incorrect.
That's why the ÷ symbol shouldn't be taught in school. Use fractions for everything
@@ThomasTheThermonuclearBomb It's still an issue if you use / for division, which some publication style guides recommend if it makes a nested fraction easier to read.
@@codahighland If you must use the / symbol, parentheses are absolutely necessary on both sides like 1/(3+4) or (1/5)*6
Good rule of thumb for square roots, if you introduce the first square root is +/-. If the question gives you the square root, usually it’s just +
For D, i like to change parenthesis into the X. For some reason, no one will tell you that 12 ÷ 4x is 12 ÷ 4 × x
That's because 4x is a monomial. So you treat it the same way you would do 12 ÷ 4
That's why this debate is so stupid, NO ONE in real life would write 12÷4x if they mean 12x / 4
That's what confused me so hard.
I thought it was basically 12/4x where you just simplify and get 3/x where x=3 so it's just 3/(3) and therefore 1.
Well 12/4x is exactly 12/4*x. What are you even talking about? I never thaught 1st year arithmetic is so hard
@@sychuan3729
No it is not, for no one writes 12/4x expecting it to be read 12x/4. No one. That doesn't happen.
D is the worst because order of operations is an arbitrary linguistic layer applied over the top of mathematics and not itself pure mathematics, so all arguing about it in a mathematic context is inherently insipid.
This is why I would guess c as the worst debate... because there is no debate
As far as I can tell, the only actual debate is whether 0.333... = 1/3. Because once the naysayers inevitably are forced to agree, the debate is over.
Suppose you're throwing a dart at the real interval [3,4) , and that the outcome X (= the number where your dart lands) has a uniform probability density across the interval.
Then
Pr(X ≠ 5) = 1
(as 5 lies outside the interval, hence it's impossible to be the outcome),
but
Pr(X ≠ π) = 0.99999....
(as it's possible, though highly improbable, that the dart lands on the number π = 3.14159... ; the chance that the first decimal of X and π don't match is 0.9 , the chance that at least one of the first two decimals of X and π don't match is 0.99 , the chance that at least one of the first three decimals of X and π don't match is 0.999 , etcetera.)
Since the probability Pr(X=π) must "clearly" be greater than the probability Pr(X=5) (as the event X=π is possible while the event X=5 is not), the probability Pr(X ≠ π) must "clearly" be less than the probability Pr(X ≠ 5) .
Therefore, I think it can be argued that 0.99999... does not equal 1 .
@@yurenchu 0.9 repeating is just another form of 3/3, due to base 10 numbers not being able to resolve 1/3.
If 1/3 = 0.333... and
1/3 * 3 = 3/3, and
0.333... * 3 = 0.999... then
1 = 3/3 = 0.999...
There's no odds or probabilities involved. Equations overcomplicate the whole thing.
Think of it like a display glitch in our number system. It's not that it's "close enough" or being rounded or anything like that. It's literally equivalent, just displayed differently.
@@yurenchu Actually the probability of X ≠ π IS 1. Read about the concept of "almost surely". The probability that the dart will land exactly on pi is 0, although the set of points it could land for that result to be true is not empty. It's just weirdness that comes from probabilities with infinite sets, but mathematically, you would say that P(X ≠ π) = 1. Of course, you are always encouraged to note that that 1 is representing an "almost sure" probability and not a certain one.
But your argument actually ends up helping the case for 0.999... = 1, because when you know about the concept of almost surely, you know that P(X ≠ π) = 1, and your creative way to calculate the probability by each decimal, also gives the 0.999..., so they have to be equivalent.
@@ric6611 You guys are missing my point. I'm not saying that it cannot be argued that 0.9999... equals 1. I _know_ that 0.9999... equals 1 within the common mathematical framework that we learn in school. And I went to school too, so I too learned this and am fully aware of this.
My point is that arguably it could make sense to devise an alternative mathematical framework in which 0.9999... does not equal 1. Such a mathematical framework.would allow us to distinguish between different scenarios such as the ones that I sketched above.
Therefore, I can see that there would exist (non-trivial) debate regarding option C.
0^0 = 1 since 0^0 is the set of all maps from the empty set to the empty set, where there is exactly one such map. It's also the IEEE standard. The often-made flaw is to assume that x^y requires to be continuous, and then argue with lim.
Using limits to prove the value of a function just shows that the person does not understand limits and functions. 0^0 = 1
Could you point me to a resource that explains that definition please? I remember being told during my studies that 0^0 is 1 "for a reason you'll learn about later", but I never pursued far enough to encounter it, and I've always been curious. This seems very interesting and I'd love to learn more
@@procop314 Maybe one of the more intuitive approaches is starting with really understanding what a power set 2^S of a set S is, then more generally a function space in set theory, and finally an exponential object in category theory. Thereafter, you'll have a very solid basis for knowing that S^0 must be 1 for any set S, including finite ordinal numbers S. Once you understand the set theoretical necessity, understand the natural numbers on the basis of a von Neumann definition of ordinals. Then get an understanding of how natural numbers can be considered as being embedded into rational and real numbers.
Persue this path step by step. Your understanding will get more and more solid.
@@geraldeichstaedt thank you very much for the reply . I'll look into it when I can. Have a nice day kind yt commenter
The order of operations for (D) has nothing to do with giving priority to parenthesis. 3(4) is implied multiplication, not explicit multiplication, and does take priority in this expression so the answer is in fact 1. The confusion with this problem is the result of calculators designed for the United States vs the rest of the world. Calculators designed for the US market will allow you to enter an expression using implied multiplication, but will auto correct and add the multiplication sign making it explicit multiplication when you hit "=". Calculators designed for the rest of the world will not add the multiplication sign and give priority to implied multiplication over division and explicit multiplication (As stated in the order-of-operations section of their respective instruction manuals).
If you want proof, enter the expression exactly as it is written into any calculator designed for the global market and you will get the answer of 1. Enter the expression into a calculator designed for the US market, and if the calculator will display what you entered as well as the answer, you will see that it adds the multiplication sign and returns the answer of 16.
This is not because we do not give priority to implied multiplication in the US. We in fact do just like the rest of the world. The mindset is that you should not (and usually can not) use implied multiplication when programming, which you are essentially doing on a calculator that allows you to enter the entire expression before hitting =. Matlab for example will not let you use implied multiplication.
In any text book written for the US, implied multiplication does take priority and you see this all the time with coefficients. For example; 1 / 2y would never be interpreted as 2 / y.
2 / y would be 1 / 2 * y which uses explicit multiplication and has the same priority as division but the order is left to right.
the most annoying math debate i had was with someone who would go “aha! just a theorem!” to everything, yet never had anything to add.
the worst part is we weren’t debating math, i was just trying to explain some math for a foundation of my argument.
How would you evaluate: 12÷3x
It feels very wrong to say this is (12÷3)x now that it's a variable, juxtaposition to me seems like "treat this as one entity"
Edit: So the debate basically reduces to whether 3(4) is implied multiplication like 3x or normal multiplication like 3*x. Implied seems like a better convention to me 🤷
They're essentially the same if only one term has a variable.
3X is implied multiplication, 3(X) is not, 3(X) is just normal multiplication, following the syntax of multiplication (left to right) it would be ((12/3)*4).
Some people might argue that
12/12=1, 12/(6+6)=1, 12/3(2+2)=1, 12/3(4)=1 but that's not how it works.
(6+6) Is after a division sign meaning the value of (6+6) is inverted into 1/12, by pulling (3(4)) out of the parentheses you would have to invert the equation into 12/3/4 and using the division syntax (left to right) you get ((12/3)/4) which is 1, it would be written as
12/12 = 12/(3(4)) = 12/3/4 = 1
@@arno_grnfld455 in my entire career, 3(4) has always been implied multiplication, no different than 3x.
@@arno_grnfld455What is "implied" vs "normal" multiplication? It's juxtaposition in both cases. The reason we add parentheses to get 3(4) instead of just writing 34 is because 34 is a different number.
@@TheUnlocked no, 3(4) is normal explicit multiplication, you can shift the 3 around and multiply it for example, 7(x+y)*6(-2x+y) = 42(x+y)(-2x+y). It is not tied down to a variable like how 3x is (3*x) (implicit multiplication)
Implicit or implied multiplication is like 3X where 3 cannot be seperated or shifted around, e.g. 3x/4 ≠ 3/4x, the 3 (or 4) is glued to the variable like: (3*X)/4, 3/(4*X), if 3(4) is implicit multiplication, you'd write it as (3*(4)) not 3(4) which is just normal multiplication. In this case 12/3(4) would follow the normal Syntex of left to right, (12/3)*(4), if 3(4) is implicit multiplication, 12/3X, X=4, it would work like 12/(3*(4)) instead
I found out there is no agreement for D. Much of the world, especially academia and Europe, follow the juxtaposition of multiplication which gives 1 because multiplication is implied. North America, especially teachers in the U.S., follow strict PEMDAS, which gives 16.
Even calculators don’t agree between the two and some have been known to document the order they use and some companies have been known to change between the two orders over time (and back).
In short, don’t write notation like this. It’s confusing and I swear it’s used just to start flame wars. And if order is confusing, add parentheses for clarity.
One more argument that I've seen in the past: 0 is a natural number. Some people say yes, some people say no.
It's not an argument, it depends on what you want the natural numbers for.
0 is natural. If i say 1/x is x for all natural i can say 1/x, x є N, but 1/0 not well defined, then i can say 1/x, x є N*, not including zero
I say no. 0 is a whole number but not natural number. A natural number is a number that you can creat with a string of 1's added together.
@@TheMassacreOfTheBanuQurayzahQu Are you American? In my experience, it seems to be Americans that don't want 0 to be a natural number. Besides, 0 is the sum of a series of 1s. It's just the series with 0 terms. :)
@@omp199 Hmmm. I suppose it would be better to say Positive Integers for 1,2,3,... And use Nonnegative Integers for 0,1,2,...
At least that's how Prealgebra, the Art of Problem Solving book gets past the natural vs whole debate.
For D I recently learned that there are some conventions where multiplication by jusxtaposition (when you have a parenthesis adjacent to a number, with no explicit multiplication sign) comes before both regular multiplication and division. Some calculators even have this as a part of their order of operations. Under this order of operations, PEJMDAS, the correct answer would be 1, but most people don't use this order of operations.
A is a matter of what branch of math you're working in
B is a matter of following a definition of a radical function
C is a matter of understanding this in term of limit question
D is a matter of confusing notation
I'm thorn between B and D as being the worst of those 4
The notation on D is not ambiguous. Some people are confused by it because they either cannot or refuse to understand basic order of operations, or because they incorporate misguided assumptions into the calculation.
@@anewman513 okay, the notation is technically not ambiguous, but it's for sure impractical. there are better ways to write down the same arithmetic operation without causing so much confusion about something that hardly has anything with the math itself
@@anewman513 It is ambiguous because some countries basically teach PEJMDAS by telling students to only remove operators if things are "grouped"
So someone who learned that would see 12/3(4) as 12/(3(4)) because omitting the multiplication is an implied grouping.
For variables most people do this most of the time.
Usually when I see someone write 1/2x they don't mean (1/2)x.
@@anewman513 Multiplication by juxtaposition is still commonly used, so there are two answers.
@@drdynanite - Yes, I agree it could be written in a way that even the most easily confused persons would not be confused by it. Though, the problem is with the persons who are easily confused, not with the notation itself
D is the worst debate because it just a notational thing that boils down to how we want to define it, there are no deep secrets hiding within this. B I have never heard of and is also just notation.....
if anyone wrote D on an exam or in a paper, I would consider that reason to flunk them. (They have misunderstood the central requirement in math to explain yourself clearly, not just be technically right.)
I'd also like to offer this counterpoint: 12 / 3x in this case it's fairly obvious that the x should go in the denominator, but it's actually the same rule as in (D), an implicit multiplication.
C drives me crazy the most, my friend asked me why it's equal to 1, did all the same stuff you did, and he just said, nah, they aren't equal cause this says one and this says 0.999...
it made me want to rip my hair out
D is definitely the most annoying. For every one time you see any of the other three, you will see 10 posts with 100x more engangment each about D.
Can you please consider making a wordless definition of the limit shirt? Meaning only quantifiers and other math notation exclusively. I would purchase it so fast, by far my favorite calculus topic!
D is just poor notation, while B and C are misconceptions, so only A worth a true debate
A isn't even a debate, it's just context-dependent. For limits it's an indeterminate form, for power series it's 1, otherwise just define your terms and run with it.
A is worth nothing. When you are in multivariable calculus and you have 1 limit from the x-axis and one from the y-axis do you say "let's debate"? NO. It's undefined. If sometimes it's convenient to DEFINE it (VERY LOCALLY FOR THIS PROBLEM) to get continuity, fine. Otherwise, UNDEFINED.
Well C is a good debate because the answer is 0.999 does not equal 1 😬😬
I so disagree with C
@@harrisonewer 0.999 does in fact not equal 1
you might have missed the ellipsis, however, as 0.999... does equal 1
Honestly, I don’t think it’s a question of what comes first. Rather, it’s a question of how that division symbol should even function. Obviously the fraction brackets are self explanatory, they take the everything at the top and divide it by everything on the bottom. I personally think that the division symbol should take everything on the left and put it over everything on the right, so it ends up working similar to a fraction, thus the answer being 1 for me. Now I am fully aware that isn’t currently the convention, and there are multiple reasons why it might be a bad idea to do it that way, but that’s just my opinion.
"I personally think that the division symbol should take everything on the left and put it over everything on the right" - it hasn't mean that for well over 100 years - it was changed no later than the late 19th Century. It's the term on the left divided by the term on the right, same as any other operator works (which didn't use to be the case).
"so it ends up working similar to a fraction" - it already DOES work similar to a fraction! 3+1÷2+6=3+½+6=9½. Under the old meaning that would've been 3+1÷2+6=(3+1)÷(2+6)=4÷8=½
"thus the answer being 1 for me" - yes, correct, but for the modern reason, not the old reason (the answer to this particular expression in fact has ALWAYS been 1, since there's only 2 terms in it anyway - 12 divided by 3(4)).
"Now I am fully aware that isn’t currently the convention" - yes it is. Under ALL rules, both old AND new, this expression has ALWAYS been equal to 1, as we were only dividing by 1 term anyway!
"there are multiple reasons why it might be a bad idea to do it that way" - no, there are multiple GOOD reasons for doing it this way, which is why it was changed more than 100 years ago, but as I said that change doesn't affect THIS expression anyway - the answer to this has ALWAYS been 1.
A: maybe also add 0 to the options
B*: ³√(-1) (principal root vs real root)
B**: arcsin(2)
D*: 2x / 2x (options: 1 or x²) to see who picks (2x/2)x instead of (2x)/(2x) by that "order of operations"
E: f(x)ⁿ for f that can be written without brackets (e.g. sin(x)² ln(x)³)
E*: f⁻ⁿ(x) for f which fⁿ(x) is used as (f(x))ⁿ and f⁻¹(x) being an inverse function
F: any notation which limit exists but written without limit (e.g. x³/(5x-x²) at x=0) (whether it equals a value when it gets only one possible value)
F*: sum of divergent series (e.g. 1+2+3+...)
F**: step function at 0
G: non-integer factorials without using gamma function (e.g. i! , 3.5!)
G*: analytic functions / continuations
H: Division as inverse (e.g. matrix, modular arithmetic)
I: Symbols that can act as both prefix and suffix operators (e.g. 3³3, 3!3)
I*: Symbols that can also used multiple times at a row (e.g. 3!!3)
J: Iterated binary operations without ending mark, unlike integration (e.g. Σx+1, Π2x)
J*: Operators which are not associative, but the operation orders or associativity directions are not well known (e.g. P → Q → R)
Impressive list of ambiguities.
B*: + set of all complex roots
D**: =2^2^2^2
D***: =−2^2
Funny part::
• ²3²
• !3!
WA somehow assigns a value to the latter.
Follow-up to B*: do irrational equations such as ³√x = −2 or ³√(x−6) = x have solutions over the field of complex numbers, and if so, what definition of the cube root should we use.
For (B) I think its important to point out that the ± is evaulated separately from the radical. We can see its OUTSIDE the radical, so it is its own thing, done after you get the radical's output. The radical only gives one output because it is a function. That output is defined as the principal root. See:
±√2 means plus and minus the (principal) square root of 2. √2 = ±... would be wrong.
not really math "debates" its usually just people who struggled to pass high school maths thinking they suddenly understand anything about the subject and get really loud about it
or people who think math is countable and can somehow be plural
@@zachansen8293or people who don't understand that maths is also a singular noun
@@methatis3013 its not, mathematics, not mathematic
You know maths is not plural for mathematics, right? Right?@@zachansen8293
@@imPyroHD maths _is_ a subject, mathematics _is_ a field
i feel like the problem with C is that a lot of people dont grasp that 0.999... is non-terminating. any proof stating that 1-0.999...=0.00...1 doesnt get that their "proof" contradicts the nature of the statement, because to imply that you're left with a fractional part after taking the difference is to imply that 0.999... terminates, which is a contradiction to how its defined
The fact there’s people on this video that say “it depends on context” or whatever for D is so dumb
It may depend on a tool you are using to evaluate the expression, and conventions may vary across different tools, so, yeah, it sort of depends on context (of conventions).
Some advanced calculators (e.g. CalcES aka Scientific calculator plus 991on Android) simply let you choose how you want them to treat implied multiplication: as *1/2π = 1/2∗π* or as *1/2π = 1/(2∗π).* And letting users choose the way they want their implied multiplication being evaluated is a wise decision - it sort of solves the problem by providing both available options.
In a similar fashion, the value of =2^2^2^2 or =−2^2 also depends on conventions being used, and different tools may give different results, And some tools will simply refuse to return a value for the second expression, stating it is ambiguous and "parentheses must be used to disambiguate operator precedence" - exactly the case with the expression from the video.
It's the correct response though.
It's simply ambiguous notation. A trick.
Academically, multiplication by juxtaposition implies grouping but the programming/literal interpretation does not.
That's the issue.
You can't prove either answer since it comes from notation conventions, not any rules of maths.
Wolfram Alpha's Solidus article mentions the a/bc ambiguity and modern international standards like ISO-80000-1 mention about division on one line with multiplication or division directly after and that brackets are required to remove ambiguity.
Even over in America where the programming interpretation is more popular, the American Mathematical Society stated it was ambiguous notation too.
Multiple professors and mathematicians have said so also like:
Prof. Steven Strogatz, Dr. Trevor Bazett, Dr. Jared Antrobus, Prof. Keith Devlin, Prof. Anita O'Mellan (an award winning mathematics professor no less), Prof. Jordan Ellenberg, David Darling, Matt Parker, David Linkletter, Eddie Woo etc.
Even scientific calculators don't agree on one interpretation or the other.
Calculator manufacturers like CASIO have said they took expertise from the educational community in choosing how to implement multiplication by juxtaposition and mostly use the academic interpretation. Just like Sharp does. TI who said implicit multiplication has higher priority to allow users to enter expressions in the same manner as they would be written (TI knowledge base 11773) so also used the academic interpretation. TI later changed to the programming interpretation but when I asked them were unable to find the reason why.
A recent example from another commenter:
Intermediate Algebra, 4th edition (Roland Larson and Robert Hostetler) c. 2005 that while giving the order of operations, includes a sidebar study tip saying the order of operations applies when multiplication is indicated by × or • When the multiplication is implied by parenthesis it has a higher priority than the Left-to-Right rule. It then gives the example
8 ÷ 4(2) = 8 ÷ 8 = 1
but 8 ÷ 4 • 2 = 2 • 2 = 4
With part b, its important to note that sqrt(x^2) is abs(x) BY DEFINITION. Thats where the plus or minus comes from. From "cancelling" the absolute value. Too many people believe it's just voodoo, which you kind of lend credence to by saying "oh, its when we solve this equation." NO! Sqrt(x^2) = abs(x). That is where the plus or minus comes from. Please bprp, I rely on you to note this kind of nuance since you are an authority, so i can point people to your videos when people get real resistant to being told they are wrong.
The definition is actually not a mere abs, the y-th root of x is the z with minimum principal argument that solves z ^ y = x. It just happens to be abs when dealing with non-negative reals. But, for example, cbrt(-8) is not -2, unless you are restricted to real numbers.
Every problem needs to say x ∈ ℍ or x ∈ ℝ or whatever. If it doesnt then whoever wrote the question wrote it for a classroom, not for youtube or for the real world.
The identity sqrt(x^2) = abs(x) is not a definition, it's a provable property.
@adayah2933 Depends how you're defining the operation "sqrt".
Maybe you should do a video about the Monty Hall Problem. Pretty sure that might come in pretty high on your list of "most debated math topics" if you kept getting comments about it :P
4:27 The response I hear most often for that is along the lines of "A number between 0.999… and 1 is 0.9999… And a number between _that_ and 1 is 0.99999… You can always add another 9, you just keep getting more specific." Which, of course, comes from a fundamental misunderstanding.
The intuition that these people listen to says that 0.999… is not 1, but (1 - 0.0…1), but that's not really a valid expression, is it?
The problem with D is that it when you deal with variables or symbols (𝜋 etc), implied multiplication does take priority.
Take this question from a recent GCSE maths paper for example: simplify 12x⁷y³ ÷ 6x³y. The correct answer is 2x⁴y² not 2x¹⁰y⁴. And nobody would see 1/2𝜋 and think it means 𝜋/2 instead 1/(2×𝜋)
Yes, because I have learnt algebra, I have learnt about implied multiplication. Because I have learnt implied multiplication I will use it when solving D even though there are no variables in the equation.
Bro got πremium instead of regular π
I don't think D is actually 16. Like A, there is no agreement. Also, there is no doubt that 5 / 3a is 5/(3*a), so why should it not work for 5 / 3(4)? Some calculators explicitally ask you which meaning do you want to give it. So I think there is no answer agreed upon, just like 0^0.
More fundamentally, the reason to read is to understand. It's not useful to look at an article and go "Ha! They wrote it wrong!" in preference over "so that's what they mean".
@@0LoneTech Thank you for bringing up that point.
I agree with your conclusion of disagreement.
And I'm pretty surprised that bprp answered it like it's clear and uncontroversial
That is wrong:
5/3a not equal to 5/(3a)
5/3a = 5/3*a = (5/3)*a
@@strongbrain3128 You have only asserted your conclusion, not addressed the question. Your reasoning would be "I refuse to distinguish implicit and explicit multiplications." This is one possible consistent position, but quite rare; it conflicts with established use.
It’s definitely D. Only because people are so adamant about debating about what is the correct answer and what is the correct order of operations.
A lot of them don’t even realize that all of the order of operations are just an agreed upon convention on how to solve mathematical problems. They are arbitrary. Some people just added another arbitrary rule and that is to solve juxtaposed multiplication before explicit multiplication and division.
This is what makes the question somewhat ambiguous. Since there is a rule (juxtaposed multiplication) that is not agreed upon by everyone. But instead of just agreeing that it’s ambiguous, people will fight tooth and nail for their way of solving it.
I fight tooth and nail for better notation.
But imo worst is still c because there are multiple proofs and there isn't any argument other than "nuh uh", but people still fight it.
@@michaelsorensen7567 Suppose you're throwing a dart at the real interval [3,4) , and that the outcome X (= the number where your dart lands) has a uniform probability density across the interval.
Then
Pr(X ≠ 5) = 1
(as 5 lies outside the interval, hence it's impossible to be the outcome),
but
Pr(X ≠ π) = 0.99999....
(as it's possible, though highly improbable, that the dart lands on the number π = 3.14159... ; the chance that the first digit of X and π don't match is 0.9 , the chance that at least one of the first two digits of X and π don't match is 0.99 , the chance that at least one of the first three digits of X and π don't match is 0.999 , etcetera.)
Since the probability Pr(X=π) must "clearly" be greater than the probability Pr(X=5) (as the event X=π is possible while the event X=5 is not), the probability Pr(X ≠ π) must "clearly" be less than the probability Pr(X ≠ 5) .
Therefore, I think it can be argued that 0.99999... does not equal 1 .
@@yurenchu if your dart is magnetically confined to whole numbers, the odds you'll hit 3 instead of 4 (in a random throwing) is 1 in 2. If your magnetic confinement is expanded to include tenths, the odds of hitting 3.1 is 1 in 11. That's a significant decrease in odds, right? If you expand to hundredths, the odds of hitting 3.14 is 1 in 101. We're already at a less than one percent chance to hit the approximation of pi that fits into our magnetically bounded random throw.
However, if you abandon the constraint and get infinitely small dart tips, the odds of hitting pi exactly are zero. You WILL get off by a little bit one way or another, EVENTUALLY.
@@michaelsorensen7567 The 4 is not included in this (half-open) interval, and instead of "odds" you actually mean "probability" (or "chance"); but okay, those points are not really important.
I'd say that the probability of hitting pi is not exactly zero -- or at least not the same zero as the probability of, say, hitting 5 .
To put it another way: suppose I throw the dart, and it lands on (pi + 1/7). The _a priori_ chance of hitting (pi + 1/7) equals the (also _a priori_ ) chance of hitting pi. But since the dart did hit (pi + 1/7), it proves to be actually possible; so that chance cannot be equal to zero. Or at least not be equal to the same zero as the chance of hitting 5 (which is physically impossible).
Sure, the chance of hitting (pi +1/7) is infinitesimally small, but it's (arguably) _not_ 0 .
@@yurenchu so now you're arguing 0≠0?
4:19 i didnt realize you could prove it like that, i always thought it was equal to 1 just because is supposed to be 3/3 = 0.9999..., but when the denominator and numerator of a fraction are the same then it becomes 1
"3/3 = 0.9999..." Huh?
Great, now you've added "which is the worst math debate?" to the list of the worst math debates...
This should be fodder for an XKCD if it isn't already!
I think you're wrong on D. There's a very significant difference between two objects concatenated and those same objects separated by a multiplication sign. Suppose the (4) were a variable like x. Then you would have 12÷3x, which is clearly 12 / (3x) not (12/3)x. This is the same situation - just replace the x with (4).
That's because a monomial is one term. 3*4 isn't one term.
As the other person said. Monomials are one term and implicit multiplication is two terms. So the are not the same.
People keep bring up variables as though "Ha! Found this example that proves you completely wrong!" When really those two things are not comparable.
D) you've been miscommunicated. We don't multiply the parenthesis first because we "misheard" doing the inside first, afaik that's just your strawman. The actual reason is due to juxtaposition which is considered of higher order than a multiplication dot •.
Example: 12÷3x vs 12÷3•x where x = 4
The first statement has a juxtaposition of 3 and x, wheras the 2nd statement has a multiplicative dot between 3 and x. Thus the first statement is 12/12 = 1 and the second is 12/3*4=4^2=16
Now do this with a parenthesis instead as you can juxtaposition those as well
12÷3(4) = 1 ≠ 12÷3•4 = 16
Or you can just view the equation as a fraction:
12
-
3(4)
@@jasonnelson9141 EXACTLY just use fraction notation, it clears up all of the confusion.
They are both 16 for the exact same reason. You do it from left to right. 12÷3x=12÷3×x=4×x
@@MrPassigo But 3x is one term. 3(4) isn't, so the notation is ambiguous.
@@jasonnelson9141 3x and 3(4) are of the same priority due to juxtaposition
The thing with b is that it depends on the context. Without any, it should be assumed that you're taking the principal root. In another context, it can be a multivalued function.
Exactly. For a cube root *³√−̅8̅* Wolfram Alpha gives:
• a principal complex cube root: *³√−̅8̅ = 1+𝒊√3̅*
• a principal real-valued cube root: *³√−̅8̅ = −2*
• a set of all complex cube roots: *³√−̅8̅ = {1+𝒊√3̅; −2; 1−𝒊√3̅}*
All three possible definitions of a cube root.
C) I think people have this notion that a real number only has ONE decimal representation. And I find that very understandable. Pretty sure that's the whole crux why people argue about this at all. The idea of 1 = 0.999... would break that notion.
You may be right about that!
-quietly cackling about p-adics -_-actually-_- having only one expansion for every number-
It's also that they give symbols different meanings in their heads instead of referring to clear definition. Hence we get stuff like "it's infinite 9s", as if "infinite 9s" was somehow a number.
I have to disagree on D, 3(4) should be done first like 3x because usually juxtaposition takes higher precedence than multiplication and devision, it should be 1
Or you could just be a normal person and write the whole thing in fraction notation.
Type it into any online calculator and check to see that you're wrong lol.
@@thewhat2 And yet read some academic journals' publication style guidelines and you'll see that in academic papers, it would be interpreted the other way and YOU'd be wrong. But the keyword is "some". It's just a matter of convention, and different places have different conventions. The trick is to realize that it is a dumb argument because you're literally arguing for which definition is better, and the fact there's a disagreement at all shows that neither is good and you should adopt a better way to notate it (which already exists).
D is less of a math issue and more of an issue with how some people interpret the syntax. It’s not that bad.
The first one is also an easy hit or miss because like you said, there’s no agreement.
I can’t decide whether I hate B or C more.
I’ll say C only because that not equal 1 is a mistake I can’t understand how they’d make (if they are a beginner the +-1 on B is an easy mistake) but 0.99999… not equaling 1 is not justifiable in my eyes (at least not comparatively) so it’s the one I dislike the most. B is a close second however.
No agreement means undefined. So easy. Then in particular cases you can use it however you want (once you DEFINE it).
@@AlejandroMéndez-j6j exactly, it’s hit or miss depending on how you define it.
The problem isn't how people interpret the syntax. It's how they leap from there to "any other interpretation is wrong, including the author's".
As a tutor, A is my least favorite. If only for the reason that I don't like saying anything "wrong." So when I bring up "Any number to the power 0 is 1" I always have to throw in "except 0, there's actually some disagreement on what that should be."
I blame D on teachers who are themselves mistaken.
You misinterpreted the argument for why D should be equal to 1. The actual argument is that putting 3 and 4 next to each other like that makes it a multiplication by juxtaposition, which is usually considered to have a higher priority than division. Many calculators can't agree on that one.
12/3(4) is not the same as 12/3*4. We all agree the latter is 16 but the former uses juxtaposition. The order to use for a juxtaposed operation depends entirely on the context. If you are a college level North American student you are expected to ignore the juxtaposition and treat it as a regular multiplication because that's what they decided to teach, as a convenient simplification which unfortunately introduce this confusion. If you are writing a post graduate scientific math paper for peer review, on the other hand, you are expected to use PEJMDAS, because that's the international standard for math publications in science. Also, many countries outside North America introduce PEJMDAS already at college level and thus scientific calculators sold in those regions are programmed to take juxtaposition into account and perform it first.
There is no right or wrong, it depends on the context, because PEMDAS and PEJMDAS are just different conventions that have different applications, in general the former is more scholastic, the latter is more scientific. That's also why it's a popular debate: different people have been taught different and there is no truth.
The problem with the root, is that people don't understand that it's not the root itself that gives us two options
x²=1
√(x²)=√1
|x|=1
x=±1
you either use absolute value
x²=1
x²-1²=0
(x-1)(x+1)=0
x=±1
or the difference of squares
-1/12 is mising from your list.
That wouldn't fit on this list.
I wouldn't say it's a big debate, though. While it's certainly confusing, there aren't a lot of actual arguments about it, mostly just people going "huh?"
It's also been thoroughly debunked multiple times in super long, very detailed videos.
@@ivansmashem The reason it's a thing is because of the zeta (ζ) function. The analytic continuation of it is very useful, but also gives weird answers like that.
There's not much to argue about.
@motobike3904 The Riemann zeta function doesn't give an answer of -1/12 for the analytic continuation of positive integers, though. That requires setting the parameter to -1, which is not valid, as it doesn't keep the analytic continuation.
It's officially gobbledygook and nothing other than someone saying, "Hey, what happens if we start plugging in invalid numbers?"
The analytic continuation of the function is indeed useful, but not when the parameter is -1. In that case, the result may be interesting, but it is pretty meaningless.
There's a difference between the sqrt(x) function and x raised to a fractional power. Sqrt(x) has a restricted domain. (Unless I'm wrong)
No, they are the exact same thing. Both have only positive values (on positive values) (ok, unless I'm wrong as well).
If x^(1/2) would have 2 answers, you would be unable to say (x^(1/2))^2 = x for example.
I don't understand the last one, why is 3(4) not prioritary if something like 3x would be calculated first? It seems that just changing the 4 for an x will totally change the order. Maybe I have been tought wrong but I would solve 12÷3x as if 3x was in parentheses because I have been thought, even if that may be wrong, that implied multiplication by parentheses are prioritary
If I have a items that must be divided into b lots of c people, then we are looking for x per c to get how many items each person gets... a ÷ (b × c) = x/c.
If I have to figure out how many items I need to give c people so that for every lot of a, each person would get b, then the required items are x... (a ÷ b) × c = x.
When given 12÷3(4), there is an assumed "= x." The second solution is therefore correct, since we simply assign the values of a, b, and c to the equation as shown. Since we are looking for x and not x per c, then we get 16 instead of 1.
@@theJACKHAMMER13 this got me even more confused, whynis adding =x changing anything?
@Draconic404 Basically, we have a known quantity in 4, so it acts differently to an unknown quantity in a variable like x.
Because we are solving for = x, we can do the order of operations left to right without implied multiplication. The ambiguity of the equation is solved by the word problems since known quantities can be seen as tangible things like items, lots, and people.
@@theJACKHAMMER13You're talking complete nonsense.
It doesn't say (12÷3) x 4, it says 12÷3(4).
@MrCmon113 Put it this way. When you read it how, do you say it? Twelve (12) divided by three (3) times four (4)? Or twelve divided by the quantity three times four?
If the former, the answer is 16. If the latter, the answer is 1; however, you would need to demonstrate a reason to call 3(4) "the quantity 3 times 4" instead of just "3 times 4." There isn't a feasible way to do so without adding more symbols such as parentheses to the equation, so this answer is incorrect.
It is easiest if you write the problem out left to right as you would say it to make it more obvious. 12 divided by 3 (=4) times 4 (=16), for example.
On the other hand, 12 divided by the quantity 3 times 4 would look like this--12÷(3×4)-- when written as spoken. Clearly different notation, yes?
Does that help?
I was taught to do PEMDAS left to right one by one, I never knew you combined PEMDAS into PE(MD)(AS).
I believe your answer for D deserves an "Incomplete" mark. It's correct, inasmuch as PEMDAS is the final word on order of operations. But PEMDAS is *not* the final word on order of operations everywhere. In some contexts "multiplication by juxtaposition" (eg,, signifying multiplication by just putting two objects next to each other) is given a higher precedence than regular multiplication.
Don't believe me? Spend some time going through the manuals for many different calculators, preferably of different brands and sold in different countries. You'll find that each manual (usually) has a whole section on "order of operations" and that some do assign higher precedence to "multiplication by juxtaposition" and some do not. Also note that in the submission guidelines for many scientific/mathematical journals, authors are instructed to observe the convention that "multiplication by juxtaposition" takes precedence over regular multiplication.
I would argue that D should also get a "no agreement" answer on this basis.
So is it 1
Seen too many people claim PEMDAS solves D without knowing that multiplication by juxtaposition exists.
@@oc-steve It's remarkable, considering BPRP himself published a video on how PEMDAS is incomplete, and as usually presented it doesn't even specify what the value of -1+1 is.
To add to my previous comment,
the definition of 3(4) is not 3*(4) but is (3*4). Use this and the answer to D is clearly 1.
3(4) is not implicit multiplication, you cannot force it to be that's not how the math works. If 3X, X=4 then it is implicit multiplication, you write that as (3(4)) not 3(4) because 3(4) is just normal multiplication
@@arno_grnfld455 Both 3(4) and 3x are examples of juxtaposition, or what you're calling 'implicit multiplication'.
@@arno_grnfld455 You keep using that word. I do not think you know what it means.
Implicit means unstated, as in unwritten. There's no multiplication symbol in 3(4) so if it involves multiplication, it is implicit. This is a plain fact of the notation only.
If I see 1-(2+3), I do not read "parenthesis without multiplication". It doesn't matter how many pairs of parenthesis you put around an expression, it's still the same expression; the point of the parenthesis is to keep it so.
Just like how the order of operations in common algebraic notation is implicit unless made explicit by parenthesis. You can always include parenthesis when doing variable substitution, e.g. x=4 in 12÷3x.
Juxtaposition is another clear fact. 3 and (4) in 3(4) are juxtaposed as there's nothing written between them. Again, this is notation, not arithmetic operation. 1 and 2 are also juxtaposed in 12, and in that case it is part of decimal notation.
BPRP presented one incorrect reasoning for arriving at 12÷3(4)=1. Using that to assert 1 is incorrect is the fallacy of denying the antecedent. He then implied that 12÷3(4)=12÷3×4, which is a begging the question fallacy. The question is: does implicit multiplication read exactly as explicit multiplication? The form is therefore a÷bc, and the numerals and parenthesis are distraction.
@@0LoneTech I agree with all the points in your comment, but I think you have missed one important point. Someone who doesn't know what "implicit" means is, with almost complete certainty, not going to understand what you mean by "the fallacy of denying the antecedent" or "a begging the question fallacy". You have to write to the level of your reader; otherwise you are not going to get through to them.
@@omp199 Good point. I am not particularly good at rhetoric (the art of speaking convincingly) but more writing to let my thoughts out sometimes.
I should also have included the observation that the parenthesis in 3(4) were there to preserve the 4 rather than make 34. They could equally be around either (3)4 or both (3)(4). I had written the reasoning motivating that use (that juxtaposition can mean other things than implied multiplication).
Hi very clear, 2 comments:
(1) I think it depends on the type of 0. If we are talking about finite math: combinatorics, graphs etc, and dealing with integers or natural numbers, then 0^0 will always be 1. But if we are talking about reals / complex numbers, then 0^0 usually is defined to be the limit of something, and the result may be undefined. It's not that there is no agreement on the math: instead the generally agreed meanings vary depending upon the context.
(4) If I write: 12 - 8 + 8 - 3 - 9, then there is no confusion. We just evaluate from left to right to reach the answer 0. It can be just the same for multiplication/division: 12 / 3 * 4 / 2 / 2 = 4. There's nothing different structurally going on. This also means that people (including YOU! heh) should feel free to write 27 / 3 / 3 = 3. It's certainly very convenient, but for some reason there is a taboo on this that most people aren't even aware of.
At my grade school, when we learned order of operations, we were told to do PEMDAS in exact lettering order, so instead of going from left to right, we would always multiply any terms before dividing. Pretty weird in hindsight.
When a number precedences a parenthesis with no symbol between they’re one term. I will fight you over that. No one would read or write 1/2x and think .5x
"No one would... _write_ 1/2x..."
Exactly. That is simply bad notation. So is 1÷2x. It should be written as a fraction as either:
1
----
2x
Or
½x
These are two different expressions. Let x = 5. The first expression then equals ⅒, and the second 2 ½. This is because the numerator and denominator are evaluated independently. Using either the / or the ÷ in this context creates ambiguity.
@@JustifiedNonetheless "That is simply bad notation" - it's NOT bad notation. It's a Term. ab=(axb) BY DEFINITION.
"So is 1÷2x. It should be written as a fraction" - fractions and division aren't the same thing. 1÷2x is 2 terms (separated by an operator), ½x is 1 term. Terms are separated by operators and joined by grouping symbols (brackets, vinculum, exponents).
Regarding the 0.999... == 1, I was thinking that if it's not equal to 1, then there must be some value x>0 such that 1-x > 0.999.... One could somewhat easily prove that x does not exist, since there will always be some extension of the geometric series that disprove an arbitrary x.
Yeah, I spent way too many years grading for discrete mathematics.
X=0.0...1
Obviously 😝
@@ThanatoselNyx Okay, now prove that's greater than zero... ;-)
0 < ε = 1/ω
Neither ε nor ω are Real numbers (though they're exactly as real as the badly named so-called "Real" numbers), nor is this particular use of 0.999.... If you insist that 0.999... is the Real number associated with the given geometric series, then the only option is 1.
@@angeldude101You can have multiple notations for the same number. This is another notation for one
@@kadanseward3022 And you can have multiple things with the same notation in different contexts. In the context of the ℝeal numbers, 0.999... and 1 refer to the same number. In the context of the hyperreals, 0.999... _might_ be its own thing that is legitimately less than 1 by an infinitesimal amount. In the context of the integers, 0.999... is nonsense.
C is definitely the worst because there’s no debate. one side is objectively wrong
But the only one which is kinda a debate is A
The other ones all have objectivly right answers too
ikr everyone is so dumb
In the usual construction of the real numbers, its obviously 1, yes.
That makes C the best though - it's a debate where one side is right and the other side is wrong, so it's an opportunity for the wrong side to learn something (if they have an open mind).
The others, especially D, are pointless debates that can go on forever because they're just arguments about conventions. Nobody can ever "win" the debate for A, B or D because they aren't about a mathematical truth, they're just about humans disagreeing about arbitrary choices.
@@forbidden-cyrillic-handle the way ur describing it makes it seem like the most ok one but remember even tho it is an agreement issue it is still the order of operations we r talking abt sonething u should know from private school
D: The reason for me is that terms can be brought out from within the paranteses. (2*2) = 2(2) = 4(1) = 1(4) = 4. Something infront of paranteses is part of the the paranteses. x(y) = (x*y). This is where the confusion comes from I think. Noone would write 1/2(4) if they ment 1/2 * 4, simular noone would write 1/2(4) if they ment 1/(2*4). As a programmer I would always write it as (1/2) * 4 or 1/(1*4) to make it clear.
(4) = 2(2) = 4
same as (2x) = 2(x) = 2x
surprisingly, his solution for D isn't correct.
admittedly, it's not really recommended to write it down like this, but it looks like that the international physics literature overwhelmingly agrees upon that, say, "2x" refers to an "implied multiplication" which cannot spilt up any further and thus has predominance over anything else of seemingly equal (!) order. or in order words, "2 × x" is treated differently from the other notation.
no one has to believe me, but that's also how most scientific calculators work (which isn't by mistake but design.)
but what I actually tried to say is that in those rare instances where it actually says, "x / 2y," the expected operation to be performed is, "x / (2 × y)."
at the end of the day, it's just a convention and doesn't break actual math.
also, I wouldn't even call it a math debate as unlike all the other examples, it's not even about math at all, but sheer syntax, which belongs somewhere else, as far as I'm concerned.
one way to tell them apart is: math is the part which works for aliens as well, while syntax is invented by humans.
100%
2y represents a number, it isn't the same as saying 2 times y. However 3(4) isn't a representation of a number, it's simply a way of writing 3 x 4
@@TheSketchGuy672 that is were we disagree, 12/3*(4) and 12/3(4) is not the same if you use PEJMDAS.
12/3*(4) is equal to 16, 12/3(4) is equal to 1. This is what the J in PEJMDAS means. It consider juxtapositions (multiplications without written sign) to be of higher importance than explicitly written operations.
@JeppeAchtonNielsen what the heck is pejmdas.
Option D is not so clear. The RUclips video "PEMDAS is wrong" by "The How and Why of Mathematics" tells some examples why multiplication by juxtaposition should be made before division (at least slashing fractions). It is indicated like that in an article by the AMS, another Physical Review Style and Notation Guide, and is usually used when using x, like 1/2x, which is interpreted as 1/(2x) and not x/2.
The debate surrounding D is pointless when you realize that fraction notation exists. By just using fraction notation it gets rid of all ambiguity.
Yes, surprised by this video being able to explain why .9 repeating is a series equal to 1, but not using multiplication by juxtaposition.
D is a matter of multiplication by juxtaposition where 3(4) takes precedence over 3x4. It used to be taught that way 100 years ago, and it is coming back. Some calculators are programmed now to do #(#) before doing x / left to right. My calculator can be set to do it either way.
Another reason for defining 0^0 to be 1 is that in general for non-negative integers m and n, m^n is the number of functions from a set with n elements to a set with m elements, and there is 1 function from the empty set to the empty set.
D is notation problem
The problem with D is that academic textbooks above a certain level almost unanimously disagree with him. It might be 16 in a grade school, but in any setting that matters at the higher level, the answer will almost always be 1. No one at that level is even attempting to work out a grade schooler's mnemonic device because it isn't relevant to people's understand of math at that level. (The division sign used doesn't particularly matter, so I won't address the difference in division signs.)
Let's say you have 12 / 3b. Hard liner PEMDAS users will say that it's equal to (12 / 3) * b, which is equal to 12b/3. So hard liner PEMDAS people believe that 12/3b = 12b/3.
That is prima facie absurd. The only reason to have written 12b/3 as 12/3b is if (a) you're a hard liner PEMDAS addict who wants to intentionally confuse people for some reason, or (b) you understand that 12b/3 is not the same as 12/3b. The fact is that anyone in an academic context will easily understand what 12/3b means without having to apply a children's mnemonic device that was created to dumb down math enough to teach kids the basics.
Kids are also taught that they can't take the square root of a negative number, but we all know that's not actually true. Why is it so hard for some people to understand that the "rules" they teach kids are actually just helpful guidelines to ease lower level learning?
For those who think that 3b where b = 4 isn't the same as 3(4), consider how we typically substitute b with 4 in 3b. Mathematicians at a higher level tend to be lazy and avoid writing unnecessary notation as much as possible. It is simply understood by convention. Some fields aren't as lazy and demand that it be made more explicit, but the general understanding is that 3(4) and 3b where b=4 are the exact same thing.
Only people who cling to PEMDAS as some golden rule even as they enter spaces where PEMDAS isn't even a consideration would keep prattling on about how 1 is never correct. 16 might be correct in lower level education, but it's not useful in the slightest to cling to children's mnemonic devices at higher levels.
Dear anyone who disagrees with him on D.
As someone who studied in advanced mathematics. You are exactly wrong. As for pemdas...I haven't thought about it in years.
It's just built in to our calculators. It's built into the axioms of math. I mean what you are implying is nonsense.
The reason why 12/3(4) = 16 is because it follows naturally from the definition of mathematical operations.
A. Multiplication acts on two adjacent numbers. (key words for adjacent and two)
B. Multiplication is associative.
C. Division is multiplication of the inverse of a number.
D. A number multiplied by its inverse is the identity and every number excluding 0 has an inverse.
etc.
12/3(4) = (12/3)4, by necessity of the axioms of multiplication.
@@insainsinwell that's just not true, calculators themselves are split on what to use for their orders of operations. with some only changing to pemdas over pejmdas due to lower level education requesting it.
It's not a settled issue either way
D is interesting because we're talking about the parse tree and which needs to be evaluated first. If I was writing that in C, if you looked at 12 / 3 * 4, you'd have to know how it'll be parsed and evaluated which isn't obvious at first. I can never remember. In the interests of clarity for my future self and my coworkers, I'd write it as (12 / 3) * 4. I wonder if teaching parsers would help people understand PEDMAS. :)
Now if only people would have that interesting discussion instead of screaming "you're wrong" at each other.
Yeah but it's 3(4) Not 3*4...
At least im Europe 3(4) means (3*(4))
These math debates are so much fun to dive into-each one has its own twists and turns! Personally, I think this video really breaks down the logic behind each debate brilliantly. I’ve been exploring similar tricky concepts, and SolutionInn has been super helpful for clarifying the details.
Isn't 12/3(4) = 12 / (3(4))?
Yes, in countries that learn multiplication by juxtaposition
There is no universal way to interpret x÷y(z), so their are multiple valid solutions. The problem is poorly written
No, 12/3(4) = 12 / 3 * 4 = 16
To get it to equal 1 it would need to be
12/(3(4)) = 12/12 = 1
When you stop using the division sign and use fractions, you are putting brackets around the numerator and denominator. This is not using fractions however, just the division sign so there is no brackets.
@@forbidden-cyrillic-handle Considering new Casio calculators are still being made and sold, all of which using implied multiplication, I'm gonna have to disagree with you on that. It's what my algebra and my geometry teachers in highschool taught me, and it's the calculator I own. I got an A in every math related class, so I certainly wasn't napping through them. It's my favorite subject after all.
If you manage to somehow get to a point where you have 0^0 you honestly don't deserve an answer
That's not true. 0^0 comes up all the time when dealing with power series, and should always be interpreted as 1.
@@ultimatedude5686 I've never seen a coherent argument for why it should be zero, in any context. Most of the time the argument is "BuT zErO!!1" which doesn't really cut it. It's 1.
@@cooperisedyea, “Nothing to the power of nothing equals something” makes so much sense in comparison.. right. The right answer is 0, unless you’d like to prove me wrong in a video, by holding a 0 object, putting it to the power of a 0 object, and somehow creating an object. Go ahead pls. You can’t multiply nothing by nothing and create something lol. If that was the case, then why can’t we solve world hunger by simply creating food from no resources? The only slight chance 0^0 has a chance of being 1 is in some BS philosophical debate where you say crap like “Oh but the existence of nothing is counted as a state of existence, making it 1” which is also wrong
@@ultimatedude5686 But doesn't it require dividing by 0?
As far as I understand:
x³ = x × x × x
x² = x × x
x¹ = x
x⁰ = x/x
x ^(1) = x/x²
Etc.
Wouldn't that mean that 0⁰ =0/0?
@@donedonefree6828 I don't think that's true. Why are you representing x^0 as x/x and x^(-1) as x/x^2? I would represent x^0 as simply 1, and x^(-1) as 1/x.
The issue here is that you're defining x^1 = x as the base case. If we instead said the base case is x^2 = x*x, then by your logic we could say that 0^1 is undefined as it would be represented as x^1 = x*x/x. If we instead define our base case as x^0 = 1, then everything works out.
This is a valuable definition because it comes up in practice. For example, the Taylor series for e^x is written as x^0/0! + x^1/1! + x^2/2! + x^3/3! + ... This converges for all real x, but unless we define 0^0 as 1, it does not work when x=0. This kind of thing happens all the time with power series.
Doesnt the pedantic following of BODMAS/PEMDAS as an exhaustive ruleset imply that the mixed fraction 4½ should be interpeted as representing 2?
I think that 4½ should instead be written as 4 + ½, or 9/2, or 4.5. In fact, I have not seen mixed fractions since elementary school, they are simply a bad way to represent fractions
@@vampire_catgirlthey do that on street markets
@@fucku2b Street markets? I'm not sure what you mean
@@vampire_catgirl just put it in Google images
yes. BODMAS is not math. It's like FACE (in music) or ROYGBIV. It's 1. correct
I literally forgot how to write the divide symbol when tested by my friend a few days ago. This just shows how trash it is.
We simply used the / symbol throughout college. Literally never seen anyone remotely knowledgeable about math use ÷. I vaguely remember seeing it used in kindergarten and elementary school.
No that's not the argument for 1 AT ALL for the D, the argument is that this implicit multiplication can be seen as prioritary over division, replacing putting many parenthesis. ex : 1/2x instead of 1/(2*x), which is much longer and uglier. Is it just a notation and is purely SUBJECTIVE, so no, the answer is not simply 16, it depends on the convention you are using.
Are you saying that it CAN be one?
@@cswsgaming8841 Yes, it's all about notations, which are conventions, which are subjective.
Another silly one is whether dy/dx is a fraction or a notation. =P
Or that e^x is actually e being raised to the power of x
It's a quotient if you know enough math (with differential forms, whatever that means). If not, it's just notation.
Physics gang rise up, derivatives are fractions brrrr...
@@vampire_catgirldude what?
@@Termenz1 e is a function, the exp function, and "raising e to a power" is just putting that number through the function. It looks like how normal powers work, which is why the notation is to raise e to a power, but that only makes sense for real numbers; you can also raise e by complex numbers and matrices
D is bad notations. If a teacher asks such a problem it's on them. Put another set of parenthesis and ambiguity is gone.
There is no ambiguity. Also, the plural of "parenthesis" is "parentheses," so it is a "set of parentheses."
@@robertveith6383 causing confusion is ambiguity, which that problem does. It's a 'bad' question. Parenthesis, parentheses, idc I wasn't bright at grammar.
about D : if you see 2a:3b. That is definitely means (2a):(3b), but not ((2*a):3)*b. So the answer in D is 1.
It's much better to explicitly write (2a)÷(3b) to avoid any ambiguity (as with Wolfram Alpha, Maple, Mathcad or some online, mobile or handheld calculators).
@@cyberagua "It's much better to explicitly write (2a)÷(3b) to avoid any ambiguity" - there ISN'T any ambiguity, as per Maths textbooks.