PEMDAS is a Lie! (Or why it should really be PEMA)
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- Опубликовано: 1 окт 2024
- PEMDAS is the typical way that the order of operations is taught in school (though I guess GEMDAS is gaining some traction: groupings instead of parentheses). The order of operations is important as it's part of the grammar of a number sentence and ensures that everyone will agree on the value of an expression. However, hidden inside the acronym PEMDAS are some "lies" about arithmetic that tend to mess us up. This video is designed to dispel those "lies" and help you understand the reason that PEMDAS exists in the first place. In particular, this video explains why the rule for MD and AS is always left to right and why the rule should really be PEMA.
Partway through the video, I reference the associative property of addition and multiplication and remark that there is a link in the description. Here is a link to a khanacademy lesson on the associative property if you want to learn more about it.
Khan Academy video on the associative property: • Associative law of add...
After you finish watching the video, there are bonus points for anyone who can come up with a great acronym for PEMA and add it in the comments.
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This is so fundamental to the way math is worked out, why the f*** was it not taught? I graduated in 2013, they never cared to teach us why it is a way, only that we got high marks on our test.
I am continuously disappointed in the education that I received.
It's takes 12 years to learn a lie and a life time to rewind and learn it all over again the proper way.
This literally makes sense, and a lot of different things click now.
I'm so glad that this helped clarify some things for you and that you enjoyed the video. I'm sorry about your troubles in your math education. I agree that we should teach things like this a lot more. This is often one of the first things that I share when I teach new students at Utah Valley University and they also have usually not seen it described this way. The only thing that I can say about math education in the US is that a lot of teachers want to see some change too that would move more towards thinking like this, but math education is highly regulated by state and federal governments (for good reason, unbalanced education around the country has its own issues and we've tried some absolutely horrible things in the past too like New Math in the 70s), so it's slow to change, but hopefully starting to move in the right direction.
I hope you find some of other content equally interesting and fun!
@@scholarsauce I made the mistake of going down the rabit hole of differnet calculators giving different answer. After watching more videos on how math legitimetly works, I can see why I always failed math in school. I was stuck in Algebra from 7th grade until my 1st year of college. Thankfully I did a Criminal Justice degree so the math stopped at Algebra, but I still only managed a 71% in that class.
Now that I am getting heavily into programming (video games with Godot 4), the "contextual math" makes sense. I made "contextual math" up, but its math that directly describes 'physcial' objects that need to be manipulated in Vector space (position, scale, rotation(in basis), delta, normalize, etc) or the objects themselves need their quantaties adjsuted.
For my day job I am an Armed Security Guard EMT, so my math skills don't need to be superb, just my hand-eye coordination.
what is really going to shock you, everything before and after the a in pemdas is lie too. it is all addition in the background. subtraction is addition, division is inverse multiplication, multiplication is addition, exponents are multiplication, which in turn is addition, and when you have it all as addition, you don't need any brackets.
I don’t know what you were taught in school, but I learned in primary school that dot operations go before line operations. And if they have the same priority, we go from left to right.
@@Nikioko so you don’t know that subtraction is just adding a negative number? Or the multiplication is repeated addition? Or that division is multiplication, so really just addition? Pretty much all of PEMDAS is just addition at its core.
This concept of PEMA instead of PEMDAS also helps you understand a little better how they optimize computer code and reduce instruction sets for processors. You dont have to do subtraction or division so the processor can optimized for the real functions instead of the shorthand.
@TurtleWaxed & then one learns that computers are horrible at division. Maybe the relevant programmers also suffered from bad maths education & coded for the shorthand instead of the real functions? This would not be noticed by employers suffering from the same thing.
'In confidence going wrong'...
@@Scott-i9v2s 60 years ago computers were "horrible at division". Modern microprocessors have algorithms as complex as trig functions and exponentiation in microcode, let alone division of 128-bit floating-point numbers. Learning "PEMDAS" or what ever rules is completely pointless when dealing with a well-defined computer language that has its own definition of operator precedence and sequencing. You do what the language specifies, or get the wrong answer. And not one language I have ever seen in 60 years of programming indicates multiplication by juxtaposition. Furthermore, modern compilers can optimize the code/expressions you write far better than you can, including between separated lines of code.
Computers are actually really smart these days.
@@BernardGreenberg Point 1: we have the difference between computers back then & computers now. Point 2 is "dealing with a well-defined computer language that has its own definition of operator precedence and sequencing": the phrase "well-defined" has at least 2 meanings: a) internally-consistent; b) consistent with mathematics. Point 3 is the mix of points 1 & 2.
If a calculator program is based on PEMDAS-or-equiv, then the program's design depends on the programmer's interpretation of PEMDAS. The program's operation depends little on whatever the calculator's operating system is (assuming that a dedicated-function calculator's program & its OS are not the same thing).
Furthermore, would you agree that:
2a
is an example of "implied multiplication by juxtaposition" (when '2a' is NOT a hexadecimal number or the name of a variable or such)?
I can indeed not remember as programmer ever writing "2a", but always "2*a".
I’ve never had such a good time watching a math video like I did just now ! You are very entertaining , funny and clever . Thank you for making this video . 😊
Thanks! I'm really glad you enjoyed it! Please tell your friends about our channel!
Why do so many people follow pemdas by the letter? It's supposed to be fundamental, that everything is understood best from left to right. Multiplication and division, addition and subtraction. Notice the and, not the separation of them. Multiplication and division, go hand in hand, and the same applies to addition and subtraction.
Thanks for commenting! I agree that it's built from fundamentals and problematic to take it verbatim like that. Hopefully, this video helps with that some. I'm glad you enjoyed the video and hope that you'll find some of other content interesting too!
Really good vid, but I would like to see multiplication by juxtaposition included as well (PEJMA). For example you see the classic crap running around Facebook 6 ÷ 2(2+2), the "2" leading the parens is tied to the value in the parentheses explicitly, meaning you do the 2(2+2) as an expression before the division kicks in. Calculators are switching back to that method, and higher maths do it as well.
She's far more eloquent than I at covering this... ruclips.net/video/4x-BcYCiKCk/видео.html
Thanks for watching my video and I hope enjoy more of my content. However, I respectfully disagree that juxtaposition has higher priority than regular multiplication. On the contrary, this contradicts the definitions of the operations and frustrates a lot of upper division math. Moreover, computer algebra systems written by mathematicians do not prioritize it for this reason. Check out Maple or Wolfram Alpha, for example. I just had a discussion with another viewer about this topic in the comments of this video and rather than repeating myself, I'll just encourage you to read it and see what you think.
Thanks again for watching my content. I hope you subscribe and check out more of my content!
the Division sign means change the value up to the next multiplication sign to a exponent (inverse) power, so it becomes 6 * 2^(-1) * (2+2), then you can commute the multiplication how ever you want.
@@scholarsauce With respect, check any textbook that teaches basic algebra and you will see equations like this:
2x ÷ xy =?
In none of these textbooks would it be resolved as 2 * x ÷ x * y = 2x ÷ x * y = 2 * y = 2y, which is what it would mean if juxtaposition were not a higher order operation. Instead they resolve it as (2 * x)/(x * y) = 2/y.
Ergo, by demonstrated practice juxtaposition _is_ a higher order operation. Can't really selectively choose when to apply the order of operations, now can we?
(Sidenote: this was a recognised contradiction even back in 1917 before people started using PEMDAS as a mnemonic, citation: "Discussions: Relating to the Order of Operations in Algebra" by N. J. Lennes. A deep dive into the issue was done by the channel The How and Why of Mathematics in their video "The Problem with PEMDAS: Why Calculators Disagree", which is what that commenter linked, and in it she explores different calculator companies' policies and how they were developed - apparently CASIO switched from PEJMDAS to PEMDAS in the 90s and early 2000s because specifically North American teachers rejecting juxtaposition, but have since switched back because it was _only_ North American teachers calling for it not to be a higher order operation. And specifically _only teachers,_ not mathematicians, engineers and the like from North America, who also demonstrably use juxtaposition as a higher order operation in practice - THaWoM has gone into demonstrating that as well through scientific publications and the like in other videos about the issue as well. Your comment at the end of the video about "four operation calculators" is going to steer you wrong when using scientific calculators, because it actually varies by brand not by whether they're basic or scientific calculators. Even Texas Instruments who solely pander to NA teachers' PEMDAS obsession recognises that written questions - their example was 1/2x and whether that's half of x or 1/(2 * x) - do not match their own calculators' order of operations, because written questions use juxtaposition as a higher operation. Current generation CASIOs actually put the implied brackets in after you enter the question so _6 ÷ 2(1 + 2)_ becomes _6 ÷ (2(1+2))_ on the screen and it solves it as 1.
Also, as a programmer, I can tell you no programmer would take the risk of relying on _any_ programmed order of operations to save the day because it is subtly different how they run the operations in every language's compiler even before we factor in people's mathematical ideological leanings. And your appeal to what computer algebra systems written by mathematicians say is... probably inaccurate, given the American Mathematical Society have come out explicitly against what you're saying here, the American Physical Society just put multiplication as a higher tier operation than division while keeping addition and subtraction equal tier, and the American Institute of Physics say never to write 1/3x unless you mean 1/(3x) so... they assume juxtaposition has priority too, and even in that other discussion you cited with that other commenter your example systems didn't exactly work the way you expected and in algebraic contexts used juxtaposition.
In short, what educators believe mathematicians say and what mathematicians actually say... actually two different things. Calculators pander to educators, not mathematicians, but actually most educators across the world don't agree with you. And you don't want to put any toes into the field of computer science in this discussion because, it's just awful even the operators themselves process differently depending on context.)
@@Zahaqiel With respect, I'm speaking as a mathematician, not just a math educator (you can be both by the way). I have a PhD in mathematics, am a tenured professor at a university, and have been published in academic research journals in both mathematics and physics. So, I'm part of the group one is quoting when one invokes "mathematicians say...".
In response to your question about whether algebra textbooks do what you say, I actually pulled one of my beginning algebra textbooks off my shelf and can find no instance of the type of thing that you're claiming that they interpret that way. On the contrary, they avoid inline fractions entirely, which I'm sure is to avoid this very confusion. Sure there are some textbooks that I've seen that are inconsistent in some examples with what they claim is the order of operations, but that doesn't make one side of that argument better or not. I mean why not assume that the inconsistent juxtaposition example was the mistake and the stated order was correct. Or that the author wouldn't have corrected it to be consistent with their stated order if the inconsistency was pointed out. There's mistakes in textbooks all the time.
Besides, to your argument that people use it so it must be right: "demonstrated practice" does not a good mathematical reason make. There is no good mathematical reason to prioritize multiplication denoted by juxtaposition higher than multiplication indicated by any other means. That makes it a memorized rule with no context, which you seemed to be against in other comments you've made here. Why confuse the issue just because people can't be bothered with writing parentheses?
And I've seen the How and Why of Mathematics video that you're talking about and yeah, according to her, the AMS, an organization I've been a member of in the past by the way (now I'm a member of the MAA), apparently used to include it in their mathematics publication style guide. I personally have been unable to find any evidence to support that claim and their style guide now certainly makes no such claim. I suspect that if it once did, the reason why it doesn't now is because too many mathematicians cried foul about it. Granted that's speculation on my part, but it's based on my experience interacting with a ton of mathematicians. Regardless, the fact that the AMS no longer claims this, if they ever actually did, is significant too.
I do appreciate how much you like the How and Why of Mathematics' video and I think she is pretty thoughtful about most things too and like her channel. Ironically though, I had a similar reaction to her ability to claim what "mathematicians think" that you had to my claim. In the description of her channel, she claims to have graduated in physics (which I take to mean that she has a bachelors degree in physics, but she doesn't specify) and only took math in her first two years of college, which to be honest means that she didn't get very far into the undergraduate math curriculum and is probably unaware of a ton of advanced mathematics (and indeed her channel only seems to concern itself with ideas in math up to calculus, which isn't a bad thing, but there's a lot more to math than just that). So my reaction to her was similar to yours of me: how does she even know what mathematicians think about this when she isn't part of the mathematician community? On the other hand, I actually am part of the mathematician community, do research in math myself, and interact with other mathematics researchers all the time, some of whom are quite well known. None of the mathematicians that I've posed this question to have disagreed with me. Most agree that one should never write such an expression as 1/ab without parentheses, but that if one did it would mean (1/a)b not 1/(ab). Of course, that's not to say that there aren't any that agree with you or even that I've asked a random representative sample, I just haven't found one that agreed with you about juxtaposition yet. And I can really only share my experience and what I've learned after decades of studying and researching mathematics. And as a mathematician, am I not at least as qualified to claim what "mathematicians say" as a programmer or a person with an undergraduate degree in physics?
Anyway, you're welcome to continue to disagree with me, I don't really mind. I love the passion that people like you have for issues like this and I think that that's more important than who's right or who's wrong. Thanks for watching my video and taking the time to comment. I hope that you'll enjoy some of my other content. Take care!
@@scholarsauce The thing is, the data seems to suggest it's a regional thing. In fact, in Europe in the 80s, multiplication apparently was given precedence over division as their standard.
Unfortunately, that's not how the PEMDAS extremists take it (generally arguing that anyone who disagrees with them is "bad at math" - Mind Your Decisions' channel is rife with that kind of commenter).
And while there's no mathematical reason to assume juxtaposition to have precedence, there's also no reason to assume it doesn't. It's _all_ a notation convention, for which there has never been a global community consensus established. The one thing I will point out is that consistently polling indicates strict PEMDAS-adherence position (no juxtaposition, M/D as the same step read left-to-right, A/S as the same step read left-to-right) is the minority one, again which often gets taken by those people as meaning that most people are "wrong", rather than what a data scientist would take it as which is... that's interesting, wonder if there's data to explain why that is? Which, one could argue, is very similar to being told that whole textbooks consistently do a thing, and responding with brushing it off as probably just individual errors.
With regard to you being unable to verify THaWoM's claims about the AMS style guide... did you check her references? She provided source links on all of her claims, including the Internet Archive's copy of their Guide for Reviewers from Aug 15, 2000, which includes the full text and original URL.
It says:
"Formulas. You can help us to reduce production and printing costs by avoiding excessive or unnecessary quotation of complicated formulas. We linearize simple formulas, using the rule that multiplication indicated by juxtaposition is carried out before division."
And if you aren't aware of any mathematicians that say different, how's about Howard Ludwig, David Halliday or Nathan Hannan, who have commented online/support the comment online specifically that professional mathematicians and physicists don't follow PEMDAS and instead that stacked exponentiation and juxtaposed implicit multiplication are additional factors that PEMDAS does not adequately account for and are used by professionals in ways that are contradictory to PEMDAS? In fact, Howard's comment on the subject may go some way towards explaining why the AMS doesn't use the above quotation anymore - because the standard policy of publishers these days is to avoid stating equations in ways that have any ambiguities in them at all, rendering it (and PEMDAS) unnecessary.
All of the above, is very easy to find stuff _if you actually want to look for it._ It took me less than five minutes to track just those three down as expert attestations against what you're saying. I was literally trained on textbooks you've brushed off as not existing or being errored. It took me no effort to find attestations from different countries about completely different educational practices regarding orders of operations. So it kinda seems like you've leaned into confirmation bias and not bothered to look further.
Subtraction is an addition of a negative number, and division is a multiplication of an inverse value. This said, it becomes quite obvious why those have the same priority in the order of operations. They also share commutative, but no associative properties.
In some sense the associative property and communities property are the same thing? Negative is to addition what the “inverse value” is to multiplication. I wonder if there is some connection between negatives and “inverse value”.And you know the numbers 0 and 1 are kinda similar too… if I take any number and add it’s negative I always get 0. If I take any number and multiply it’s inverse vale I always get 1.
Pemdas says multiplication AND division, then addition AND subtraction. Addition and subtraction are same order operations and therefore occur in order from left to right. The a and the s are in that order in the mnemonic only because it rolls off the tongue better than writing it as pemdsa. The same issue goes for the m and the d. Pemdas rolls off the tongue better than pedmas, pemdsa, or pedmsa although some do occasionally use pedmas as an alternative to pemdas. Where people err with pemdas is forgetting the and between multiplication and division, and between addition and subtraction. The ands indicate same level operations. Same level operations are read from left to right in the order in which they are written.
Yep, I learned it as PE(MD)(AS), then later learned to remove division and subtraction to remove one layer as that will make the multiplication layer and addition layer both commutative within the layer itself.
So BIMA/PEMA are at least easier to SAY than BODMAS/PEMDAS...
@@Scott-i9v2s you have to remember that these mnemonics are taught in schools prior to learning how to write things to make it commutative within the layers. It's a beginning point, not the end point for the understanding of how the operators interact with each other.
@@ianbelletti6241 OMG! Even worse than I thought that it was! I am SO GLAD that no school that I went to had such RoTs in their curriculum!
The RoT 'PEMDAS' literally says' multiplication BEFORE division, then addition BEFORE subtraction'. Just as it says 'parentheses 'BEFORE exponents' & 'exponents BEFORE multiplication', &c. NOWHERE in it is ANY mention made of the word 'AND'.
Implying that an 'AND' is meant there means that the RoT was 1st explained properly. Which means that said explanation MUST have included that division is multiplication-by-its-inverse & that subtraction is addition-of-its-negative. Which ALSO means that any teacher who actually USES the brain should have KNOWN that the RoT's name made no sense as-formulated. And therefore would refuse to teach such nonsense to students.
Using any RoT as a BEGINNING: no wonder that maths-illiterate teachers are helping further dumb down the students! IQ = 80 & falling...
@@Scott-i9v2s your teacher was bad if they taught you that. It's multiplication AND division (same level), and addition AND subtraction (same level). I was taught it with the word and in the right places when speaking the full meaning of PEMDAS. It's easier to say and remember PEMDAS as opposed to PEMaDAaS.
By using a Slide Rule you can divide a number by multiplying it by the inverse of the divisor. The Inverse Scale on a slide rule (CI) is used for this. You can do a chain of multiplications and divisions this way on a slide rule. As clunky as a slide rule is, doing operations on it can clarify a lot of things in mathematics.
That is really cool! Thanks for sharing! Now I want a slide rule even more.
that is basically how division really works, 15/5 is really 15 multiples of 1/5 so 1/5+1/5+1/5... division is addition, much like subtraction is addition, and multiplication is addition.
Note that one in the same way can do subtractions with a 2nd ruler placed inverted against a 1st ruler. Thus doing ADDITIONS of negative quantities.
The slide rule uses logarithmic scales. You multiply two numbers by adding their logarithms. Or divide by subtracting the divisor‘s logarithm from the dividend‘s.
@@Nikioko Yes, log scales are used for multiplication(/division). But linear scales can be used for addition(/subtraction). My mind shuts down however when considering what type of scales to use for exponentiation... A log-of-log scale (ie [log]^2), & for tetration a log-of-log-of-log scale (ie [log]^3)?
And then you have implied multiplication aka juxtaposed multiplication. And then you have style guides that say one thing or another depending on the publication and may treat / and ÷ differently.
Btw when I was taught math in Sweden they never mentioned PEMDAS or any similar short hand 🤔
Thanks for commenting. Yes it is all kind of confusing out there with lots of established rules. Personally, I think we should use the rule that is derivable from all the definitions of the operations. That's what I presented here. To me it's what is the most natural way to interpret an arithmetic expression that doesn't rely on a memorized rule that doesn't have a mathematical reason behind it. It's what you would come to if you broke it down into its deepest constituent parts which is how mathematicians tend to handle everything in mathematics.
Thanks again for watching my video. I hope you find some of my other content interesting too.
PEMDAS
1. Parentheses (inner to outer)
2. Exponents
3. M/D - whichever comes first from left to right ( just like you're reading a book)
Note: Do all Multiplication and Division before you Add and Subtract
4. A/S - whichever comes first from left to right (just like you're reading a book)
I was taught PEMDAS. If PEMDAS, which is a Mnemonic (Memory Aid) for the order of operations, was wrong, that would mean every teacher who taught PEMDAS pre-internet (Google) would be wrong for teaching it. Google didn't come out until September of 1998. There are places in France that use PEMDAS.
If PEMDAS didn't work, I guess you're saying the formula for converting Fahrenheit to Celsius is wrong?
C = 5/9(F-32)
No one needs mnemonic devices to learn how the Order of Operations functions.
I was actually taught it as PEMA or Please Excuse My Attitude.
we know that x - y is the same as -y + x so there is addition from subtraction.
we also know that a negative exponent means invert the number so x divided by y is the same as (y ^ -1) * x. That turns division into multiplication.
Those two substitutions removes 99% of the confusion with PEMDAS. The tricky part is if the problem is grammatically correct and understanding you can have -x + -y also if you already have x divided by y with a negative exponent assigned to y. So 4 divided by (3 ^ -2) would really be 4 * (3²).
The obelus sign should be banned from being teached in school as it is already banned from the standard mathematical norm. I never heard of pedmas but it is super confusing; I just know multiplication takes precedent over addition and that's all I need to know; I use parentheses and fraction to express a division (which are self explanatory). The trick "math problem" that show up periodically on social networks with an implicit multiplication is a good example of why obelus shouldn't be used; and why pedmas even understood as it should pe(dm)(as) is not enough to understand conventional modern maths; as the expression A / B(C+D) where / is an obelus is understood differently wether you're a PEDMAS rigorist or someone who uses complex maths in their daily life (in the first case you will get (A/B)(C+D) , in the second A / (B(C+D)) ; where both camp will be adament they have it right. And remember that convention are conventions; they are contextual; not a divine mathematical truth bestowed upon us; and in this case it appears very clearly.
I'm with you for the most part and certainly agree that we shouldn't just rely on a rule. I only take issue with your comment that A/B(C+D) is understood as A/(B(C+D)) by people who do complex math in their daily lives. As a math professor, who I think would qualify as doing complex maths daily, I do not interpret it that way. In fact, I haven't found another mathematician that really does either. Some point out the inherent ambiguity, but that's the closest I've seen. It seems to be somewhat subject specific. Physicists and engineers seem to interpret it the way you said, but most mathematicians I think would argue that the expression itself is not great and that parentheses should be used for clarification. But if pressed to pick an interpretation, I think the vast majority of mathematicians would pick the interpretation (A/B)(C+D). But it could be regional too, I haven't done a study or anything. It's just the sense I get from talking about this with other mathematicians.
Thanks for the thoughtful comment. This issue seems to garner far more attention than I expected. It makes me want to make a few more videos delving deeper into the idea. Anyway, I hope you enjoy my other content too, which I personally think is more interesting than arithmetic quirks. Thanks again!
@valdir7426 The problem with the banning (ditto with ANY change) is the time that needed to be processed by the system. Update the schooling needed by future teachers. Then wait years for them to graduate.
Temporary fixes of the current batch of teachers also takes time.
&c,.. &c... &c...
STILL, that is NO excuse to NOT re-educate the currently teaching teachers ASAP.
@@Scott-i9v2s This is definitely a huge problem to any change in the education system. We have been facing this very problem as we've been discussing changes in our state's curriculum.
@scholarsauce I constantly get into trouble with inline written maths-statements if I process 1 term & then FORGET to immediately RESTART scanning the WHOLE statement for the next appropriate term to process.
So just teaching/remembering 'scan left-to-right' is NOT sufficient. The WHEN to do that part of the convention is MUCH more important.
WHY is this so? Because the EXPRESSION of the statement has changed.
Even then one can still (with full confidence) go wrong, depending on how-blindly one adheres to some convention.
Anyway, I have just proved (once again) to my own satisfaction that knowledge of the MEANING of things leads to more-appropriate results than blind adherence to rote.
PLUS that (some) current conventions CAUSE more problems than they resolve.
Note that this is the case with both inline statements AND formal maths formatting.
In Australia children are/were taught BOMDAS Brackets order multiplication division addition subtraction, PEMDAS seems kind of foreign.
I was taught math BC - before calculators. The emphasis was on simplification, with the goal of reducing all the elements of an expression to common terms.
A lot simpler, easier, and more reliable than PEMDAS.
dgkcpa1 Reliability based on consistency. The RoT 'PEMDAS' is NOT consistent.
With simplification being the major goal of maths, removing from it the not-sense-making shortcuts (like subtraction, division, reciprocals, &c) makes sense to me. In this case that would be at least 3 fewer rabbit holes to stumble into.
> all arithmetic operations are binary operations
except for the negation, square root, factorial, and maybe some other unary operations
BTW, 3 - 6 x 3 + 8 ÷ 4 becomes 3 + -(6 x 3) + 8 x (1/4), but that obviously doesn't change the result
Yep, this helps also with the 'ambiguous' case you see on the internet such as 6 ÷ 2(2+2); which converts to 6 * 2^(-1) * (2+2) and now you can move the factors and distribute without harm.
@@ingiford175 it doesn't when there's the `ab => (a*b)` rule
Not ONE math teacher in all my grades, Not ONE, ever explained PEMDAS to the point where there were 'left to right' rules with MultiplicationDivision & AdditionSubtraction which means PEMDAS can also be PEMDSA, or PEDMAS, MEDMSA. It wasn't until after Al Gore invented the internet did I 'discover' my math teachers were incompetent.
I'm really glad that you enjoyed the video and that it helped. I'm sorry that your experience with math teachers was, shall we say, a bit wobbly. I'll admit that there are a lot of places in modern math education that frequently have similar issues. If there's anything else that ever totally confused you or made things overly complicated or even not really work, let me know and maybe we can make a video about it too!
Al Gore? Whodat?
SMH. Why misrepresent PEMDAS?
Nobody teaches it as multiplication before division or addiction before subtraction. It’s taught as having the steps “parenthesis”, then “exponents”, then “multiplication and division”, then “addition and subtraction”.
You are creating rather than clearing confusion.
On the contrary, I have run into students and teachers even this past week or two that were taught that. In the comments here, there's even someone who posted a screenshot to a textbook that taught that. So I think it's very relevant.
However, my point with this video was less about that and more about the reason behind the left to right rule. Without this explanation, that's just a memorized rule and convention, which it isn't. Moreover, this makes a connection to more advanced algebraic settings where we talk about algebraic structures like groups and rings and division and subtraction are precisely treated like this. As such, I think this is a very useful take and the students who I explain it to in person find it very useful and enlightening.
Thanks for watching my video and for your comment. I appreciate the passion that so many people have for this topic.
People are making this more complicated than it.
"Red riding hood nightmare" ROFL...
Yeah, this was a fun one to make!
1) left to right
2) Parentheses and brackets
2b) the fraction bar acts like a bracket
3) Exponents
4) implied multiplication like 2x or 2(x)
5) multiplication and division
6) add and subtract
PEMDAS is wrong:
a) the fraction bar is forgotten
b) Implied multiplication is not taken into account
c) Multiplication is not superior to division
d) Adding is not superior to Subtract
I appreciate the comment, but neither 2b nor 4 are part of the order of operations. There is no good mathematical reason for those. The order of operations is derivable from the definitions of the operations. Those are just some random rules someone arbitrarily added without thinking. They aren't taught nor have I found a single mathematician that thinks they should work that way.
But it does show that we should always be more clear. I address c and d in the video and give a good mathematical reason why which also explains 1.
Nonetheless, you're free to think what you want, but 2b and 4 are definitely nonstandard among mathematicians.
This is impractical is so many ways.
Love to use this for a weekly payroll and have a massive line of employees pissed that their paychecks were short. Do a inventory with scattered items and this wouldn't give you the real amount of items.
Your first example is 2+3×4
Day 1
2 hrs worked
Day 2
3 hrs worked
= 5 hrs x 4 dollars per hour = 20 dollars to be paid but your way would pay 14, oops
Given your example would simply require parentheses around the 2+3. As I said in the video, parentheses are a way to artificially prioritize a calculation. There's nothing wrong or impractical with the order of operations, you have just misinterpreted how to correctly represent your practical example using arithmetic. The correct way would have been (2+3)×4 which correctly counts it as 20 as you pointed out. 2+3×4 simply does not represent the scenario you described.
BTW, somebody here stated what comes down to the 1st number in a multiplication ALWAYS expressing the quantity of times that the 2nd number should be counted. While implying that switching the roles of the 2 numbers is NOT valid.
The commutative aspect of multiplication that emerges from deductive application of the underlying maths makes said assertion senseless. Elevating said assertion to a convention would be a disaster.
The Commutative Property allows you to move multiplication around ANYWHERE within a TERM as long as you do not affect the denominator of a division operation...
When dealing with inline infix notation only the numeral to the right of the obelus is in the denominator unless WITHIN a grouping symbol... The only exception to this is the Algebraic Convention given to coefficients and variables that are directly prefixed with no delimiter and forms a composite quantity by this convention...
Parenthetical Implicit Multiplication and Coefficient/Variable composite quantities are not mathematically the same...
Rings only have two binary operations. 👍
Gracias, perfecta explicación. Yo tenía algo de idea que venía por el lado de que en una división de por ejemplo en este caso 5:10x2 es igual a decir (5/1):(10/1)x(2/1). Supongamos que la barra estoy dibujando la fracción. Pero obviamente la explicación correcta es la que está en este video. Ya que esas operaciones como división, raíz, resta son operaciones inversas a las otras.
¡Gracias por el comentario! ¡Me alegro que hayas disfrutado mi explicación!
I agree that a,4 letter acronym explained correctly works be better than a 6 letter acronym.. However it's disingenuous to call PEMDAS a lie...
PEMDAS represents 4 levels not 6 exact steps. There isn't a math book on the planet that states PEMDAS represents 6 exact steps and failure to understand that fact doesn't make expressions ambiguous, it makes the individual who lacks the correct knowledge, clueless....
With all that ranting you should have also explained the Commutative Property... 10-7+2= 12-7 & 10÷2×5 = 50÷2
The funny part about mixed Multiplication with Division is that if you evaluate the Division first you always get the correct answer 10÷2×5= 5×5= 25 5×10÷2= 5×5= 25
Yes, Multiplication and Division share equal priority and can be evaluated equally from left to right as they are inverse operations by the reciprocal.
Yes, Addition and Subtraction can be evaluated equally from left to right as they are inverese operations. Subtraction is just addition of a negative number.
But to restate PEMDAS is not a lie just because someone fails to understand the acronym and its use correctly. And failure to understand the basic rules and principles of math correctly as intended doesn't make an expression ambiguous.....
I see what you're saying, but I have actually taught several students who prioritize multiplication over division regardless. And in one of the many conversations on this page, if I remember right, there is a link to a textbook that explicitly states as much. Another famous link is to the American Physical Society, I think it was called, I have it written down somewhere and their journal style guide says that it uses the convention that all multiplication is evaluated before division. And don't even get me started on the idea that multiplication denoted with juxtaposition has higher priority than multiplication denoted another way.
I also find it troublesome how students will misunderstand how things commute when there is a division or subtraction sign, that the sign has to move with the number it's glued to.
So while yes the PEMDAS is a lie line is a bit of clickbait, it is true that the acronym is misleading on its own. Hence why I think PEMA is a better one.
I appreciate the comment! I hope you'll check out some of my other stuff too and subscribe!
@@scholarsauce understandable... I prefer GEPS ....
Grouping symbols (inside only)
Exponents like powers and roots
Products and the inverse
Sums and the inverse
But the idea is the same....
There were a few poorly written math books prior to the 1940's that did advocate all multiplication before division. Complete ignorance...
As for the APS... STYLE GUIDE says it all. It's a statement to the reader of a change to the STANDARD Order of Operations that the Journal decided to embrace. They have suggested it was for formatting and printing purposes... But a good enough reason if you ask me... Besides the APS I doubt deals with grade school arithmetic expressions....
If you want to take this a step further, really it could be GA - Grouping and Addition. Your calculator only knows signed addition. The grouping, () [] or {}, just lets us know the problem is solved in an order other than purely left to right. This means Exponents and Multiplication are short hand for how many times a specific addition should be repeated. For example, 5^3 = 5*5*5 = ((5*5)*5) ((5+5+5+5+5)*5) = ((5+5+5+5+5)+(5+5+5+5+5)+(5+5+5+5+5)+(5+5+5+5+5)+(5+5+5+5+5)) = ((25)+(25)+(25)+(25)+(25)) = 125
Yes. If you resolved everything down into addition, you totally could. In fact computers probably do something like this.
The only reason why I would not want to do this pedagogically is that some of the things that I mentioned in the video still hold in places where you have an addition and multiplication that don't have that relationship, such as with vectors or matrices. And teaching it as PEMA builds intuition in these other places.
But you do have a great point.
@AxGryndr @scholarsauce Actually--& taking into account what scholarsauce replied--one could make the mnemonic GC (Grouping & Counting). Which ought to reduce to solely G. But by that point the students should know enough that they need no mnemonic/RoT, & certainly not as a base to teach/learn from.
Umm... methinks now that such a mnemonic is not to help STUDENTS, but to help TEACHERS stay on-target. So that TEACHERS can reduce the amount of time/effort needed for the subject.
Another probably significant factor is that thing called 'social studies' present in elementary schools in mainly the USA. Where a SINGLE teacher is required to teach MANY DIFFERENT subjects.
This seems to go beyond the scope of maths, but if it is part of the root-cause of maths problems, then it IS relevant---even here.
I have been seeing some very unclear videos lately of why PEMDAS is wrong, it seems to be a movement. They all cheated in their explanations of why it's wrong to prove their point. This was a valid and clear converstion and the best video about the problem I have seen. Thank you.
Thanks! I'm really glad you enjoyed it! I hope you find some of my other content equally interesting.
we were taught as BEDMAS, and were told that multiply and divide are equal and add and subtract are equal and when they are equal it is left to right.
BEDMAS is the British version, right? (Brackets, exponents, ...)
@peterodonnell4404 @scholarsauce Thanks, Peter O'Donnell, for the link to your article "Beyond PEMDAS: making sense of the arithmetic order of operations". I shall 1st reread it a few times to understand its full import.
Clear is already that consistency in maths-conventions is DARNED-important. (But then I was already convinced on that point.) So important that I think that revamping of the ENTIRE conventions-set is worthwhile. I shudder at the magnitude of such a project. But a complete restart from scratch seems to me MUCH BETTER in the long run than fixing the current state bit-by-bit, the-latter being way too prone to introducing even-more ambiguities.
(IBM had back-then PTF-tapes, officially named 'Program temporary fix'. We-users understood it to mean PERMANENT temporary fix; an in practice more-appropriate oxymoron when following the chain of fix-on-fix-of-fix...)
I could just as easily say addition is not a real operation as you said subtraction is not a real operation, since addition is just subtracting a negative number. Same with multiplication; it's just dividing by a number's reciprocal. The benefit to this is that if we start with subtraction and division, we can define negative numbers and fractions just by subtracting and dividing positive numbers. If we start with addition and multiplication, then we still have to define subtraction and division before we can define negatives and fractions in terms of positive numbers.
Sure, I guess you could if you wanted. Basically, it still comes down to just two operations. Subtraction and division are probably avoided as being "the operation" since they don't have as nice of properties; for example, they aren't commutative. That's why I say that they are the shorthand. Also in more advanced settings like in group and ring theory, we only talk about addition and multiplication. Subtraction and division are just forms of those operations. So that way it's more consistent with how we describe this structure in more advanced settings.
Thanks for the comment! I hope you enjoy more of my content!
The 'Why is 0! = 1?' video by Eddie Woo of 9 years ago seems to also have issues based on misunderstandings of conventions.
Or... maybe teachers are sending out signals about conventions-issues--& therefore also issues with curriculum? If so, then Peter O'Donnell's article 'Beyond PEMDAS: making sense of the arithmetic order of operations' would be way too limited in scope.
A SERIES of papers/articles/videos on the problems that ambiguous conventions cause could/should/would be useful (in getting [the quiche-eating 🙂] ) mathematicians to speak out against disastrous curriculums/curricula(e)?
That's a great idea. There's a whole bunch!
@@scholarsauce I am trying to imagine what a maths curriculum might be, were the teaching of all the stuff related to the 3 big shortcuts (subtraction, division, reciprocals) dropped. But more important is of course how mathematicians would view such a no-shortcuts field. More-precisely, a no-USELESS/AMBIGUOUS-shortcuts field.
At 2:40 You did not literally meant "literally". Going shopping for grandma and returning with 20 apples instead of the 14 she wanted is far less consequential than eating her.
At 3:35 Either the expression is ambiguous or it has a correct answer -- 4:08 -- You literally can't have both :o
It's true that 7 - 3 is the same as 7 + (-3), but unfortunately everyone who has a brain and not a CPU works out 7 + (-3) by changing it to 7 - 3 and then doing the _subtraction_ (did I just use a dirty word?) to get 4. Same goes for division. Nobody would evaluate 12 / 3 * 2 by changing it into 12 * (0.333333...) * 2 -- at least I hope not. Nope, 12 divided by 3 is 4 and we multiply that by 2.
Doesn't anybody teach mental arithmetic any more?
How about if I whispered to you that subtraction and division are very real, but fractions are not -- they are just incomplete divisions that haven't been resolved yet?
Finally if you think PEMA along with negation and reciprocation solves all the issues, how would it handle the stacked exponentiation 2^3^3? Just askin'.
At 2:40 I said that grammar in math is literally more important than grammar in English. And I meant it literally. So I used it correctly.
At 3:35 without the order of operations, it would be ambiguous. That's the point of the order of operations to make it not ambiguous, which is the point that I was making. With the order of operations, it doesn't have a correct answer.
As for the subtraction and division being shorthand, that's actually exactly how we use it higher mathematics. And contrary to your point, it actually helps with mental math because it gives meaning to the operations. Adding a negative number moves to the left on the number line as opposed to adding a positive one. And the reciprocal thing makes a lot of algebra considerably more clear. So I disagree that it causes problems with mental math. And, honestly, anyone who tries to do arithmetic with decimals is just asking for trouble. Truthfully the reciprocal thing is just the idea that division and fractions are the same thing, so it's really not as bizarre as you're making it out to seem. I teach this to students all the time and it practically invariably improves their capabilities with it and makes a ton of algebra things way more clear.
Finally, as far as the stacked exponentiation, the expression you gave using carets for the exponents is ambiguous. Does it mean 8^3 or 2^27. If you're going to write it with infix notation, then yeah, you need some parentheses to make it clear. However practically no one writes exponents using carets. It's always with superscripts. Superscripts are an implied parentheses (just like the numerator and denominator of a vertically aligned fraction). Which makes it entirely what is meant. If the second 3 is a supersuperscript on the first superscripted 3, then its 2^27. If it isn't a supersuperscript on the first superscripted 3, then it would have to be written as a superscript on a parenthetical (2^3) and again, it's totally clear. It would be silly to supersuperscript it to the first superscripted 3 but mean 8^3.
Thanks for watching the video and commenting. I hope this answers your questions about it.
@@scholarsauce Grammar in math is no more important than grammar in English or in many other subjects. The devil is in the detail. You may think that instructing people to eat your grandma is less important than buying the wrong number of items for her. The rest of us don't. So, no, you didn't use it correctly.
I hope you re-watch your own video since you went straight from telling us baldly that the expression 2 + 3 x 4 is ambiguous to explaining that it isn't, because multiplication has precedence over division and therefore there is a correct answer. If you want to make the point that with the order of operations it does have a correct answer, then you're better off not starting by bluntly telling us that it is ambiguous.
Nobody, from higher mathematics to junior school, ever works out 7 - 3 by turning it into 7 + (-3); everybody simply learns by rote that 7 - 3 is 4. Even infants learning simple maths can count 7 apples and take 3 of them away to leave 4 apples. They don't visualise 3 negative apples that annihilate 3 of the 7 apples leaving 4. That's the meaning of the operation of subtraction in its simplest form, removal of quantities, not addition of negative quantities. If we put heat into a system, its temperature goes up. To make the temperature go down (even to negative values), we take heat out of the system. We don't add "negative heat" to the system, and it's not helpful to think like that.
The reciprocal thing is useful in simplifying division by fractions, and very little else. If I ask someone to work out 12 divided by 3, they either know it's 4 (a lookup table of learned responses), or they count how many times they can remove 3 from 12 and work out it is 4 times. Not a single person on Earth takes the reciprocal of 3 (=0.33333...) and multiplies it by 12 to give 3.999999.. which they then interpret correctly as 4. Well, maybe you do, so please feel free to enlighten me on that.😮
I appreciate your experience in teaching maths and I'm happy to hear that you get good results with your students, but the plural of anecdote is not data. I was teaching mathematics before you were born, and my experience differs from yours.
Unfortunately not every platform enables the use of superscripts, and this is one of them. So you kind of have to get used to the limitations of typing inline. I'm glad you appreciate that it would be silly to interpret the stacked exponentiation 2^3^3 as 8^3, because that would surely be redundant to writing 23*3 since 2^9 has the same value. It should only be sensible to interpret 2^3^3 (particularly when described as "stacked") as 2^27, since that value has no alternative formulation. Also, as you pointed out, we always evaluate a "tower" of exponents from the top, which of course turns out to be right-to-left. But there's nothing sacred about the direction of operations, and I really hope we can agree on that.
Anyway, thanks for taking the time to reply to my (mainly rhetorical) questions. I hope you will pardon what I am in the habit of using as a pedagogical device, and I don't expect you to agree with me. I do appreciate you listening, though.
I have dyscalculia and have never been able to understand mathematics, but I was actually able to (just about) understand this. The way you explained it really makes a lot of sense and I'm hopeful with practice my math skills will improve. Thank you!
I know it must be really difficult to deal with dyscalculia in learning math, and so I'm really glad that this video helped some. It's good to hear your willingness to be up to the challenge. I hope that it works out for you! Good luck!
Hi, I'm curious on dyscalculia. I got once a doctor to say that I had dyslexia. But I have never had issues with reading or understanding text, but I've always had trouble with spelling. So I assume there are grades to all this.
How does dyscalculia work? What are the challenges, is it the mathematical concepts that are the trouble or is it with interpreting the written formulas?
Because if it's the mathematical concepts then I would argue to some extent (remembering the grades of dyslexia) that you need a better teacher who can bridge the gap. But if it's with interpreting the formula then it's something else.
@@dbtest117 Is the Wikipedia page on dyscalculia of any use?
@@Scott-i9v2sProbably but that is not the same as getting the information from someone experiencing it.
The mistake most make is they think that it's only about what is inside the parentheses. They continue to take precedence until they're GONE.
Once you resolve what's inside, the parentheses are gone. Parentheses are only intended to prioritize the computation inside them not any operation outside them. Maybe I misunderstood you, but parentheses are only about what's inside them. That's why you use them. They indicate that that is its own quantity or number, nothing else.
3-5+4 =2. You don't even know how to add and you want to teach us something
You apparently didn't finish watching or understood the point of the video. The video explains why this is ambiguous if you follow the order of operations as stated (PEMDAS) and do addition before subtraction. It is offering an explanation of why addition and subtraction are on the same level and should be done from left to right. But also why the rule is left to right and how that is not arbitrarily chosen but induced by the meaning of each number. Try finishing the video and you'll see what I'm talking about.
@scholarsauce I think that you said that you are an educator. If that is correct, then my compliments for being for me the 1st (USA-native?) educator to make sense of the whole PEMDAS issue. (Which, btw, would for me be BIMA in ENGLISH-English, ie Brackets, Indices, Multiplications, Additions), since 'bracket' is the name of the set of which the curved type (AKA parentheses) is just 1 member.
But aside from the bracket-niggle, you explained the issues of distributivity & associativity in a very clear manner. So we finally see a USEFUL form of the RoT (Rule of Thumb), namely PEMA/BIMA. AND we can now trash PEMDAS & its as-ambiguous sisters.
Your explanation of WHY the words 'division' & 'subtraction' are shorthand should make things clear to even those who hate maths, since said hate is probably caused by people who should NOT be in the field of education. (Probably Mrs Grundy types who think that using brackets/parentheses is cheating?)
Again, my sincere compliments!
Thank you a ton! I am an educator and comments like these really warm my heart. I am very happy that this explanation made sense too and that you liked the video. I noticed that you commented on another of my videos too. I'm glad that you're enjoying my content.
Feel free to share my content with others that you know that might enjoy it as well. Thanks again for the kind words!
@@scholarsauce The "problem" with saying "makes sense" is that the sense-making is a PERSONAL thing, with the sort-of assumption that it THEREFORE ought to make sense to everybody...🧐😕
@@scholarsauce Maths makes much more sense to me than human languages. The main reason being that maths is a matter of DEduction from axioms, in contrast to human languages (where everything is highly opinionated habituation-based INduction).
I am working on designing a joke where a Flat-Earther, a theist, & a linguist meet in a bar. I am now sure that a mathematician will NEVER enter that bar when those 3 are present--which could be the punch-line of another joke involving said 3 persons. I might make yet-another with Achmed-the-terrorist's reaction to said 3-some.😎
Assistance in the writing of them will be appreciated!
Really wish all the jackanapes like the editors of a few journals would stop putting in their appendices a certain way to interpret a/bc and just BAN IT ALL TOGETHER. Basically they should just write "use a fraction bar or parentheses or we're not publishing your paper."
india : bodmas
Sir you make lot of sense. Excellent job
Thanks! I'm glad you enjoyed the video!
9:11 yes
Great explanations and a funny approach, thanks! What I find disturbing is, that under other videos to this topic there are examples discussed as this one:
6/2(2+1)=
Resulting in either 1 or 9, depending of whether you accept "2(" as a valid grouping because, how pro guys argument, "it is that way in algebra". But, is it really?
If I e.g. write x+2(3x+4), "2(" is not a grouping in itself because the multiplication just is at higher priority as the addition. And if I have something like x/2(3x+4) I would simply write the term "2(3x+4)" as it is in the denominator, right? So in my eyes, above example has 9 as its correct result.
Prove me wrong 😉
I agree with you that 6/2(2+1) should be interpreted as (6/2)(2+1) and that the answer is 9. This is because both the division marked by the / and the multiplication indicated by juxtaposition have the same priority for the reasons I cited in my video.
Some groups artificially make juxtaposition multiplication of higher priority, but I do not see a good mathematical reason for it. Everything else about the order of operations is mathematical, but that rule is just arbitrary. That's why there's confusion though.
But as for me and every mathematician that I've posed this question to so far, we all agree with you that it should be 9.
By the way, I'm really glad that you liked the video and I hope you enjoy our other content and please subscribe if you haven't already. Thanks!
@@scholarsauce Thank you for pointing out, that this "juxtaposition rule" is of arbitrary reason. Yes, there are "habits" around the mathematical and engineer world, and (slightly different) when it comes to computer science (guys here are more into placing brackets, just from programming background). But: we should have a set of rules which is under no circumstances to be "interpreted" differently just because of anybodies background. Just my 2 cents.
Oh...and thanks for pointing the subscription thingy out again...I totally forgot that but of course, Id did now :)
It would help when someone doing a video on PEMDAS would actually know how PEMDAS works!
Here is the order of operations:
1) Solve everything within Parentheses first
2) Solve for Exponents
3) Solve Multiplication AND Division at the same time as they appear from left to right
4) Solve Addition AND Subtraction at the same time as they appear from left to right
3-5+4 will always be solved by doing the subtraction first, then the addition! The only time that you would do the addition in this problem first is if Parentheses were used as such 3-(5+4)!
Did you watch the entire video? I was motivating exactly what you're saying by explaining why it has to be from left to right for both addition and multiplication.
@@scholarsauce Yes I did... Still you approached it as Multiplication has precedence over Division and Addition has precedence over Subtraction. That isn't true. Multiplication and Division have the same precedence, that is why they are done in order as they appear left to right. Same as Addition and Subtraction. There is no need to convert division into multiplying the reciprocal, nor converting subtraction into adding a negative, that is only adding in unnecessary steps. PEMDAS is better represented as PE(MD)(AS). PEMDAS ?is no different than BODMAS: P(B) = Parentheses (Brackets), E(O) = Exponents (Orders), MD(DM) = Multiplication and Division (Division and Multiplication), AS = Addition and Subtraction. The U.S. uses PEMDAS, the UK uses BODMAS, but they mean exactly the same thing.
@@jasonpenn5476 I totally agree that they have the same precedence and are done left to right; that's absolutely how you should interpret the order, but it isn't the whole story. Just because I started with the confusion that MD and AS follow the stated order doesn't mean that that's what I was saying was correct. Thought, strictly speaking the acronym, on its own, does indicate that Multiplication is first or for BODMAS that division is first so if you didn't know Multiplication and division had the same precedence, the acronym wouldn't help. So, I started with the confusion of the acronym because some people are confused by the ordering of the acronym. As an educator, I come across it a lot. But I quickly point out that the correct way is to resolve the MD and AS from left to right. Moreover, I pointed out that the reason these operation pairs have the same precedence is because division and subtraction aren't real operations, but shorthand for multiplying or adding something and that's why the left to right convention is used, which is indeed true and how they are defined mathematically. The point of the video was to illustrate why the rules are what they are and that they aren't just arbitrary memorized rules. This provides considerably better insight to what is actually happening that not only helps algebra computations make more sense, but it helps in more advanced settings like vector calculus, linear algebra, and even ring theory.
The "division and subtraction are shorthand" thing is good to know because it's what's actually happening structurally. I recommend the use of PEMA because the PEMDAS or BODMAS acronyms obfuscate this point on their own. It also reflects more advanced structure and so it builds a better mathematical foundation than just memorizing a rule because someone said it was right.
TL:DR, just because I started with a confusion that some people have that you don't, doesn't mean that I was saying that that confusion was correct. I was pointing out that the acronym is confusing on its own and provided a way to make it less so that helps one understand the arithmetic at a far deeper mathematical level.
I hope none of that came off rude, it wasn't meant to be. It just sounds from your comment that you didn't understand the point of what I was saying in the video. I never claimed that multiplication before division was correct, I only claimed that's what the acronym says.
@@jasonpenn5476 I totally agree that they have the same precedence and are done left to right; that's absolutely how you should interpret the order, but it isn't the whole story. Just because I started with the confusion that MD and AS follow the stated order doesn't mean that that's what I was saying was correct. Though, strictly speaking the acronym, on its own, does indicate that Multiplication is first or for BODMAS that division is first so if you didn't know Multiplication and division had the same precedence, the acronym wouldn't help. So, I started with the confusion of the acronym because some people are confused by the ordering of the acronym. As an educator, I come across it a lot. But I quickly point out that the correct way is to resolve the MD and AS from left to right. Moreover, I pointed out that the reason these operation pairs have the same precedence is because division and subtraction aren't real operations, but shorthand for multiplying or adding something and that's why the left to right convention is used, which is indeed true and how they are defined mathematically. The point of the video was to illustrate why the rules are what they are and that they aren't just arbitrary memorized rules. This provides considerably better insight to what is actually happening that not only helps algebra computations make more sense, but it helps in more advanced settings like vector calculus, linear algebra, and even ring theory.
The "division and subtraction are shorthand" thing is good to know because it's what's actually happening structurally. I recommend the use of PEMA because the PEMDAS or BODMAS acronyms obfuscate this point on their own. It also reflects more advanced structure and so it builds a better mathematical foundation than just memorizing a rule because someone said it was right.
TL:DR, just because I started with a confusion that some people have that you don't, doesn't mean that I was saying that that confusion was correct. I was pointing out that the acronym is confusing on its own and provided a way to make it less so that helps one understand the arithmetic at a far deeper mathematical level.
I hope none of that came off rude, it wasn't meant to be. It just sounds from your comment that you didn't understand the point of what I was saying in the video. I never claimed that multiplication before division was correct, I only claimed that's what the acronym says.
Can't they just fight it out in a ring somewhere?
Was that an intentional math pun? If not, curiously, there's an abstract algebra structure called a ring where only addition and multiplication are considered as operations. And the real numbers form such a ring. So in a sense, they do fight it out in a ring somewhere.
Gotta raise a complaint because your phrasing will confuse the gullible/ignorant. Saying "subtraction isn't a real operation" or "division isn't a real operation" is just plain untrue. It'd be equally valid to say "addition isn't a real operation, it's actually shorthand for subtracting implied negative numbers" or "multiplication isn't a real operation, it's actually just shorthand for dividing by the reciprocal". Which kinda demonstrates why phrasing it that way is a bad idea - you aren't actually informing people that you're giving them an advantage and how that advantage really works, you're just lying to them to trick them into exploiting that advantage _while still remaining ignorant of what's really going on._
For example: addition and subtraction _are_ commutative if you don't have any other operations going on other than addition and subtraction. 3 + 1 - 7 is the same in any order: 3 + 1 - 7 = 1 - 7 + 3 = 3 - 7 + 1... it's all the same equation, it'll always equal negative three. So why go to the bother of switch the operators under that circumstance?
Likewise, claiming that the "definition" of division is multiplying by the reciprocal of the divisor is also a lie. You've just chosen that method to achieve the goal of division, but it isn't the definition - because _that literally requires you to do a division anyway, what do you think fractions are?!_ >.< All you've done is trick people into burying a division elsewhere in the equation. Rather than elucidate what's going on, you've made a false claim that obfuscates what's really going on while tricking people into taking steps that'll still get them where they want to go. That's not helping people, that's kneecapping their understanding.
If you actually explained the limitations of each operation and why you might want to exploit certain operators instead of other operators, people not only get your message but also _why your message is going to be useful_ and not only that but when you can maybe ignore it and why. You know, that thing that people consistently complain maths teachers don't give them? Context? Understanding? That thing the absence of which makes people consistently do poorly in maths classes?
TL;DR: What you're suggesting people do is good. The way you're suggesting it is awful. Stop teaching maths like a math teacher. Maths teachers are consistently rated the worst kind of teacher because _they usually don't explain things they just insist on rules absent any context which _*_isn't actually teaching._* Be an engineering teacher who teaches maths instead, those folks have to explain what they mean. :P
I didn't read your entire novel but there is only one operator -- addition. Subtraction is addition of a negative number. Multiplication is repetitive addition. Division is repetitive subtraction. End of story.
@@garymartin9777 And here we have someone proving exactly my point.
Too hopped up on dogma and ego to recognise that his definition for division only works when the numerator is larger than the denominator and won't get you any decimal answers, and actually you need a quotition-based or partition-based definition for division to make sense in all contexts which is why long division is a thing.
Again, shit people would know if maths teachers could stop lying to make things seem simpler than they are.
I appreciate your passion for this topic, but I have to fundamentally and respectfully disagree with you. Subtraction and Division being defined this way is exactly how we do it in more advanced contexts and indeed many beginning algebra textbooks define it this way. In fact, it's used in computer algebra this way, or at least it's so claimed by the Wikipedia page on the order of operations.
I teach math like a mathematician, because that's what I am. I hold a PhD in mathematics and teach at a university. And this is the context for the operations. For example, you can find addition and multiplication occurring in other places too such as with matrices. And how do you subtract two matrices? You add the additive inverse of the second one to the first. You can also divide by a matrix by multiplying by its multiplicative inverse if it has one. So not in every setting is division a fraction. This sets up and provides a wide context for students to see structure in other places as with polynomials or linear algebra. It also leads into more advanced ideas like groups and rings. In fact, from a mathematical perspective, the real numbers are a ring with the two operations, addition and multiplication, such that the multiplication is commutative, has an identity, and every nonzero number has a multiplicative inverse. That's the actual structure of the real numbers. Such a structure is a called a field and there's lot of other fields out there that have the same structure, and hence have division (i.e. multiplication by the multiplicative inverse), but might not have fractions.
And while it's tempting to say that what I've presented about subtraction not being an operation is the same idea by rephrasing addition as subtracting a negative number. It's not quite the same thing. For example, what I mean is that subtraction can be phrased as adding the additive inverse. The additive inverse of a number a is the number b such that a+b = 0. If you made subtraction the base operation, then the subtractive inverse of a would have to be the number b such that a-b=0, which means that b=a and every number is its own subtractive inverse. In that case, you can't refer to the opposite sign number in an algebraic way and it loses context and structure by doing so.
Moreover, you quite incorrect that subtraction is commutative. If subtraction were commutative, then 7-1 and 1-7 would mean the same number, which they don't. Or in your example claiming that addition and subtraction are commutative with each other, 3+1-7 would have to equal 7+1-3. That's what commutative means. It means that you don't move the operation, just the numbers. If you think that the - has to stay with the 7, which it does, then you are not treating subtraction as a commutative operation.
So, quite the contrary, I believe I've actually given much better context to what's going on with these operations and what they mean and the structure they provide than you think.
But you're welcome to disagree with me, that's up to you. I hope that the above doesn't come off rude; that's not my intention. Thanks for watching the video and being passionate about this. I hope you find some of my other content interesting too.
@@scholarsauce On division:
3 / 4 = 3 * (1/4) = 3 * (1 * (1/4)) =...
If you define division _as_ multiplying by the reciprocal then regular division cannot be resolved, only infinitely expanded. At some point you have to accept that you need to do a division some other way, or else you will just be infinitely expanding the equation.
But, it _is_ a very good method of simplifying the process of doing a division, assuming you use a different definition for division. That's why computers use it as the first step to processing division... but also, the fraction still needs to be resolved _which is a division_ and usually computers do that by guesstimating the solution successively to converge on the correct answer by comparing to a multiplication which should output one of the input values if correct, and then assessing the error variance when compared to the originally input values before re-guesstimating a possible solution in the direction that should reduce the error variance until the test multiplication results in one of the inputs and the error value is nil, which (as ought to be pretty obvious) is not a process any human really uses when doing division. It's simpler for computers to do that with a smaller division is all, so the equation is first re-arranged for that purpose. The rearrangement is not the division itself. (By the by I'm assuming you're referencing Newton-Raphson, I should also point you at Goldschmidt for a completely different computational process of division which is equally inhuman, but also pretty common... basically never look at computers for a human-compatible way of explaining things, it's really awful...)
As for matrices... matrix multiplication isn't commutative like regular multiplication is. Also... most people would just say you can't divide a matrix by another matrix, and that while you can multiply one by another's inverse doing that is only division-esque? So in most respects those operations in that context are kinda different in a way that probably shouldn't be generalised to real numbers? The outcome looks similar, but the process is decidedly not the same.
And yes, you're right, I mistakenly said commutative when I meant that given the context the equation is the same regardless of the order the operations are run, which is not the same thing, total my bad. Programmatic calculations are typically stepped through vertically with a single operation to a line (primary school style, which is objectively the best mathematical notation style if we are all being honest with ourselves), not horizontally with all the operations in the same line, and when you do equations that way the operational order is more clearly prioritised as what matters. Brain in the wrong space - it differs when you add in multiplication and division, it doesn't if you do it with solely addition and subtraction.
But the fact still remains, when you say "x operation isn't real" you aren't actually helping people. Case in point: the reply to my comment before yours. It's certainly valid in certain circumstances to claim that division is just repeated subtraction... but of what? If it's of the divisor, then... only if the divisor is smaller than the number being divided. And you'll probably have a remainder, which is fine for primary school but... not exactly a well-defined answer. If you try to divide the remainder by the divisor... you can't subtract the divisor from the remainder, so you get an unresolvable fraction. And fractions (as I may have already mentioned) are a division, which proves the definition to be incomplete. And also, it's kinda not the subtraction itself, it's _the count_ of the subtraction, which is a difficult concept to parse into a simple explanation too. So... he's attempting to approach the concept of partition but has no viable way to explain it, because he's trying to stick within the limit of his secret special lore that actually some operators "aren't real" and should only be described in the context of the operators that he's arbitrarily decided to claim "are real", instead of trying to address them as their own things.
And I've also witnessed the exact inverse of your claim about subtraction by the way, for exactly the same reason - someone claiming that there are no negative numbers only subtractions, because they had a _good_ education and _understand PEMDAS_ (thanks PEMDAS, another failure in understanding resulting from that gosh darn acronym!). And again, equally dogmatic about it because fabulous, secret powers were revealed to him the day he held aloft his magic sword and...wait no, that's something else. It eventually wound out putting them in the position of being unable to explain how to resolve multiplications by negative numbers... because the multiplication always comes first and every time you try to factor away the negative numbers to defer the subtraction to the AS step of PEMDAS... you just get more negative numbers to multiply and the multiplication step _never ends._ And then to cover for that they wound out having to claim that the definition of a negative number must be just a positive integer multiplied by -1, which is equally an infinite recursion because how does one define -1 under those circumstances? As 1 * -1?
_Any_ time you say that a mathematical concept isn't actually real, you're inviting people to switch to a dogma rather than an understanding because mathematical correctness is often considered to be a prestige form of understanding. Any "secret knowledge" answer is going to make people into idiots, because they will value it as something they hold over people who do not hold the secret. In both the above cases, for the exact reason of being adamant that they knew something other people didn't, they couldn't admit/see that their claim itself was wrong even when taking the claim to its ultimate conclusion results in something that can't be resolved.
That's what that teaching style does to people.
Arguably it is better to define operators in terms of what their goal is (which is another reason not to compare anything to matrices, because goddamn do maths teachers not know how to explain to people who've never encountered matrices what is going on there!), because there's just so many ways to arrange an equation to get to the same outcome. You kinda got close to explaining things that way in the end with your reference to stepping in one or the other direction on the number line for addition and subtraction... just need to take it one step further to explain multiplication and division that way too.
*Or, to put that all in a really simple form:* If your explanation can't get someone from 1/4 to 0.25, you haven't actually explained division. At best, you've explained something that maybe rote memorisation or a calculator can fill in the gaps for.
GEMA has been taught for decades, don't change what works.
Aren't eggplants always purple
I had to look this up. Almost all varieties are some shade of purple, but there are a couple that are green or white, like the Thai eggplant.
Thanks for watching the video, I hope you enjoyed it!
I think… I remember in grade school ( the early 60’s, sp my memory isn’t great ) that division comes before multiplication and subtraction comes before addition. I get a lot of correct answers that way.
AND I GOT THE ANSWER RIGHT!!! Division before multiplication and subtraction before addition. PEDMSA IS the formula!
@@eddiepierce7028 except it isn't as explained in the video; a substraction is an addition and a division is a multiplication; and a division is expressed as a fraction in actual maths used by people beyond elementary school
I had to think about this for a bit as I couldn't see a counterexample at first. However, if you have two divisions in sequence it becomes a problem.
2÷4÷3×6
runs into problems. You can do the divisions first, but only get the correct answer if you do them from left to right. And this is due to the issue I point out in the video. If you do the divisions and subtractions from left to right, that is equivalent to PEMA, but you're still back to a left to right rule without a clear reason why.
Kind of cool. And made for a fun afternoon of conversations with a few other mathematicians. Thanks for the thoughtful comment! I hope you find my other content equally interesting!
@@valdir7426 Re 'a division is expressed as a fraction in actual maths used by people beyond elementary school':
Maybe because said people also learned fractions in elementary but never really got used to expressing division as a multiplication with a negative exponent?
The example by @scholarsauce :
2÷4÷3×6
is methinks a no-brainer when expressed as:
2^1 * 4^-1 * 3^-1 * 6^1 =
2^1 * 2^-2 * 3^-1 * 3^1 * 2^1 =
(2^1 * 2^-2) * (3^-1 * 3^1) * 2^1 =
(2^[1-2]) * (3^[-1+1]) * 2^1 =
(2^-1) * (3^0) * 2^1 =
2^-1 * 1 * 2^1 =
(2^-1 * 2^1) * 1 =
(2^[-1+1]) * 1 =
(2^0) * 1 =
1 * 1 =
1
I used bracketing only to focus one's attention on the groupings of same-base numbers--which leads to what SEEMS many more steps than usual in traditional division notation.
Note that with this notation one as-easily simplifies the factors common to what in traditional division are called nominator & divisor.
When I was taught PEMDAS, it didn't stipulate M before D nor A before S, but it said M or D first and A or S first. We were taught that we could interchange the numbers and the operations MDAS before them before we carry out the operations.
For example
(1)5÷6×12=5×12÷6=60÷6=10
(2)3-10+15=3+15-10=18-10=8
(3)3-5÷6×12+24-45÷9=3-5×12÷6+24-45÷9=3-60÷6+24-9=3-10+15
=3+15-10=18-10=8
As long as you move the operation whenever you move a number like you have that works fine, for the same reason I mentioned in the video. Indeed, in practice this is exactly the best way to work with it to be fast and accurate.
Thanks for watching and commenting!
I have been saying this for decades!!! Thank you.
So this addresses the rudimentary errors on the Facebook posts, but where the nerdy debates happen is with multiplication by juxtaposition which I noticed was left out of your examples.
Example being:
2/5(3+2)
Does it become:
2x(1/5)x(3+2)
Or:
2x(5x(3+2))
With arguments that the multiplication by juxtaposition causes the division symbol to apply to the whole part.
Yes, this is a video that is on my list to make.
Personally I think that 2/5(3+2) should be interpreted as (2/5)(3+2). But ultimately, as long as there is confusion as to the meaning, no one should ever even use the expression and should use parentheses to indicate what they meant.
Thanks for the comment and for watching my video. I hope you enjoy some of my other content too!
@@scholarsauce I agree with that two but I’ve seen other videos that imply there is strong historical context for the juxtaposition priority.
@@FosukeLordOfError Yeah I've seen them too and indeed there is. Historical context makes things hard to change, but it's not a particularly compelling argument from them that we should keep that rule. If we want the order of operations to be understandable, avoid confusion, and be based in mathematical principles rather than merely poorly thought out memorized rules, we should push to get rid of the higher priority juxtaposition thing. That'll probably be the content of my follow-up video to this.
If nothing else controversy sells and maybe it'll get me a bunch of views! Gotta build this channel. Lol.
Thanks for the fun conversation!
@@scholarsauce The juxtaposition higher priority rule is rather a thing of convenience. For example, substitution of a value of a variable into an expression turns clunky and/or ambiguous unless you can maintain the inherent structure of algebraic expressions even after the substitution. Requiring every such substitution to have extra parentheses around them would be a pedagogical nightmare to convey to pupils.
@@rechet Thanks for your comment and for watching my video! I appreciate your consideration of how this is taught to students as part of your reasoning. That's really important. However, as a university math professor, I do indeed teach that all substitution should be done with parentheses around them. It is critical in fact for things in college algebra and calculus, and just things that are order of operations related. It is not a pedagogical nightmare at all, but is usually received well and students perform calculations as a result. It honestly helps them understand a calculation considerably better. So I disagree with your conclusion that such a thing would be a pedagogical nightmare, quite the contrary, I would argue that it's essential and a skill that many students are missing by the time they get to college.
Juxtaposition is just another way of writing a multiplication symbol. Anything beyond that is not based on any mathematical reasoning and leads to more confusion than its convenience is worth. At least no argument that I've seen in favor of prioritizing juxtaposition has been particularly compelling that this rule is based on anything other than convenience rather than anything mathematically natural. But that's just my opinion. I can see several sides to this argument. Thanks again for commenting. I appreciate the positive interaction even if we happen to disagree.
The reason my six-year-old has no trouble with this is because I taught her from the beginning that subtraction is simply the addition of a negative, and that division is multiplication by a number smaller than one, specifically, the reciprocal of the number being divided. (Of course I also taught her other ways of thinking of the same concepts.) She often rewrites any subtraction in the form a+(-b) automatically. However, multiplication and division are still problematic, if only due to usage. Standard representation of formulae in the fields of mathematics, physics, and engineering contradict this explanation, and as cited on another channel, so does at least one mathematics style guide. In standard usage, for example, a/bc would normally be interpreted as a.(bc)⁻¹, not a.b⁻¹.c. So the question comes down to the other side of what I have been teaching my daughter: the need for rigorous communication.
I love that you were teaching your six-year-old about that idea early! I really think that teaching us to view things like that goes a long way in helping the next generation. I just wish we could do it more.
I agree that multiplication and division are a problem due to usage. I think it's really unfortunate too, because the definitions of all the symbols are pretty clear and allowed for no ambiguity when we first started using the order of operations to resolve the potential issues with using infix (the operation is between the numbers) notation. The ambiguity is created, I think, solely by people wanting to unnaturally prioritize something else in the computation without using parentheses. For example, wanting a/bc to actually mean a/(bc), but without writing the parentheses. But that's the very point of parentheses, to artificially prioritize a computation, so why not use them? It's really odd to me that some usage guides try to do it this way, but I don't personally think it's good practice. Of course, I would probably never write a/bc in the first place. I would write (a/b)c or a/(bc) depending on which I meant to be certain that I was understood.
At any rate, thanks a ton for watching my video. I'm glad that you enjoyed it and I hope that you'll find the rest of our content enjoyable too!
@@scholarsauce : Yes, I wish that more of us would focus on teaching the concepts of mathematics. I have found that most teachers see mathematics as a means to an end, rather than a pursuit in its own right. Consequently, so do most students. And this is, I believe, why mathematics education more generally has been reduced to a series steps to follow pedantically, neither understanding nor caring what is happening.
Once I realized that exponents, division, multiplication, and subtraction were just shortcuts to expressing the addition operation, maths became so much more intuitive
@@glensmith491 I hated to memorise things in my head like 1*7, 2*7, 3*7 etc (what ever this form of teaching is called in english). Because it was obvious from the start that 2*7 = 7+7 etc. I wonder, are these things not thought out in some pleases if these things are something you realise after you've learned to use exponents or is it common that not all figure this out early on? I mean it's so fundamental.
@peterodonnell4404 I have become sick-&-tired of the nonsense spouted in language style guides, & given how we-people are in general, I am fairly certain that the better place for ANY style guide is the tip (in US-English known as the trash dump). In the field of language one cannot get away from everything being an opinion, be it spouted by a so-called authority (with or without university-degree) or the average user of language. In maths however opinions should be able to be ignored, given that it is based on axioms & deductive logic. So yes, I agree fully with the need for RIGOROUS communication in ANY scientific field. While adding that ANY communication with a flat-Earther is a waste of time & energy.
Where at about 8:10 he says addition is associative (which it is) I believe that he meant to say or should have said commutative. And again at 12:00 when talking about multiplication.
It's both associative and commutative, and both of these matter to his point. Let ♣be an arbitrary mathematical operator, that takes in two numbers, and produces a single output, where "A♣B" is read aloud as "A club B".
If (A♣B)♣C is the same thing as A♣(B♣C), then this means the club operator is associative.
If A♣B is the same thing as B♣A, then this means the club operator is commutative.
There are associative operations that are not commutative. Matrix multiplication is one such example.
@@carultch Thanks! Great explanation!
@John Paul Colthrust Yes, I probably should have been more specific here. In the two computations displayed, I only employed the associative property because I didn't change the order of the string, just the order of which two I resolved first. However, there is a third way to add/multiply them that is also just fine, but would require the commutative property. And I do kind of handwavily say that the associative property basically means that you can add them in any order, but to really get that, you would need both the associative and commutative property. Ultimately, the associative property lets you pick which two adjacent numbers you'll apply the operation to first, while the commutative property allows you to change the order of the numbers themselves.
Thanks to both you guys for clarifying this. I hope you'll enjoy more of my videos!
@@scholarsauce this video is excellent and for me most serendipitous! My partner and I home-schooled our daughter from age six through 12 to spare her the curiosity-destroying, love-of-learning-killing rote learning of the traditional school system. I believe that there are “moments of danger and crisis” in a child’s mathematics education, which emerge every time the child’s universe of numbers is expanded: from just the positive real (counting) numbers to include, in turn, positive rationals (fractions), directed numbers (negatives), all real numbers, complex numbers, etc, and also when more-advanced mathematical objects: vectors, matrices, tensors etc are first introduced. I believe that these crisis points must be handled VERY carefully, and that the failure of educators to do so is largely responsible for the fact that far too many people are intimidated by and hate mathematicians.
This video will be a valuable resource in my helping her with a bit of discomfort that, just this week, she told me she is feeling over the addition and subtraction (but, interestingly enough, not the multiplications and divisions) of directed numbers. Thank you so much!
@@johnpaulcolthrust8207 This is awesome! I am so glad that you found this video that helpful! I totally agree with you that we all should be careful with those crisis points like you mentioned. Good luck with everything and I hope you can continue helping your daughter as you've been. That's awesome! If you ever find that you wish there was a video on something mathematical, let me know and I'll see if I can make one!
I’d say PEMDAS is more of a “half-truth” than a lie. It TECHNICALLY isn’t wrong, but doesn’t account for ambiguities and/or unspoken rules.
That's fair, but where's the clickbait in that?
@@scholarsauce I didn’t say it’s clickbait :)
@@mimitogami Oh I meant that as a joke not as a rebuttal to you calling it that. I definitely called it a "lie" as a bit of clickbait. I agree with you that it's more just about clearing up ambiguities. Lol. :)
@@scholarsauce oh sorry. My tone-deafness strikes again-
Anywhos, great video ! Also, unrelated but you kinda look like a Chem/Physics teacher in my school?
@@mimitogami That's fun. I hope that they're a good teacher. I'm glad you liked the video! I hope you find my other content interesting as well!
4:40 --- true. The expression is really 3 + (-5) + 4
Yet another BEDMAS video. You can be as semantically clever as you want but EVERYONE agrees what the BEDMAS order means, whether it’s semantically correct or not. It makes no difference as long as everyone agrees, which they do.
The issue is that not everyone agrees with the order or doesn't understand what the order means.
So, I've actually written an algebraic parser before, and to summarize what's going on with this terrible infix notation we're stuck with, because five hundred years ago someone abbreviated the Latin word for "and" (i.e. _et_ ) with just the letter "t"... as in, III _et_ III _sum_ VI (3+3=6 in modern notation) (" _plus_ " means "more" in Latin by the way)... anyway...
The reason why parentheses (or an overbar) are required at all is because the composition of binary operations forms a binary tree. The leaves (terminal ends of the branches) are numbers (or letters), and the two leaves join together with a binary operator at the fork. If you are a dumb computer with almost no memory, you walk down the tree until you find something that is not a binary operator-- a number or letter-- and then you look at the other side branch from the operator just above this number... if the thing on that side is another operator, then keep walking down that branch of the tree until you reach a terminal number or letter... When you find a number on both sides of an operator, now you actually do some math! Combine the numbers together into a new number, and stick it on the tree at the location where that operator was located (overwriting the operator, and essentially deleting the two input numbers). Now keep walking back up the tree combining pairs of numbers like this until you reach the root of the tree, and the last number you write into the root of the tree is your final answer. Ta-da!
I'm trying to keep this comment short, so I'm not going to talk about stacks, and reverse Polish notation... but prefix and postfix notation is superior because you never need parentheses, there is only one unambiguous way to evaluate it, and stupid facebook clickbait math problems would not exist.
Anyway, parentheses are a hack for serializing a tree structure into a single line.
Furthermore... math notation was created by humans for humans, and humans do not read math notation in the way a 1960's computer with only two CPU registers reads algebra. This PEDMAS crap is a recent invention, and historically every famous math book did something slightly different, but you could understand it from context.
Anyway, personally.... I interpret "/" as a ratio, or fraction, which has been squished into a single line entirely for typographical reasons. When you're writing a fraction with a typewriter, or using the cheapest print shop in town, things like fractions are a pain in the ass to physically write in ink on paper. Until a century ago, everyone wrote math by hand in chalk or with pens, and book publishers were manually setting each block of text with physical lumps of metal. (Not Linotype) (I'm suddenly reminded of why Knuth created LaTeX.)
Anyway... "/" has implicit parentheses, because the left and right sides are a numerator and denominator. foo/bar is implicitly (foo)/(bar) You can't use more than one "/" without parentheses around that "fraction"
I have a lot more to say on this topic, but I'm trying to keep this comment short.
Here's an example:
6 2 1 2
\ / \ /
÷ +
\ /
×
... becomes ...
6 2
\ /
÷ 3
\ /
×
... which then becomes ...
3 3
\ /
×
... and finally ...
9
So, this tree is 9=(6÷2)×(1+2)
The other popular interpretation goes like:
6 2 1 2
\ \ \ /
\ \ +
\ \ /
\ ×
\ /
÷
... becomes ...
6 2
\ \
\ \ 3
\ \ /
\ ×
\ /
÷
... and then becomes ...
6
\
\
\
\ 6
\ /
÷
... finally reducing to ...
1
Or, in other words, 1=6÷(2×(1+2))
The argument is over which of these two trees to use. And this is all a historical accident that started back in the 1400's
This is like arguing about how to parse Roman Numerals... which, by the way, are also more ambiguous than they teach in grade school.
How about 8 minus negative 4 ? Are we going to write it like 8 + (- -4) ?
Rather it's written as 8+ (-(-4)) the - means the "additive inverse of". The additive inverse of the additive inverse of 4 is 4 and so it resolves as 8+4.
@@scholarsauce I see, thanks for the reply
@@gemorp8506 You're quite welcome!
Next, you need to include Juxtaposition in your abbreviation.
PEJMA because when you get to algebraic unknowns it becomes extremely important. It also adds a layer of complexity, and therefore confuses many people. confusing
Thanks for watching video! I appreciate the suggestion, but I respectfully disagree that juxtaposition should have higher priority than multiplication denoted any other way. I know some groups state this, but I don't agree that it should. Everything else about the order of operations has a good mathematical reason for it, but the juxtaposition thing is just because, for some reason, people don't like using parentheses or vertical fractions when they should. Artificially granting it higher priority is just a memorized rule with no good reason behind it and personally I think that that is what causes the confusion. I may make a video about how some argue that juxtaposition should have higher priority, but if I did it would likely be to argue that it is a bad practice, leads to confusion and ambiguity, and has no mathematical basis. The juxtaposition thing is something that I've never seen discussed among mathematicians and seems really odd to me as a mathematician. That said, I would like to investigate a little more into why people use this and where it came from and whether any mathematicians actually use it (I've yet to find one) or if it is just relegated from physicists and engineers before making a video.
I do appreciate the comment and I hope the above didn't come off rude, it wasn't meant to. Thanks again for watching my content and I hope you find some of my other stuff interesting too.
@@scholarsauce in rebuttal:
I don’t feel that it would be learning a new rule, rather than keeping the rules consistent throughout mathematical notation.
By grouping them together, you are implying that they are one term, IE: 3b. Yes, this is implied multiplication, but it is considered one term because it must be resolved before much further manipulation can occur. The same should hold true here in the case of juxtaposition with a parentheses.
To my knowledge, this debate also exists within the programming of various calculators as well, with most manufacturers specifically stating that it is predominantly US sales driving the “PEMDAS” or “PEMA”functionality, while the rest of the world relies on “PEJMA.”
I would be fascinated by what you find if you took the question to the wider scientific community, as you have yourself suggested.
@@Wyrnikh I agree. I've been talking to a few of my colleagues at my university and the mathematicians are all in my camp if they think about it, though some of their initial first glance computations vary with a minority of them.
So yeah, it'll be interesting to see where everyone is at.
@@scholarsauceWe in the sciences and engineering have always done multiplication by juxtaposition first, since as wyrnikh has pointed out these terms are grouped (associated with each other) so they must be resolved first. You would be hard pressed to find a scientist who doesn't follow this rule.
@@ronruszczyk7935 It's a pretty inconsistent rule to be honest and forces the creation of a type of grouping that has higher priority than multiplication but lower priority than exponents, since otherwise 4x^2 is ambiguous and meaningless. You'll probably interpret it as 4(x^2), but if juxtaposition is a grouping as in at the P level which occurs before exponents, then this should mean (4x)^2=16x^2, but that should mean (16x)^2 and so on, so it's not really a grouping. At that point does 4x^2 even mean the same thing as 4xx? The problem is that if you define juxtaposition as a grouping, you run into problems immediately because you don't clearly define where the group is delimited. The same issue comes up with how the juxtaposition crowd defines /. If 4/ab means 4÷(ab) then what does 4/a/bc mean: 4÷(a÷(bc)) or (4÷a)÷(bc) or 4÷a÷b×c? If you think the first, then does the symbol / also define a grouping after it? And how far does it go? Why should you not treat 4/a+b as 4/(a+b). You see because people want these things to be groupings but don't define how those groupings are delimited, it requires you to come up with a ton of caveats to get expressions to mean what you want. And none of those caveats are based in anything mathematical, just convenience because people want to avoid writing parentheses. What's the point in doing that? What advantage does that provide? Why would we do that for this one instance when the entire rest of the order of operations are induced by the definitions of the operations? Mathematicians don't mean anything different by / and ÷ and they don't mean anything different between juxtaposition and ×. Because that's the only interpretation that is consistent and doesn't require a bunch of caveats and arbitrary rules to make sense of expressions. All the confusion on this issue comes from a poorly defined notion of juxtaposition by those scientists and engineers. I know that you say that scientists and engineers all think like this, but I have been hard pressed to find even a single mathematician (and I mean university professors with PhDs in math) that think it makes sense to prioritize juxtaposition. Just because a bunch of scientists and engineers, who I guarantee you did not think about the implications of that rule very long, think you should do it, is not a good reason to. Especially when all the mathematicians who have thought much about it are crying foul and saying it causes all sorts of weird inconsistencies with the expressions.
I get what you're saying, but in my professional opinion as a mathematician with a PhD in math, that rule is not founded in any mathematical idea and creates all sorts of inconsistencies with what expressions mean.
@scholarsauce I came up with 'PEMA' a few weeks ago after watching 2 videos wherein all kinds of fairly nonsensical comments on BODMAS/PEMDAS/&c were propounded. Most seemed to try to validate those RoTs, which imho is the wrong way to go.
Just NOW I noticed that this video of yours is already at least a year old... But then, I also know that whatever *I* come up with can NEVER be original--which I am perfectly OK with. Yes, I suffer from a serious case of Confirmation Bias...
I'm glad that this supports what you found too. I don't think I'm particularly original here either. This is mostly just trying to apply the way we use these notions in more advanced mathematical settings (like groups and rings) to real number arithmetic because I think it offers some insight. At any rate, it's always good to hear that others come to the same conclusion.
Thanks for commenting! I hope you enjoy some of my other content too!
@@scholarsauce My main interest in life is improving things from the root up, NOT piling fixes on fixes that end up making the mess even-worse. For instance maths is based on axioms, logic, deduction, consistency. Anything that has a negative effect on them is for me anathema & needs addressing at its root. Like teachers who have less-than-sufficient knowledge of the subject that they teach.
OK, here follows a summary of my thoughts on the PEDMAS/&c nonsense:
My guess is that BIDMAS, PEDMAS, &c originated with people who lacked sufficient insight in & knowledge of both maths & language. They were probably teachers to boot. These 2 lacks caused those people to think that students would use maths more-easily if they had a quick-&-DIRTY RoT (Rule of Thumb) to follow.
Said lacks caused these people to NOT realize well-enough that maths is axiom-based, DEductive by design, logical, & consistent throughout. Plus that human language (mistakenly AKA 'Natural languages') lacks known axioms, is by design NOT-logical, NOT-consistent, opinion-based, & INductive.
I KNOW that many/most teachers wish the best for their students & work VERY hard to achieve that goal. But those lacks ensured that despite the creators' good intentions, ANY RoT for this goal that they came up with was doomed to fail from word 1. Simply because NO RoT is ever needed when the subject is taught correctly.
ALL versions of this RoT exhibit the same defects:
0: The creators had insufficient knowledge of the subject matter. Ditto many/most perpetuators (AKA teachers) of the RoT.
1: Things are included that are in the formal sense NOT maths operations (namely reciprocals, division, subtraction, juxtaposition, bracketing).
2: NOT making clear in its FORM that the RoT referred partly to sequencing, partly to paralleling, AND partly to conventions.
3: ASSUMING that the student knew enough to "fill in the gaps".
4: Adhering to a badly constructed curriculum.
An off-the-cuff question: What might the ROOT be of the badly-constructed-curriculum issue?
@@Scott-i9v2s I agree with you that there are a lot of problems exactly as you present.
Actually, I have a lot of thoughts on what the biggest issues are in today's math education curriculum. I have a video on this planned, but finding time to make it has been difficult lately.
The short of it is that a lot of this is caused by an over-reliance on algorithms and computational proficiency over conceptual understanding. That over-reliance is borne out of this artificial race to calculus that has plagued math education for the better part of a century. If we can break that chain of racing to a topic, we can take more time helping students appreciate mathematical thinking in all its forms and gain some ability in it too within topics that interest them. As opposed to stuffing as much precalculus in their heads as possible so that most of them can take college algebra and stop before making it to calculus and find out what they did it all for. I think that's the root cause of all the RoTs without context and/or understanding.
@@scholarsauce Ermm... I think that I lack understanding of some of the terms there. The 1st is ' artificial race to calculus', probably also because I am not sure what is meant with 'calculus' & 'precalculus'. Wordbooks (AKA 'dictionaries') are not helpful; they confuse the issue even-more. Note that I might well know exactly what it means if the Dutch word for it is used. Much of my formal education was in Dutch schools, so despite being bilingual in US-English & Dutch, I might actually know a subject, but not the English name for it.
Another is 'racing to a topic'. Is that an idiom or are the words meant literally?
I shall make a guess anyway, namely 'teaching to an exam''. Which, when push comes to shove, can be very neatly defined as resulting in 'a RoT without context and/or understanding'. Or as ''I know HOW to do something, but neither the WHY nor WHAT it means'.
BTW, an 'over-reliance on algorithms and computational proficiency over conceptual understanding' describes both 'RoT' & my understanding of 'teaching to an exam'.
If I have mostly said what you have said but with different words, then you expanded somewhat on 'a badly constructed curriculum' . Wright/rong?
@@Scott-i9v2s I think the dutch term for calculus is differentiaalrekening. It's the mathematics of rates of change (derivative) and areas under a curve (integrals) and limits. The US math education system is built fundamentally on getting students to the calculus level as fast as possible. Our precalculus stuff is all the algebra, geometry, and trigonometry necessary to do calculus. I hope that clarifies it. It was surprisingly difficult to find a Dutch word for this, which is probably why the term is so unfamiliar. Language is hard. I applaud you on being bilingual, that's a quality Americans seem to lack too often (myself included).
Teaching to a test definitely shares some traits with that and that race to that topic causes some teaching to a test, but it's more than that. Algorithms get forced on students not just for the test but so that they'll be ready for the next idea that prepares for calculus. All of it is taught in a very linear fashion, which isn't necessary and I think is quite bad for overall understanding.
I agree with your thoughts on teaching to a test being bad, but the problem, at least in the US, is more than just that.
Excellent video. I've advocated what you mentioned as a first step, before working the parenthesis: convert the divisions to multipications, subtractions to additions, and convert any resulting fractions to decimals.
As an example:
3+6-7(6÷2)
3+6+(-7)×(6×0.5)
3+6+(-7)×(3)
3+6+(-21)
9+(-21)
-12
I'm really glad that you enjoyed the video and love that you're teaching it that way! I hope you'll find the rest of our content enjoyable too!
Yes, this works since it follows the commutative property law in math that says you can change the order of operations in multiplication and division. The fourth and fifth steps in your example could have just as easily been written as:
3+(6+(-21)) -- meaning to add the 6 to the -21 first
3+(-15) = -12
@@petepalmere210 The only issue I see is when converting the fractions to decimals. In my example it was easy because 1/2 converts into into a terminating decimal, however with a fraction like 1/3, which converts to 0.333..., it might be easier to leave it in fraction form, with possibility that the denominator could be eliminated over the course of working the problem.
@@Solitaire001 Yes, I agree with your example even 1/3 would work as a fraction since division can be solved by multiplication of the dividend by the reciprocal of the divisor. So, in your example had it been 6 divided by 3 it simply would convert to (6 x 1/3). It would be problematic though to covert other fractions such as 1/5.
@@petepalmere210 1/5 would not be an issue since it converts to 0.2. However, 1/7 would be an issue since it doesn't start to repeat until at least 10 digits.
"PEMDAS is a Lie! "
No.
Division and subtraction are not "false' simply because they happen to be inverse functions.
Might as well claim radicals aren't real because they are the inverse of exponents.
Actually radicals are just exponents. They're fractional exponents. In that way, radicals are no different than division and subtraction in that they are just doing the same operation but with the opposite number. But in the case of radicals, the order of operations already accounted for this fact, which is why radicals aren't listed in PEMDAS. Because they are part of the exponent level. All I'm saying is that we should treat division and subtraction the same way we treat radicals and not list them, but understand them as multiplication and addition of inverse numbers. Indeed, this viewpoint is exactly how these operations are defined and how they are treated in every setting and this gives insight to how they are treated in more advanced settings like in rings. The order of operations is simply a consequence of these definitions. It confuses the issue when we make it seem like division and subtraction are somehow different than multiplication and addition.
I hope the above doesn't come off rude. I certainly appreciate your comment, but I do respectfully disagree.
@@scholarsauce "Actually radicals are just exponents"
That is what I said.
Your post is just another way of wording the same point I was making.
I am glad we agree.
I was merely pointing out that anything inside the radical is grouped in the radical AND the radical itself is an exponent application.
This means that the radical affect both the P and E levels.
"I certainly appreciate your comment, but I do respectfully disagree."
We are in agreement on this aspect.
We are simply communicating it differently from one another.
P = Address what is inside the radical as if it were a parentheses.
E= Address the radical itself ad a form of exponent
@MrGreensweightHist
Since radicals ARE expressible as exponents/indices, we need them like an extra hole in the head, the root-symbol can fade away, & the concept of radicals can join subtraction & division as being no-longer-extant.
After all, why waste time-&-energy on effectively useless stuff?
SIMPLIFY!
@@Scott-i9v2s "the root-symbol can fade away"
I'm ok with that.
"can join subtraction & division as being no-longer-extant."
Nope.
Those two are still needed.
@@MrGreensweightHist Re 'Those two are still needed.':
Why? Show me just 1 case of each where they are needed because of absence of any other way to consistently express them.
The '%'-sign? Since it means 'per 100', it can be written as '100^-1'. So forget about that 1. Ditto btw re 'promille' (the symbol of which has not 1 but 2 instances of '0' at its end) which means 'per 1000' & can be written as '1000^-1'. Or, if one wants to get fancy, write it as '1k^-1'. Note that in both rewrites the '-1' is not (& never was) a subtraction, but is a negative.
It seems to me that some people have a special way of handling division when the obelus is involved especially relying on PEMDAS with equal precedence to multiplication and division and a left-to-right order of operations. They will argue that the expression in the video with the use of the obelus is not the same as 6/2(1+2) or more precisely 6 over 2(1+2)...
6
-------------
2(1+2) This can only equal 1 no matter what order of operations one uses. Unless there is some rule for top-to-bottom operations, one can resolve the grouping first to get 3 and then multiply by 2 to get 6 which would render to 6 over 6 =1, or divide the 2 into six first and get 3 over 3 and still get 1. So there you have it...unless someone can show me proof that the fractional notation is not equivalent then the answer should be 1. For the record, I've given a college-level reference in other posts to bolster my position. I find it troubling that the 2 next to the group is ripped away as if it had no connection with the group at all; as if the expression should be (6:-2) x (1+2). So notwithstanding the ambiguity of the subject video expression, the obelus appears to be the ogre here.
I appreciate your comment, Pete, but I respectfully disagree. Your resolution of the expression 6/2(1+2) seems to assume that multiplication notated by juxtaposition is somehow different and of higher priority than multiplication notated by the × symbol. This is simply not true. It certainly doesn't make it a group; you would need parentheses for that. I'm a university math professor and have taught math at the college level for over 15 years, and I have never seen that taught in a math class and indeed it seems quite foreign to most people that I talk to. In fact, I've only seen it in non-math journal math style guides which are apparently not really thinking too much about the ramifications of that arbitrary choice. Indeed, it frustrates the main point of the order of operations which was to make algebraic expressions using infix notation unambiguous. Thinking that multiplication means something different simply because you notate it differently is what's causing the ambiguity. There is no good mathematical reason to do this.
Let me be clear, there is no difference between 6/2(1+2) and 6 ÷ 2 × (1+2). If you want the fraction you indicated, you would have to use parentheses to indicate that the denominator includes everything past the / symbol. That is, 6/ [2(1+2)] means what you wrote. While I've seen it claimed, I don't know where the idea came from that multiplication notated by juxtaposition is of higher priority than multiplication notated other ways or somehow makes something a group, but I almost guarantee it wasn't from a mathematician, who would have recognized the issues with arbitrarily making that choice. It was mostly likely by someone who didn't think about the weird ambiguities it would produce and couldn't be bothered to use parentheses the way they're designed to be used. (In fact, this gives me an idea for a new video on the order of operations!)
Another example of why it is a problem to think that juxtaposition makes something a group is illustrated by what 2x^2 means. Remember that groups have higher priority than exponentials. If juxtaposition makes something a group, then 2x^2 means (2x)^2 (which =4x^2, but wait that's another juxtaposition and so must be (4x)^2 = 16x^2, but wait....), but most people would agree that it means 2(x^2). So either juxtaposition makes it a group sometimes and not others, which is highly arbitrary, inconsistent, and creates a bunch more rules to memorize that have no good mathematical reasons behind them; or it doesn't make it a group and it's consistent and based on rules that have good mathematical reasons for them. If you want to group something, use parentheses; that's what they're for.
Ultimately, there is no good mathematical reason to treat division or multiplication differently simply because a different notation is used. To me, that's pretty arbitrary and causes a lot of confusion, the exact opposite of what the order of operations was intended to do. So really, I think the confusion is caused by the idea that / and juxtaposition somehow should mean something different than if they are notated a different way. They don't, I've never seen a math textbook teach the order of operations that way and even if some do, it's bad practice as it frustrates what the order of operations were supposed to do in the first place as evidenced by the fact that there are so many disagreements about this. If that doesn't clearly indicate that there was a lot more drawbacks to arbitrarily prioritizing juxtaposition than there were advantages, I don't know what does.
I hope that that didn't come off rude. It wasn't meant to. I actually really appreciate the discussions here and think it's awesome that this video drummed up so much interest. I hope that you'll enjoy some of my other videos as well!
@@scholarsauce No, sir, I did not take it as being rude. No worry, I enjoy the analysis and appreciate your math proficiency. However, I am not convinced, and here's why: Firstly, even TI will admit that it depends on how a calculator is programmed. TI explains: "Implied multiplication has a higher priority than explicit multiplication to allow users to enter expressions, in the same manner as they would be written. For example, the TI-80, TI-81, TI-82, and TI-85 evaluate 1/2X as 1/(2X), while other products may evaluate the same expression as 1/2X from left to right. Without this feature, it would be necessary to group 2X in parentheses, something that is typically not done when writing the expression on paper". The same source notes the following: "... in some of the academic literature, multiplication denoted by juxtaposition (also known as implied multiplication) is interpreted as having higher precedence than division so that 1 ÷ 2x equals 1 ÷ (2x), not (1 ÷ 2)x. For example, the manuscript submission instructions for the Physical Review journals state that multiplication is of higher precedence than division with a slash, and this is also the convention observed in prominent physics textbooks such as the Course of Theoretical Physics by Landau and Lifshitz and the Feynman Lectures on Physics." In fact, my college-level math book written by a Ph.D. mathematician disagrees with your explanation and order of ops, see: photos.app.goo.gl/WRUf5Wk4eBC5SQd38.
It seems to me that some have a special way of handling division when the obelus is involved especially relying on PEMDAS with equal precedence to multiplication and division and a left-to-right order of operations. They will argue, as you do, that the expression in the video with the use of the obelus is not the same as 6/2(1+2) or more precisely 6 over 2(1+2)...
6
-------------
2(1+2)
This can only equal 1 no matter what order of operations one uses. Unless there is some rule for top-to-bottom operations, one can resolve the grouping first to get 3 and then multiply by 2 to get 6 which would render to 6 over 6 =1, or divide the 2 into six first and get 3 over 3 and still get 1. So there you have it...unless someone can show me proof that the fractional notation is not equivalent then the answer should be 1. I find it troubling that the 2 next to the group is ripped away as if it had no connection with the group at all; as if the expression should be (6:-2) x (1+2). Your explanation also ignores that 2(1+2) is really a factor of (2 x 1 + 2 x 2). If you were to substitute the non-factored group into the expression the answer would have to be 1 as the grouping must be evaluated first according to PEMDAS.
Thank you for your response but either I was taught wrong (based on my college textbook reference above) or the convention has changed.
@@petepalmere210 I'm glad that you didn't take it as being rude. It's good to have a civil discussion analyzing something.
I appreciate that there are sources that argue for your interpretation. Though, as you point out, most of them were physics journals, which I'll be honest, in my experience physicists and engineers tend not to care about the mathematical implications or theory behind things and as such tend to be a little more cavalier with their notation. That's not a slight, it works for what they're doing, they just can't be bothered with weird nuances like this; it's not interesting to most of them. And I'll be honest, the Physics Review Journals are clearly showing this when they introduce this arbitrary convention, as if they think that the order of operations is just a convention rather than something derivable from the definitions of the operations.
I am flabbergasted that you have a textbook that says that when no grouping exists, multiplication always takes precedence over division. I have never seen that in a textbook or heard anyone ever teach that before. In fact, I have an algebra textbook in my office right here that actively disagrees and says that the highlighted expression in your picture should be 16 because multiplication and division are considered to be at the same priority and should always be done left to right (which as I point out in my video is to ensure that the division shorthand is interpreted correctly). In fact, in other countries, the order of operations is listed as BEDMAS or BODMAS which explicitly lists division before multiplication. Everyone I've ever talked to that teaches math says to do multiplication and division from left to right. You can even find this requirement in things like the Common Core standards, so it's definitely taught at the national level too.
I would argue that the book that you cited is unfortunately highly nonstandard. It's actually kind of bonkers to me to see it written like that, particularly in a math book. Especially since it causes problems in higher math too, where division and subtraction aren't considered different operations than multiplication and addition. Division in ring theory literally means "multiply by the multiplicative inverse" and subtraction means "add the additive inverse," just as I described. They aren't treated as separate operations from multiplication and addition at all. In fact, on the Wikipedia page for the order of operations, it mentions that this is also the interpretation in most of computer science. The Wikipedia page does also mention the odd "juxtaposition has higher priority rule" that some physicists seem to cling to and the ambiguity that that introduces. The fact that you are operating from a book that taught the highly nonstandard idea (which is also wildly inconsistent with the definitions of the operations) is likely where some of the confusion is coming from. For other examples, see this website www.mathsisfun.com/operation-order-pemdas.html and you can also see the left-to-right rule for multiplication and division in the Khan academy videos. There's also a MindYourDecisions video here about it as well ruclips.net/video/URcUvFIUIhQ/видео.html. This one actually points out that your interpretation may have been a convention in older usage, which has now been discontinued. I suppose that's possible, but it was likely discontinued because of the issues with it that I'm pointing out.
Respectfully, not a single one of the resources you cited in support of multiplication whether by juxtaposition or otherwise having higher priority provides a good mathematical reason for interpreting it that way. It's all based off of arbitrary convention. They all just basically boil down to, "I don't want to write the parentheses and it doesn't matter what choice we make anyway, so who cares?" But it does matter, because that choice alone made this expression ambiguous in a manner inconsistent with what those symbols originally meant. In my opinion, it's a really bad choice and should be discontinued wherever it's being used. I certainly disagree that it is common among mathematicians unless they haven't really thought that hard about it (or perhaps are remarkably stubborn, which does happen from time to time; I've even been known to do it occasionally...)
I will add that the best argument made in favor of your point and something that at least was founded in trying to be consistent with a mathematical rule is what you said about the distributive property. However, to that point, my explanation does account for it, but I would argue that in the 6/2(1+2) example, what you would distribute into those parentheses is the quantity 6/2. That's actually how I automatically interpret that expression when I read it. If I were to compute without thinking about it and given my years of experience teaching math, my trained gut instinct would distribute the 6/2 into the parentheses yielding (and I'll use parentheses to be absolutely clear how I interpreted it):
(6/2)(1+2) = ((6/2)*1 + (6/2)*2) = (3+6) = 9.
And I'm saying that from the perspective that I'm a university math professor with a PhD in math who has been teaching math at the college level for over 15 years (you can tell too, given how long and verbose all my comments are; I mean these professors never shut up!). I think too, that if you asked a hundred mathematicians to compute that expression, I would wager that more than 90, if not all of them, would resolve it to 9 as well. Arguably, I haven't tested that hypothesis and so just take it as my opinion; I could very well be wrong. (That might actually make for an interesting video itself, by the way.)
So, my opinion of your last statement is that, yeah, on the one hand, you were taught something that is highly not standard in mathematics literature and in fact, contradicts usage in higher mathematics, namely, that multiplication is always of higher priority than division. It's is shocking to me too that a math PhD would be so careless about it, unless the book is like really old; like over 100 years old. But you're not wrong that there is a current convention, at least among physicists, that kind of supports what you're saying, but only that juxtaposed multiplication is interpreted as higher priority than division, not multiplication that includes a sign. That is, they claim that the expression 6 ÷ 2 × (1+2) = 9, but that the expression 6/2(1+2) = 1. That to me is equally bizarre.
I'm happy to agree to disagree here. I love that you're being thoughtful about this. That's the most important thing. And honestly, I think that we can both agree that the expressions (6/2)(1+2)=9 and 6/(2(1+2)) = 1 and, honestly, that's how people should always write this expression if they want one interpretation or the other.
Thanks again for watching my videos and for the lively discussion!
@@scholarsauce No, thank you sir.
@@scholarsauce please check 1÷a(b+c) expression in Wolfram Alpha
This is novel and seems to get to the crux of the matter. Please see the two videos by Jennie on some of these acronyms. I wrote a long comment on this subject and an aside as to the problem when it comes to computer programming languages, including the definition of a b, where the remainder is calculated. Check out the two calculator functions for Microsoft Windows.
I'll check them out. Thanks for the suggestion. I hope you'll enjoy our other videos too!
Great watch! I enjoyed the video a lot. It’s nice to take a low level break from all the PDEs and Eigenvectors for once 😂
Thanks a lot! I'm really glad you enjoyed it. Stick around tomorrow for our tau day special!
I prefer PEJMA, which is simply saying that Juxtaposition (implied multiplication) has priority. It's what all the engineers and scientists use. Here's an example: 6/2(1+2) would equal 6 / (2(1+2) in PEJMA, and (6/2)(1+2) in PEMA.
Thanks for the comment! I appreciate your point of view. I'm not sure if I agree that this is what all the engineers and scientists use though. Certainly, I'm having a hard time finding a single mathematician (including myself) that agrees with it, but I arguably have a small sample size, so it might be more widespread than I realize (and maybe you didn't include mathematicians when you said "scientists"). I know that it appears in some physics journals style guides, but CASs seem to be all over the place on how it's used, with some popular ones like Maple not even having juxtaposition as an option. Given that fact, I think there is must less consensus about this than you think.
To your example, I agree with your interpretation of what both methods do and I've always consider 6/2(1+2) to be interpreted in the later way. However, I would never personally write such an expression inline anyway as I always err on being clear. As such, I would write it whichever of the two ways you mentioned depending on which one I meant.
I will say that I think the juxtaposition having higher priority is an issue for me, because there is no good mathematical reason why you should do it. It's just a memorized rule that you do because someone said to, not because there is something mathematical that is driving you to do so. On the other hand the PEMA order is entirely derivable and natural from the definitions of the operations and is consistent with how we do arithmetic in more advanced settings like ring theory. I seen no advantage to confusing the issue with an arbitrary rule like the juxtaposition thing. At best, it obfuscates and blinds students to what's actually happening since it's an unnatural and arbitrary rule. As such, from a pedagogical perspective, I think it is an absolutely terrible idea to use it.
I hope none of that came off rude, it wasn't intended as such. I'm just trying to explain why I disagree with the PEJMA interpretation that many engineers and physicists use. I hope you enjoy some of my other content too!
@@scholarsauce Yeah of course, I understand where you're coming from. It seems that PEJMA is the common standard in research papers primarily. Though it is a more "international" convention. From what I've seen, the main case its not used in is Math courses, although my experience with that is admittedly limited. Thanks for sharing, it helped me see that the conversation was more nuanced than I realized at first glance
Thank you for making this explicit
You're very welcome and thanks for subscribing! I hope you enjoy my other content as much!
PEDMAS is for kids in single digits of age who have not done expansion and factorisation and algebra and substitution yet.
Nobody in university uses PEDMAS.
PEDMAS is not a rule, it's not a law, it's not a convention. PEDMAS is an acronym for little kids to help them out since they didn't do the other more advanced topics yet.
On the contrary, as a university math professor, I can affirm that we use PEMDAS at university all the time. You're right that we don't necessarily think of it in those terms, but PEMDAS or rather PEMA is the natural implication of how to correctly evaluate an expression via the definitions of the operations. Using the definitions of operations and the axioms governing those operations and then following the implications of those axioms is precisely how we do advanced mathematics like group and ring theory, or linear algebra, etc.
PEMA is a theorem, if you will, based on the axioms of the real numbers. As are things like factoring and substitution that you mention. All of them are consequences of how the operations are defined in the real number system.
You are right that if you follow those axioms and the definitions of the operations, you don't have to use a rule like PEMA directly, but your calculations will follow that rule anyway because that's what the operations imply. So, in the sense that you're not thinking of PEMA as a rule anymore, you're right; but in the sense that you still process a calculation via PEMA whether you realize it or not, you use that at all levels, including university.
PEMDAS is an acronym, a memory tool for the Order of Operations. Left to Right is a convention that works because of the basic rules and principles of math not in spite of them....
@@RS-fg5mf it's for kids in elementary or primary school who haven't learnt enough math yet. PEDMAS can be frequently wrong and it is important for school teachers to check to make sure that it gives the correct solutions for primary school.
@@avibhagan 🤣🤣🤣 give me an example of PEMDAS being wrong when dealing with Arithmetic...
@@avibhaganstill waiting on that example of PEMDAS being wrong when dealing with ARITHMETIC...
Thank you. Peace and Grace
The makers of these rules need to get on the same page, and come to an agreement in regards to this rule.
Do we go left to right...or not?
All mathematicians that I have ever seen teach this and every textbook that is in standard use today that I've seen all agree that we go left to right. As a mathematician myself (I have a PhD in math and teach as a math professor at a university), I think that the left to right interpretation is the only rule that is consistent with the definitions of the operations. However, as I pointed out in my video, the left to right rule is due to the fact that ÷ and - aren't actually operations but are shorthand for "multiply by the reciprocal of the item to its immediate right" and "add the negative of the value to its immediate right" respectively. If you resolve these first, then it doesn't matter.
As far as I can tell, the left to right rule is what's taught. The other stuff only gains traction because some experts in non-math fields don't want to write parentheses sometimes and so have added an arbitrary rule that juxtaposed multiplication has higher priority than division. But these rules aren't arbitrary, they're induced by the definitions of the operations. So, I find this notion that juxtaposed multiplication has higher priority incredibly bad practice that ignores the meaning of the operations.
I agree though, that we should come to an agreement so that we don't confuse the crap out of everyone else just to save some people from having to write a few parentheses.
Thanks for watching my videos. I hope you'll find the rest of my content enjoyable too!
3-5+4 is not 3-9. It is 3-1=2.
Quite right, I was explaining why it's important to do it left to right. If you don't, you can get it wrong like you said.
Processing it as I explained is correct in the video, it becomes
3 - 5 + 4 = 3 + (-5) + 4 = 2.
@@scholarsauce The way you explained it using +(-) and ×(inverse) is so easy sir. Beautiful. 👏👏👏😄😄😄
@@tshepomotau4354 Thank you! I'm really glad you liked the video and found it useful.
I hope you enjoy some of my other content too!
Why do you confuse the world ?
For Fun?
a(b+c) = (ab+ac). It is not to be translated as a x (b+c). When a number is touching the outside of the brackets, it becomes a very particular function, where the numbers inside the brackets must be multiplied by the number outside (always). If the person writing the expression had wanted to imply a x (c+d), it would have been written that way. This removes any ambiguity in extended equations. The problem is not the order of operations, the problem is the meaning of the function where a number is touching the brackets and why it was written that way in the first place.
I appreciate your point of view, but I respectfully disagree. The only reason why a(b+c) = (ab+ac) is because of the distributive property of multiplication. That is, because it means a × (b+c), which also equals ab+ac, by the way. To the mathematician (at least myself, or any other mathematician I've ever talked to about this), there is no difference between a(b+c) and a × (b+c). And indeed, there is no mathematical reason for there to be a difference between multiplication notated by juxtaposition and multiplication notated another way. Thinking juxtaposition should be different than regular multiplication without a good mathematical reason is the cause for the confusion, not the other way around. Juxtaposition is used exclusively at some point to denote multiplication period and it's never intended to suddenly mean something different. In other words, no one would write a×(b+c) regardless of what they meant after about calculus. But they don't suddenly stop meaning to communicate the same thing as that. Anyway, I would argue that if you wanted a(b+c) to mean (ab+ac) with the parentheses intact, you could have just written (ab+ac), with the parentheses, in the first place. which honestly would be a way of writing it that would avoid any confusion as well.
Please don't take any of that to be rude. It wasn't intended to be. Just a friendly disagreement. Thanks for taking the time to comment. I hope you enjoyed the video and will enjoy some of my other content as well.
@@scholarsauceThanks for your response, and I don't take it in any way badly. I enjoy these different takes on the subject. 2y could be rewritten as 2 X y and x/2y could written as x/2 X y , but it's not because convention says x/2y means x divided by 2y. That was a universal choice made as to how to understand the operation when using this symbol / and that became the convention. I was brought up in the Fifties in the UK and we were taught to carry out the function around the brackets first, before any other operation. That was because we understood at the time that the function had a special meaning, just like / . We believed that a(b+c) was the exact same as a X (b+c) in the special case when the operation was on it's own and not as part of a string of other operations. Just as ay could always be rewritten as a X y, but not if was the denominator as in the case mentioned when ay had to retain its ay shape to avoid ambiguity and could not be extended. As I said, it's a question of semantics which then becomes the convention. I understand that the convention changed in the Sixties (like many bad Sixties' ideas), but there is no reason why that should always be the convention.
@@aucourant9998 I understand where you're coming from and there are a lot of conflicting conventions around, which is what causes all this hullabaloo. I prefer whenever possible to use a convention that has a good mathematical reason for it rather than just some group's preference. If there is none to be had anywhere, then whatever convention is popular is fine. I think in this case though, using the convention that I portrayed in this video, I think personally is more mathematically motivated than the other conventions I've seen. Of course, that's just my opinion and I can understand why other people prefer theirs.
Thanks for watching my video and commenting. I really appreciate it! I hope that you'll find my other content interesting too!
@@scholarsauce Thanks. Yes, we'll also just have to disagree on what is the best mathematically as a logical convention. Thanks for the debate, I've a funny feeling this is just going to go on and on until my generation of mathematicians finally dies out.
@scholarsauce I agree with your rational. I find prioritizing multiplication by juxtaposition over other multiplication to be unintuitive. I see no reason it needs to have priority and it creates needless confusion.
Amen! brother. This is how I taught my kids. So here is the question: Is this also true for logarithms? Are logarithms "real" operations?
I'm glad that you liked the video!
To your question about logarithms, that's an interesting thought. On the one hand, it is the "inverse operation" to exponentiation, just like division is to multiplication, and subtraction is to addition. However, logarithms are not ever treated quite the same as the other binary operations like the addition, multiplication, and exponentiation. This is likely for at least two reasons.
First, there isn't really a great way to write a logarithm using infix notation (where the operation symbol is between the two numbers). I suppose you could do this, but its atypical. Like log_a b could be written a log b (or b log a depending on which side you want to define the base on) or something like that so that 2 log 8 is 3. If this were the only issue, then we probably would go ahead and figure out a way to include it in the order of operations. However, I think the next issue is the bigger problem.
Second, there is no way to express a logarithm using an exponent. That is, 2 log 8 can't be expressed as 2 to some power. This is unlike division by a number n, which can be expressed as multiplication by 1/n. It is also unlike subtraction by a number n, which can be expressed as addition of -n. I think this is really where the breakdown begins.
For at least the two reasons above (more the second than the first), logarithms with a particular base are treated as functions rather than an operation.
Furthermore, the relationship between exponentiation and multiplication compared with the relationship between multiplication and addition is actually a rather deep idea. And there are several issues that cause problems. For example, addition and multiplication are commutative, but exponentiation is not. In fact, addition and multiplication are associative, but exponentiation is not. One might ask then, why does iterated addition retain these properties, but iterated multiplication does not. It turns out that part of the reason is because you can get all the natural numbers by successive additions of the number 1. Thus under addition, the natural numbers are completely "generated" by the number 1. However, to generate all the natural numbers under multiplication, you would actually need 1 and all the prime numbers. So the natural numbers have a much more complicated relationship with multiplication than they do with addition. For reasons like this, even though it looks like exponentiation has the same relationship with multiplication that multiplication has with addition, it really doesn't because that relationship partially relies on how the operations behave with the numbers themselves. These questions, of course, are far more complicated than the typical level that students are at when they learn about logarithms.
Because of these additional complications, I think it unlikely for there to ever be a good way to treat logarithms as the inverse of exponentiation the way we treat division as the inverse of multiplication. So are logarithms "real" operations? I think I'd have to say no, but not for the same reason that I say it for division and subtraction, because instead of simply being shorthand of doing the operation on some kind of inverse number, their behavior is more function-like and just not quite the same kind of animal at all.
I don't know if that's a very satisfying answer, but ultimately I guess you could just say, "it's complicated." Fantastic question, though! I love the generalization and thinking about neat weird stuff. It's awesome and something I had never quite thought of like that before.
Anyway, thanks for watching my videos! I hope you'll subscribe and enjoy more of my content!
@@scholarsauce It was very well worded and easy to understand. I thank you for the lengthy reply. I suspected the answer was something along those lines, and that's how I taught it to my two kids when they were young, even though I wasn't as sure of myself as I should have been. You explained it thoroughly and I will pass it along to them.
Warning: nauseating anecdotal story follows. Read at your own risk.
My wife and I homeschooled our kids, one result of which was that neither of them even knew about subtraction or division until they were 8 or 9 years old, at which point we taught them to immediately convert them to addition and multiplication operations. So actually, they never really learned subtraction or division in the "normal" way, at all. We also taught them the more intuitive way to add and multiply numbers -- from left to right -- and only later showed them the shortcuts that are taught as the primary methods in most schools. This made apparent the intrinsic connection between multiplication and addition. Also, when they were young, we placed a great deal of emphasis on their memorizing prime numbers, squares of numbers, and practicing factoring skills until they could do it in there sleep by the time they were 12. My kids both enjoyed solving puzzles and doing worksheets so it wasn't the torture that it must sound like. We approached teaching math that way because of an event that changed my math outlook at age 13 (1979) when my middle school math teacher pulled me aside and explained math to me in a way he never did to the rest of the class -- things like, "subtraction and division don't exist." There was a lot for a 13 year old to unlearn, but it stuck and 27 years later I began passing those lessons on to my kids, one of whom is now skewing the grading curve at University of Florida. She is a true math wizard. Anyway, the only credit I get, for myself, is having just enough wisdom to notice the importance of the important lessons and the good sense to pass them on. Anyway, for teachers like you and like my middle-school math teacher, who make math make sense, I give thanks. Thank you.
@@sailor-rick Thanks for the kind words. I love how you made it work for your kids and passed on the insights you learned. It's great to hear stories like that. Thanks for sharing!
@@scholarsauce
1: If logarithm is a function, then is that also the case for trigonometry stuff like sin/tan/&c?
2: If logarithm is a function, then no wonder that I had trouble fitting logs into the Exponents/Indices set.
Probably due to confusing myself with trying to equate the difference between:
1: A log is the RESULT of a computation.
2: A log is the EXPONENT of a number.
@@Scott-i9v2s yes, log and sin, cos, tan are all functions. The log_a is the function that when given a number x yields the exponent on the base a that is required to get to x. It can also be thought of as the inverse function to the exponential function a^x.
So both your statements 1 & 2 are correct. They really just indicate two different viewpoints of the same phenomenon.
As a programmer I already knew this :P
But I think another problem with PEMDAS is not just the equivalencies of Multiplication / Division and Addition / Subtraction but also the juxtaposition shorthand in multiplication.
Most mathematicians, scientists, and engineers agree that this juxtaposition should have a higher priority than division. Some say that it should imply parentheses:
2x = (2 * x)
but this breaks down when you involve exponents
2x^2 = (2 * x)^2
which is wrong, this should really be (2 * (x^2)).
So the rule should really be that an extra step for juxtaposition. It should be:
Parentheses
Exponents
Juxtaposition
Multiplication (and Division)
Addition (and Subtraction)
However we can also still consider it parentheses and take the order of replacement for the shorthand as follows
2x^2 = 2(x^2) = (2 * (x^2))
Which is correct.
Either way we need juxtaposition to have a higher priority than division and a lower priority than exponents.
As an argument for why juxtaposition should be higher priority than division, lets look at the following expression:
1/2x
This is always intended to be read as
1
-----
2x
For the simple reason that if they mean
1
--- x
2
they would have written x/2
In order to avoid ambiguity and maintain consistency, the optimal solution is to include a juxtaposition step between exponents and multiplication, ie higher priority than M/D
I appreciate your point of view. However, I'm not so sure that "most mathematicians" agree with this. I'm a professional mathematician teaching at a university and none of my mathematician colleagues that I've asked so far have ever considered juxtaposition of higher priority than just normal multiplication. Nor had I ever heard of such a thing throughout my entire career until some of these RUclips videos. And I have always interpreted 1/2x to mean (1/2)x for the simple reason that if they meant 1/(2x) they would have used parentheses.
That's certainly not proof of anything, but I personally find the claim that most mathematicians think juxtaposition to have this property to be rather dubious and I'm pretty skeptical of it. Though it does sound like it's pretty popular in the physics and engineering crowd.
Ultimately, I think everyone should just be more clear and use more parentheses. No one should really ever write anything like 1/2x simply because of the ambiguity from one group to the next. I'll still always see it as (1/2)x, though.
Thanks for the great and thorough comment though! I hope you'll find some of my other content enjoyable too!
@@scholarsauce I think the key issue here is shorthand. Whether we included juxtaposition as a step in the order of operations depends upon how often we want to write (1/2)x vs 1/(2x) and not just in the case of 1/2x but other situations as well like 1/2√2 and not the expression alone but the context of the expression can also be incredibly important. I personally think this tips in favor of juxtaposition taking a higher priority to division, but we can agree to disagree.
What I DO think we agree on is this ambiguity needs to be resolved :P
@@nightfox6738 Absolutely!
@@scholarsauce Just wondering... Is 2y/2y equal to 1 or y² because it sounds like you're parsing it as 2(y/2)y
Finally, I offer this as undisputable proof that the subject expression evaluates to 1 and not 9. Here's the simple logic and mathematical predicate. There is no disagreement that any number divided by itself is 1. We can logically infer that it doesn't matter whether the number is in the numerator or denominator...if we were to reverse the numbers and put the other in its place it would still evaluate to 1. So if the evaluation of the subject expression is really "one" then we should be able to reverse the numerator and denominator in the expression to become 2(1+2) :- 6 which evaluates to 1. Now I know that some will argue that 2(1+2) as a whole is not the denominator (divisor) in the original expression. But in the reversed format it doesn't matter if the expression is written as 2 x (1+2)/6 meaning 2 times the quantity (1+2) over (divided by) 6 or 2/6 x (1+2) meaning 2 divided by 6 times the group. What say all?
Since, I'm already here, I'll respond to this too, but please, anyone, feel free to respond and contradict me or whatever. The discussion is great and is getting people thinking about math, which is awesome!
Your example here still interprets the 2×(1+2) as the denominator when you take the reciprocal. But the (1+2) isn't in the denominator of that fraction, it's in the numerator. Because 6/2(1+2) = 6÷2×(1+2), which means 6 × (1/2) × (1+2) (resolving the division shorthand first as it's not an operation), then the expression would correctly be written in fractional form as
6×(1+2)
_______
2
and hence only has a 2 in the denominator. Thus its reciprocal is
2
_______
6×(1+2)
which equals 1/9, as expected because the original expression equals 9.
But this runs into the thing we were talking about before about the reference you gave saying that multiplication always has priority over division, which is just totally not standard and very odd given the definition of division is "multiplying by the reciprocal of the number to its immediate right".
By the way, @Pete Palmere, I really think that you're awesome and love your passion and deep thinking about this. I may disagree, but I definitely admire your enthusiasm.
@@scholarsauce Thanks Scholar for the kudos. Just wondering why my college text authored by a Ph.D. mathematician and Professor of Management and Statistics School of Business Administration Old Dominion University has it as I state. I'll admit that the copyright on the book is 1969 so is that the way it was done back then and convention changed over time? This same book also explains how the 2 outside the group is a factor of (2x1 + 2x2) and why I find the whole of the expression as being in the denominator (the divisor). Should I ask my college for a refund on that Mathematics for Management and Finance course? Or am I being intransigent?
@@petepalmere210 I have no idea why his college text stated it that way. I can only tell you that I've never seen it taught that way in a book before. Perhaps the author had an order of operations argument with another mathematician once and wanted to take a stand when he wrote the book. Or it's possible that it's an artifact of an older era or even a group of people that felt really strongly about it similar to what the MindYourDecisions video was talking about. Like I said, my opinion is that it's pretty bad practice and frustrates a lot of what the operations are supposed to mean.
Did you happen to take the class from the author of the book?
As far as a refund, no I don't think you need to do that. No college education is perfect or without flaws and a lot of textbooks have some funny things, though this is more egregiously different than most that I've seen. But more than that, the important idea here is to understand what the operations should be doing and be thinking about good math communication. For example, while I have argued and would still argue that for consistency with the definitions of the operations, 6/2(1+2) must be equal to 9, were I to write that myself, I would try to be more explicit and write (6/2)(1+2) regardless. Just as I would hope that someone who wanted to communicate a 1 by that expression would write it 6/(2(1+2)). Being understood is more important than a stance on the order of operations.
I don't think that you're being intransigent either. You were taught something pretty explicit and you followed it. The fact that it happens to be pretty nonstandard is not your fault.
@@scholarsauce Thank you sir for your thoughtful and sensitive replies. I get what you are saying and I am sorry to strain the point but something is challenging my logic. If 2(1+2) :- 6 resolves to one no matter what order of operations one uses then it is simply and irrefutably true that we are dealing with the same value on both sides of the expression. How can we not make the further observation that the only way for the expression to calc to 1 is if they are the same number (value of 6)? So, it can logically follow that if they are the same number, and any number divided by itself is 1, then it doesn't matter whether the numbers are in the numerator or denominator (dividend and divisor if you prefer).🤔
@@petepalmere210 You're right that 2×(1+2) and 6 both equal 6 and hence 2×(1+2) ÷ 6 = 1. The problem is that 2×(1+2) ÷ 6 is not the reciprocal of 6÷2×(1+2). When you took the reciprocal of the original expression by swapping everything to the right of the ÷ with everything on the left, you are assuming that everything to the right of that ÷ is in the denominator. That's the exact interpretation that I'm refuting. The (1+2) is not in the denominator nor does the ÷ mean divide by everything to its right. Again, division is shorthand for "multiply by the reciprocal of the number to the immediate right of the division symbol". Notice I said, immediate right, meaning the one number of group (denoted by parentheses) to its right. In this latter case, the division symbol is "glued" to the 2, just like a negative symbol would be and so, only the 2 is in the denominator, but not the (1+2) because it isn't hit by that division sign, it has a multiplication sign to its immediate left. To be clearer, the prior case resolves to
2×(1+2) ÷ 6 = 2×(1+2)×(1/6)
or as a fraction
2×(1+2)
________
6
The numerator here is [2×(1+2)] and the denominator is 6 (and hence this value does equal 1 because 2×(1+2) = 6). The latter expression resolves to
6÷2×(1+2) = 6×(1/2)×(1+2)
At that point, the multiplications are commutative and we get that this is equal to 6×(1+2)×(1/2) and hence as a fraction, it becomes
6×(1+2)
________
2
Here the numerator is [6×(1+2)] and the denominator is 2 (and hence this value is 9 because 6×(1+2) =18). For them to be reciprocals, the denominator of one would have to be the numerator of the other and vice versa, which they aren't. So the fact that one of them equals 1 does not mean that the other one does because they aren't reciprocals to begin with. Does that make more sense what I'm saying?
Yup. "Left-to-right" is just an excuse to not teach math. PEMDAS as taught these days is a horrible hoax to perpetrate on grade school kids....
PEMDAS doesn't actually specify a strict "order of operations," as if it were a deterministic algorithm. It's a mnemonic to help you remember mathematical operator precedence.
In your example, 3 - 6 × 3 + 2³ ÷ (3 + 1), there is no reason why you must do what's in parentheses first, followed by the exponentiation, etc. In this case you could actually do MSEAPD and get the right answer, because that wouldn't violate operator precedence,
That's fair and I like the idea of operator precedence. Arithmetic can be done in any order as long as any "conflict" between two operators is evaluated according to the precedence of the operator. This gets to the heart of the point and even feels a little computer science-y. I think it's true too that most experienced mathematicians tend not to spend a lot of time thinking about the order of operations when they are actually computing because they understand in on a operator precedence level, like you said.
That said, PEMA (that is, PEMDAS, but you resolve the shorthand first) does produce a deterministic algorithm, but I agree that the fact that the algorithm works is an artefact of the operator precedence and not the other way around.
@@scholarsauce Yup. Order of operations is usually (at least on RUclips) presented as much more rigid than it really is. It's not an algorithm to blindly follow; it's a set of principles and relationships.
But in all fairness to educators, it depends on the level of the students. I doubt it is reasonable to expect young students, early beginners, to jump right into exercise of the full flexibility of OofO. Skills have to be developed, and starting with the simple version is fine.
I would take it a step further and use "GMA" as the reminder acronym. This is because, as you mentioned, exponentiation is really grouped multiplication. For example, 2^3 = (2*2*2). Thus we don't need to include the "E"; it is covered by the "G".
Non-integer exponents would be handled by treating exponentiation as a function (which it is), and functions are in effect a way to group. But no one ever introduces non-integer exponents along with basic order of operations.
So I would present order of operations as follows:
1) any expression that has only addition and/or subtraction is performed left to right
2) any expression that has only multiplication and/or division is performed left to right
3) if an expression mixes multiplication and/or division with addition and/or subtraction, then the M and D are done before the A and S
4) any notation that indicates grouping overrides the above rules; that is, sub-expressions within the grouping notation are evaluated as a separate unit
Of course, I would not present it to young students (beginners) using that kind of language. There would be examples along the way.
One would also have to recognize more grouping notatons than just parentheses. Such as absolute value signs, the radical, and the horizontal line for division.
Beyond all that, order of operations is usually presented as more rigid than it really is. Given
2*3*4(1+2)
I do not have to add the 1+2 first. I can multiply 2*3*4 first, then add the 1+2, then do the final multiplication. Or given
5-2+3-4+10-2
I could first add all the positives (5+3+10=18), then add all the negatives (2+4+2=8), then subtracting the latter from the former (18-8=10). This of course is just using the associative property, but the point is, order of operations is not as rigid as usually presented.
Finally, I'd like to mention that the acronyms do not define the order. They serve only as reminders, to help the beginners remember. Calling "PEMDAS" a "lie" is not really fair. And if we were to take the acronym literally, how distressing it must be when we first learn that some places in the world use "BODMAS" for their reminder.
All good points. Thanks for taking the time to comment!
And multiplication is grouped addition, so we can get rid of the M too, right?
@@Mesa_Mike Perhaps. But I suspect it would be more trouble than it's worth.
For example, 3.4 * 4.2. How do we add 3.4 to itself 4.2 times?? And I've seen people wrap themselves into knots trying to decide if 3*4 is (3+3+3+3) or (4+4+4). Or maybe 3*4 is (4+4+4) and 4*3 is (3+3+3+3). Sheesh. This gets ridiculous. Oh, and some places in the world they say "of" instead of "times". That is 3*4 is spoken "3 of 4". That may come from envisioning 3*4 as (4+4+4). I don't know.
The problem with this is when you have something that comes between exponents and multiplication. for example should 2*x^2 be 2*x*x or 2*x*2*x? You need an order for when you apply the x^2 = x * x transformation otherwise things will get hairy again.
Thank u
I have been having this conversation for over a week on Facebook. According to you, I am right. According to two out of fifteen RUclips videos, I am wrong. The equation in question is 8/2(2+2).
Apologies, my tablet apparently doesn’t have a division sign.
Thanks for commenting. Below is my opinion about your expression.
8 / 2 (2+2) means 8 ÷ 2 × (2+2) and hence should be interpreted as 8 × (1/2) × (2+2) and resolved as 8 × (1/2) × 4 = 16 (no matter how you do it at that point). The issue with this one is that a lot of people will write 8 / 2(2+2) when they mean 8 / (2×(2+2)) (notice the parentheses around the denominator) and hence want the the thing to equal 1. In my personal opinion, to not put the parentheses around the entire denominator of a fraction when written inline is really lazy notation and is based on people wanting to cut corners when they write it rather than accurately communicating what they mean. It is NOT taught by any mathematician I've ever seen nor is there any unwritten rule that this is an appropriate way of implying that there is a parentheses around the denominator. In fact, to do so, necessarily introduces ambiguity into what is being written, which defeats the entire point of notation. The point of notation is to communicate precision clearly. Not putting a parentheses around a denominator when written inline like this example is asking for people to misunderstand it and hence fails to communicate clearly. To me, that makes it really bad practice.
The only interpretation of 8/2(2+2) as written that is fully consistent with the definitions of all the symbols used is 16.
@@scholarsauce Thank you. I am, in fact, right. 😂
Many people claimed that the answer is 1 and I understand why. I do these in my head and order of operations is not something I usually consider unless the problem is larger. Also, I'm a Dog Groomer from Alabama so, Algebra really isn't an issue for me. I even reverse engineered the equation for an answer of 1
and I came up with 8÷[2(2+2)]
but that didn't matter to anyone. People said I was too focused on it. Apparently, they don't understand that the fundamental basis for everything we have ever known about Science depends on Math. It does matter to this Redneck Dog Groomer. 😂
Keep fighting the good fight friend. ♥
@@scholarsauce If I would write the equation and I would like the answer to be 1 and not 16, I could also write it as 8/[2(2+2)] = 1, right?
@@nats50 Yes, absolutely.
@@nats50 or it could be written
8
---------------‐
2(2+2)
For an answer of one.
One of the best explanations of how to solve an expression. This video puts to bed the viral 6 :- 2(1+2) video where many could not agree on the solution. There were basically two camps: one that had a rigid adherence to PEMDAS and a certain order of operations arriving at an answer of 9 while others used multiplication by the reciprocal to get an answer of 1.
I'm really glad you enjoyed it. We will be releasing some new content soon, so please stay tuned!
Rigid adherence to PEMDAS (without the M&D, A&S equivalency plus "left-to-right" rule) may account for some of the controversy, but it seems to me that the main issue has to do with whether the implied multiplication has a higher precedence than the explicit division.
PEMDAS is too simplistic to resolve this issue.
Really. the expression 6÷2(1+2) is malformed and ambiguous, and should be sent back for clarification.
@@Mesa_Mike I completely agree that 6/2(1+2) is malformed. I have long thought that the issue is not order of operations, but interpretation of notation. So I also tend to agree with your comments on the source of the controversy, though I have a somewhat different perspective.
Rather than focusing on "implied multiplication", I focus on the concept of grouping. Algebraic notation, like "2x" is clearly interpreted as grouped. But there's great disagreement about whether arithmetic notation like "2(3)" is grouped.
I don't really think 2x is treated as grouped because it is "implied multiplication" or "multiplication by juxtaposition." I think the treatment as grouped, as a unit, came to be in the early years of the development of algebra, long before anybody named it "implied" or "juxtaposition." It's simply how we have all agreed to interpret it. Expressions like 2(3) don't have that same history, resulting in present-day disagreement.
So, although I have a personal preference for how to interpret 2(3), the fact is there is much disagreement, so much so that 6/2(1+2) must be considered malformed.
Well, let's see. The video says that division means "multiply by the reciprocal of the number immediately to my right." The number immediately to the right of 6 is 2, so we would resolve the division and get 6*(1/2)(1+2). So I suppose one could say that this video puts to bed the viral 6/2(1+2) expression -- in favor of 9. Unless one believes that, in arithmetic, multiplication WITHOUT a multiplication sign comes before multiplication WITH a multiplication sign. But there's nothing in the video to support that notion.
(Sorry, Dr. Scholar Sauce, for polluting your video with *that* debate...I seem to be suffering from a lack of self-control this evening.)
@@donmacqueen Good Point. I was wondering if using "number to the right" was just a misnomer, and if he really meant to say the value to the right. This goes to the heart of the controversy because if the 2(1+2) is seen as a discrete value then multiplication by the reciprocal would give an answer of 1. It is easier and more logical to see it as an operation in itself that should be resolved first. One reason I have for this view is that the expression 2(1+2) to me is just a factor of (2 x1 + 2 x2). Even PEMDAS would have to solve the non-factored version to 1 since the grouping would have to be done first. Also troubling is if the 2 is put after the grouping which when evaluated by itself would equal 6 but with an answer of the viral expression as 1, but according to PEMDAS we would arrive at an answer of 4.
You asked for an acronym. If you call parentheses by the English term brackets then you get the acronym BEMA.
Something I see all over Facebook is they pemdas makes people think that 8/8(2+2) the muntiplication with the 8 and parentheses is after division just because pemdas don't have any way to telly you that the implied multiplication with parentheses that most people just don't get
Parenthetical Implicit Multiplication does not have priority over Division despite the false and misleading information, subjective opinions and willful ignorance people have about parenthetical implicit multiplication....
8/8(2+2)= 8×(1/8)(2+2) NOW we have all multiplications as stated in this video... The correct answer is 4 not 1/4
@@RS-fg5mf yes every paper i have read on the subject solev it as follows... that means parenthses and juxta position before the division so 8/8(2+2) is 4...
8/8(2+2)
8/(16+16)
8/32
4
@@MautreXvids You contradict yourself...
Yes, 8/8(2+2)=4 ABSOLUTELY
8/8(2+2) does not equal 8/(16+16)
8/8(2+2) does not equal 8/32 or 1/4
For me this is weird . To say that 5 ➗ 10 ✖ 2 is not the same with 5➗ 2✖ 10 and is not the same with 5 / 10 ✖ 2 but this last one is the same with 5/ 2 ✖ 10 is strange.
I agree. Personally, I would say that the 1st & 3rd are the same (and equal to 1) and 2nd & 4th are the same (and equal to 25).
@@scholarsauce Did you deleted my response or i pressed the wrong button?
@@cosminu.4519 Sorry, I didn't delete anything.
@@scholarsauce OK, let's try this again. I agree with everything you said, but I don't think that is the issue. I am saying that the number after division is 10×2, not 10. Because the number after division is 10 × 2 we can write it as 2 × 10. So 5 ÷ 10 × 2 is the same as 5 ÷ 2 × 10 which is the same as 5 / 10 × 2 which is the same with 5 / 2 × 10.
Consider that we want to multiply this expression with 1 which should not change the result. 5 ÷ 10 × 2 × 1 This is OK. But if I put the 1 before 10
5 ÷ 1 × 10 × 2 according to you, we get a different result. Saying that multiplying or dividing by 1 an expression gives you different results is just weird.
Consider that we want to write 10 as 5 × 2 then we have 5 ÷ 5 × 2 × 2 we also get a different result. How about if we write 10 as 2 × 5 again we get a different result.
All expressions i presented are the same just written differently and are equal to 1/4 according rules that i have been taught in school.
Consider other cases like 6 ÷ 2 × 3. Because I am saying that 6 ÷ 2 × 3
is the same as 6 / 2 × 3 we can write it as 2 × 3 ÷ 2 × 3, which is the same as 2 × 3 / 2 × 3. I am pointing out that you have a lot of freedom to simplify an expression on the numerator side, not only on the denominator.
Using your rule, you would have to add parenthesis in different places depending on each case. But this is also weird ,why not add parenthesis when you want a division before multiplication so 5 ÷ 10 × 2 becomes,
(5 ÷ 10) × 2 .
In conclusion i have to say that i come across this pemdas , bomdas pemjas whatever ,recently and looks like this is a USA problem. Everywhere else in the world these expressions are solved in the way I showed which is the way i have been taught , of course me being from Europe not USA.
@@cosminu.4519what you are talking about is the Commutative Property and what you FAIL to understand is that the Commutative Property allows you to move multiplication around ANYWHERE within a TERM as long as you do not affect the denominator of a division operation....
5÷10×2 = 2×5÷10 = 5×2÷10 = 2÷10×5
The division symbol belongs to the 10 not the 10×2. The denominator is 10 NOT 10×2
5÷(10×2) = 5÷(2×10) parentheses required.