Dividing by Zero in Five Levels -- Elementary to Math Major
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- Опубликовано: 25 янв 2024
- Why can't we divide by 0? This is a common issue we encounter in elementary school when we learn to divide, but it persists through high school and even into math major courses. Let's explore why we can't (usually) divide by 0 in 5 levels, covering a range of these topics.
Level 1: Don't do it
Level 2: Still don't do it
Level 3: Just don't do it
Level 4: Never ever do it.
Level 5: Go on then, do it. Look! See! You broke something.
This comment deserves to win the internet for a day.
Level 6: We can actually do it, kind of (Riemann sphere)
Level 7: Black Hole forms
@beepbop6697 That only happens when God divides by zero.
@@soyokou.2810 Only if you take the engineering concept that 1/infinty = 0 (which, of course it is not)
*Differential Calculus* - _the art of how to sneak up on a divide by zero without triggering the alarm system_
Differential calculus plays with fire but never gets burnt lol
Division by 0... Hmm let's see, repeated subtraction indefinitely... vertical slope... oh yeah, tangent of +/- 90 degrees... and last but not least multiplying by +/- i. It's all the same. It's not undefined. It's just ambiguous. It tends towards +/- infinity.
@@skilz8098 in math, if something is ambiguous, it has not a defined value. Thus it is undefined.
@@User-jr7vf That is not the definition of ambiguous. Something that is ambiguous has multiple definitions and it can not easily be determined. We can not determine which definition is the appropriate. That's not the same as being undefined. Saying that something is undefined is stating that we are removing its definition or it doesn't have a definition.
@@User-jr7vf Look up some compiler errors for C/C++ for a function or object that is undefined and compare it to an error where there is an ambiguous function call. They are not the same thing.
My Calc 1&2 Prof once said that multiplying with zero makes everything zero, so it makes all information about a system disappear into nothing. In reverse if we could divide by zero and get a usefull answer, we could extract information about a system without any information.
I think thats a very nice philosophic approach to why it doesnt work.
In a completely unrigorous, personal, intuitive sense, having a mystical experience is like dividing by zero and getting an answer. It's like extracting information about reality when reality is a system which has no information content. The answers are useless since they can't be shared, only hinted at, but at least one feels like one has an answer to a question that was haunting them.
@@satanic_rosa would you be willing to share yours? I can mine, if you wish.
🤯My mind is blown
That's actually a reasonably useful description for why the concept of dividing by zero isn't useful for modeling physical systems -- that's not strictly a math question, though, but a question about what math is useful for modeling what things. There _are_ some people who take the ontological position that math is the ultimate reality and reality itself is defined by math (and we just have to identify the math that reality runs on), but that doctrine is _also_ not math.
depends on if we’re talking countable or uncountable infinity. Technically 0*(uncountable infinity) can equal a finite value greater than zero.
To maybe provide a concrete example, one can think on how the (Riemann) integral of a function is really just the sum of uncountably infinite many rectangles with base of length 0. If the function we’re dealing which is a constant (nonzero), then the integral become c*0*(uncountable infinity) where c denotes the value of said constant function.
“How do you divide 6 cookies with zero friends”
That one definitely has to hurt
7 year old's existential crisis... trust me, it only gets worse from here kid!
😂
Simple. I get all the cookies.
@@kwesisalim But you have no friends. You don't even like yourself. Why would you give yourself any cookies. You don't deserve any cookies.
You suck! /s
I feel bad 😢
I went from watching basketball highlights to this
Me to bro
Ball don't lie.
Congrats on your ADHD diagnosis
I went from football highlights to this
Yea last video I watched was "eating food the wrong way" to this
Level 6: Riemann sphere
Level 7: Wheel theory
Level 8: ???
God
Level 9: Profit
Level 8: Surreal Numbers ;)
@@ZeDlinG67 You can't divide by 0 in the surreal numbers. It's a field
Right. Silly me. The epsilons confused me :D
Here is level 6, graduate algebra course: You actually can divide by zero within the ring that contains only one element, because there zero is equal to one.
In fact it's a classification! In a unital ring, 0 = 1 if and only if it is the zero ring (up to isomorphism).
*their
@@M42-Orion-Nebula No, they mean "there" as in "there in the ring", not refering to the ring as possessing a zero equal to one.
@@paulfoss5385 My bad, but they should have separated it with a comma.
@@M42-Orion-Nebula Why?
What I love about the abstract algebra explanation is that, without the jargon "field", "group", etc, the explanation boils back down to a pre-calculus level of mathematics. This is the first insight for understanding that mathematicians don't use new words to talk amongst themselves as the elite, but instead define precise notions and eventually clean up what they've learned and teach it to others at a more basic level.
What I like about this video is that this is the first time I got an intuitive intro to groups and fields and how they relate. Especially once I saw the parallels to numeral systems that I've had intuition for, e.g. that the identity element is just like the radix of a numeral system and that the entire issue with "dividing by zero" is like coming back to a singularity that is simply required by a mathematical concept for the *rest* of the concept to make sense in the first place.
At least recently.
Mathematicians have put effort into formulating everything axiomatically from the same foundations, so we know exactly what we're talking about.
I'm a big fan of teaching maths like that, because it completely prevents meaningless disagreements over nothing.
Division by 0 is not undefined... it's just ambiguous. I've never prefered the teaching of "undefined"... ambiguous seems more appropriate to me.
@@skilz8098 undefined is a mathematical term which means "has no definition". Ambiguous has no mathematical meaning and is not suitable for a label of x/0
@@insising Ambiguous has a mathematical meaning. For example, the regular binary expression (1* 0*)* is ambiguous. Being ambiguous is sort of analogous to a function being non-injective, or worse, not well-defined. I agree that division by zero is undefined and not ambiguous, by the way.
I love how all these explanations are really just more careful and rigorous ways to explain that intuition we had all the way at level 1.
Study maths for long enough and you get to appreciate all the careful and rigorous ways that we have to explain the counter-intuitive.
Proof by Induction still feels wrong to me. I don't think I'll ever get over that, regardless of how much I might use it.
@@RichWoods23 the intuition I find most helpful is just "what if we did it again?" And repeated that process. At every point along the way the relationship holds, so we can talk about all objects for which that relationship applies. Unfortunately, to talk about infinite processes we need the axiom of infinity, which says that there are sets with relations that hold "infinitely" like the natural numbers, so it works because we say so, but up to that point, you can pretty safely define a relationship to hold for as long as you want.
thats bc he missed the point in the end. there actually is a satisfying answer.
I can’t lie my intuition always steers me towards division by zero equaling infinity because it approaches infinity. And the explanations for why it’s not infinity always seem to loop back to because you can’t divide by zero in the first place which isn’t that helpful. Ironically the best explanation for why it’s not infinity (for me) was not anything he said but showing the graph. And looking at it, it reminded me it’d have to equal positive infinity and negative infinity at the same time which is completely nonsensical
@@monhi64 There's also the issue that, if you assume dividing by zero equals infinity, then you can say
1/0 = infinity, therefore 0*infinity = 1
2/0 = infinity, therefore 0*infinity = 2
therefore 1=2
This logic leads to a contradiction, so you need some step in the process to be invalid, and the one most people choose is dividing by zero. I suppose it may be possible to come up with other axioms that allow dividing by zero, but the resulting number system would have to be different by necessity.
"How do you split 6 cookies evenly among 0 friends?"
This doesn't work for explaining to an elementary student because the answer is obviously "I eat them myself".
Are you a friend with yourself?
@@sikeimmike If you want to have good mental health, you certainly should be, lol
I've always interpreted the question of splitting 6 cookies evenly among 0 friends to mean something along the lines of you've just finished baking a batch of raisin cookies, but nobody wants any of them, and you don't like raisins. Who do you give the cookies to?
Then you've worded the question poorly. The question shouldn't be "HOW do you split 6 cookies among 0 friends?" but rather "if you want to split them, how many does each friend get?"
If the child is unable or unwilling to imagine a situation in which they would want to part with cookies in the first place, use pens or some other object instead.
How do you divide 6 chores up among 0 friends? @@corvus_da
My go-to for explaining it is #3. Since division is just undoing multiplication, 0x can only equal 0 so if 0x = y where y is not 0 it's an impossibility. And if 0x = 0, then x could be any value at all, so it's still undefined as to what x is.
But 0*Inf is not necessarily 0...
*Indeterminate form left the chat*
I don't really understand how they got that equation. How did you move the "0" from the right side to left in "x = 6/0"? By multiplying both sides by zero? Well then you'd have "0x = 0", not "0x = 6" as an equation. Don't know if I missed something.
@@ProferkZeros cancel out
@@Proferk In Algrebra we have that a * 1 = a, Then we also have a / a = 1, And ab/a = b * a/a. So if we combine these three and we have ab/a.
Then we get ab/a = b * a/a = b * 1 = b
So in case of 6/0, a = 0 and b = 6. multiplying by 0 gives us 6*0/0 = 6*1 = 6
Him merely mentioning “friends” in a dividing by 0 video made me anxious and nervous. for it was true.
Awww
Level 0: model division as repeated subtraction.
If a pie has 10 pieces and you keep taking away 2 pieces, how long can you do this? 5 times. Or, 10 / 2 = 5.
So if you keep taking away zero pieces, how long can you do this? You can do this operation forever.
Or 10/0 = Infinity.
This has a more intuitive feel, because if you 'never take anything away' the operation will 'never finish'
3:54 to be fair, the cookie-friend analogy breaks down much earlier than that. 6/0.5 for example. how do you divide 6 cookies with half of a friend? by giving each friend 12 cookies? it doesn’t make any sense either, yet as we all know, 6/0.5=12 is very well established.
I know some people that are not full friends. Its true actually, that they always try to get extra cookies. Needy bastards.
@@deltalima6703 lol, touché
if each half friend gets 6 cookies, then each whole friend should get 12 for sure. You don't have to have the cookies to decide how many a person should get.
@@zachansen8293 you can’t have half a person. that’s my point. if you’re trying to explain to a 5 year old how division works, you can’t explain dividing by a fraction using that analogy. it’s only a useful analogy if you have a number being divided by an integer. each half of a friend can’t get a cookie because it doesn’t make sense for there to be half of a friend. of course, you solve this problem by learning how algebra works but we aren’t teaching 5 year olds algebra.
edit: i made a mistake by saying it only works for integers. as others have pointed out, it’s even more limited than that, as it only works for natural numbers.
i can image some crazy scientist creating some dozens of 1/100 clones to divide a golden ring by 0.01 and get 100 for it. makes perfect sense.
I just wanted to say, division by zero does indeed exist the null ring if anybody is interested :)
Also in the extended complex plane, the positive and negative infinities map to a singular infinity on the North Pole of the Riemann sphere, and so there dividing by 0 will give infinity, which is particularly useful in Möbius Maps :))
Also, wheel algebra defines an element called the nullity element which is kind of like a ‘void’. In this case, we define any indeterminate form (a/0, 0/0, 0^0 etc…) to be equal to the nullity (¥ say), which u can think of as being “more powerful” than 0 and infinity combined. Any operation with ¥ results in ¥, e.g. 1/0 = ¥, 0/¥ = ¥ etc… :))))
Is this really a thing???
@@stinger4712 Yes, many math concept come from someone noticing something that shouldn't exist would make others things simpler (and several more complicated) if it existed, so they go ahead, say it exists, and see what happens.
Case in point, the square root of -1 doesn't make sense intuitively, but defining it as existing, calling that object "i", and seeing what happens, worked quite well, and was quickly associated with rotations.
Ditto for division by zero. In some very specific contexts having it exist would be useful, so in those contexts it was defined as existing and having such and such properties. But when one does that other things break, so it's very situational.
Wait, do you mean Weyl algebra? The only time I’ve heard “wheel” algebra is in the context of modular arithmetic
Well, yeah... But these are not real numbers anymore. In fact, these are not even fields.
Division by 0 might be defined in some other contexts but we're not talking about the same thing anymore... It's like saying that 1 + 1 can equal 0 just because there is a field where that's true. I think it's nice to mention how these things can happen but also we should be careful not to make people confused about what we mean when we say stuff like that (?
When I learned economics at community college, that’s when I finally understood why you can’t divide by zero.
So there’s a concept in economics called elasticity of demand. Basically it means how much the quantity-demanded of a good or service changes in response to a price change. For example, when Netflix raises its prices, people cancel their subscriptions because Netflix is not a necessity. When the price went up, quantity-demanded drops by a lot. So in economics terms, Netflix has a high elasticity. On the flip side, if something like water goes up in price, the quantity-demanded doesn’t go down much, because people need water to live. In economics terms, it has a low elasticity, or that it’s inelastic.
After learning all that, I thought about what would happen if something were perfectly inelastic. When you graph it, the line would go straight up and down. It would have a slope of x/0. This would mean no matter how high the price got, the quantity demanded would not change.
Then I thought, what kinds of things behave like this? Stuff like food and water. No matter how expensive food and water get, people still need it. But what happens when it gets to expensive that no one can afford it? The people starve.
So in economics, when you divide by zero, people starve. Kinda morbid, but that’s how I understood division by zero.
Thank you for sharing your economics knowledge. I'm sure that information could be helpful at some point, at least in my geography lessons:)
Though I think the comparison doesn't fit that well. x/0 should not make sense in any given context. If people starve at a/x it would make sense.
Furthermore I don't get how a a/x function could represent the quantity-demand of an inelastic priced product. if the X-Axis is the quantity demand, then why should the price at a quantity-demand of zero approach infinity?
If the X-Axis is the price, then why should the quantity-demand at a price of zero approach infinity? I mean that would make sense if you would't assume that the quantity is 100% inelastic.
What actually happens is that you move from economics to history and political science. People still die, but they don't always starve; they get killed in the riots or revolutions.
I learned that in my AP Microeconomics class, it's a decent way to think about it.
@@jdotoz So x/0 = eat the rich?
@@jonathanolson978 If X is a non-zero quantity of surplus necessities and 0 is the number of people they are being distributed to, yes, that does seem to be the real result.
Sean taught one of my math classes during my freshman year of college. By far one of the BEST teachers I’ve ever had and helped me enjoy math, which is a subject I usually struggle with.
I think a lot of the "it doesn't make sense" points in the earlier levels were really "this one way of describing makes it hard to talk about it, and we'll ignore the many obvious other ways that make it simple to understand". For example, you pose the game of cookies interpretation as x/y with x = # of cookies and y = # of friends so x/y is # of cookies per friend. This is an intuitive model, but it doesn't really describe the counting game that is being done very well so it makes sense that it might be confusing to know what to do with 0 friends. But instead, if you interpreted x/y with the model x = # of cookies and y = # of cookies given to each friend, then x/y is # of friends who get cookies and you have a simple model of how to calculate - you give y cookies to friends one at a time until you run out of cookies. If you have 6 cookies and you want to give each friend 6 cookies, then only 1 friend will get cookies. If you want to give each 1 cookie, then suddenly that counting game will continue until you have 6 friends with cookies. Oh! And look! Now it makes absolute sense what dividing by a half is! You want to give each friend half a cookie? Then twelve friends will get (some) cookie. And 0 here makes sense too. If you keep giving friends 0 cookies, you will never run out. So it is an unending process, or what some might call infinity or an infinite process. It's not meaningless or absurd, though, and absolutely "makes sense".
Notice also that this alternate interpretation also gives some explanation for the many strange ideas around 0/0. Because you could give no friends 0 cookies and you've already exhausted your supply. But also you could 1 friend 0 cookies, and you have the same 0 left over. Or 2 friends. Or a million. All of those answers "make sense" in this model, and help to build intuition about the higher level reasoning around these ratios. 0/0 could reasonably be interpreted as any number.
When you build counting games to explain number concepts, you want to make sure they bring clarity to the higher level understandings. When you use a model that just "doesn't make sense", that doesn't prepare the students to move to higher understandings and tends to be more of an obstacle that can turn kids off math.
Thanks, that is a new and refreshing approach! 😊
ah, you make a very valid argument, and give an excellent alternative!
My fave came from James Grime. Since division is just iterated subtraction we can count how many times 0 can be subtracted from x in this pattern for 6/2:
6-2=4 (one 2)
4-2=2 (two 2s)
2-2=0, for a total of three 2s in 6, with no remainders.
Now for 6/0:
6-0=6 ─ one 0
6-0=6 ─ two 0s, but hang on. We haven't decreased the 6 yet, and never will no matter how often we remove 0 from 6. You can never whittle 6 down to 0 by subtracting 0 from it.
I feel like this is as much a “proof” that dividing by zero results in infinity, as it is a proof that it is undefined
@@mattsains For infinity to be an answer to the problem infinite zeros would need to add up to 6. Infinite zeros only add up to zero, never any other value.
@@mattsains dividing by zero doesnt really result in infinity tho, at least over the reals. for one thing, infinity just isnt a real number. beyond that, we could make just as good of a case for negative infinity as for infinity, so thats a contradiction. however, there are systems where we actually do get infinity, for example there is the one point compactification of the reals(there is also a two point compactification), which is essentially just the real numbers with the additional definition that 1/0 = infinity, with infinity being treated as a number.
@@danielmagee8637 I never said it did, I was just pointing out that the “proof” here isn’t very compelling
@@danielmagee8637 would it be sufficient for a formal proof to use induction over the subtraction, to show that x/0 is undefined?
After taking some abstract algebra and analysis the way I see it is that for most sets of numbers defining division by zero is impossible without losing some structure in the process which leads you to now not being able to do some other things. You cannot divide your cake by zero and eat it too.
Defining things is like signing a contract. You promise to follow some rules for something and it turns out, defining zero often isn't worth it.
3:40 thank you for saying that the limit as x approaches infinity when x is the denominator is NOT the same as dividing by zero. Indeterminant and undefined are different, also and 6/0 is never indeterminant.
Intel-defined math processors actually allow it, do weird stuff, make distinctions between +0 and -0, and happily tell you that the answer is positive or negative infinity, if you flip some control bits.
As an engineer, I never had any use for this weird trickery and you should by all means avoid ever landing in a situation where you would try make that division. Mask it - ignore it - throw an exception to a higher level - skip it - or best: work consciously around it.
And 0/0, whose result is known as not-a-number (NaN), is even worse - that last one can literally "poison" a whole chain of calculations if you're not careful.
Very often, if you come to divide 0 by 0, in actuality, the result is unimportant to further calculations in which case you're best off substituting 0 for the answer, as that value will be subsequently ignored and the en dresult will be valid for what you're trying to do.
somebody who really wanted to divide by zero got wise here: "but what if it was just real close to zero?" boom we got limits :)
My preferred explanation is practical calculus. Look at speed. It's the rate of change of distance over time. It's a derivative. If distance is non-zero, but time is zero, speed must be infinite. The only way to be in 2 places separated by a distance, at the same time, is to be moving infinitely fast. By the same token, distance over speed equals time. 100 km divided by 50 km/h equals 2 h. So if you want to travel 1 km, at a speed of 0 km/h, how long does it take to cover the distance? Infinite time. Now, infinity can't be defined as a number, which is why whether you say undefined or infinity, both fit.
No, mathematically, x/0 cannot be infinity, because that means infinity * 0 = x. 1/0 = infinity means that infinity * 0 = 1, which obviously is not the case. 0 isn’t negative or positive either, so you would also be implying that it is equal to infinity and negative infinity at the same time.
When saying the 6/3 = 2, you are saying the 3 goes into 6 2 times. Similarly, when saying 1/0 = infinity, you are saying that 0 goes into 1 an infinite amount of times. If you do 0 + 0 + 0… so on, you won’t get to 1, ever. Your example is saying the same thing. You are saying that with infinite time at 0, km/h, you will travel 1 km. Even if you have infinite time, going 0 km/h means you aren’t moving. You won’t ever make it to 1km even if you had infinite time. Unless someone makes another imaginary number, this is undefined. It is not 0, it is not 1, and it is not infinity.
For the graph of dividing by zero going off to positive and negative infinity, I always thought of it as the curve wrapping around the universe and coming back from the opposite side.
I heard that exact opinion in a math video. It was off handed but the mathematician was hand waving dividing by zero and saying it might force the positive Y axis curve all the way back to the negative Y axis
I remember seeing a video where a Frieden mechanical calculator was being demonstrated without its covers on. When dividing by zero, it would just spin its cogs because there was nothing to subtract. It really drove home the point.
What about in projective geometry or the Riemann sphere, where division by 0 is defined (as long as it’s not 0 divided by 0)?
Execly
@@gabrielgabi543Didn't expect you to like math...
How does it work? 1/0 is equal to what?
Yeah, I was going to mention that too. When you extend the Reals with an "infinity" element and define division by zero as infinity (no distinction between negative and positive infinity), you get a lot of nice symmetries. Infinity is kind of like the counterpart to zero/the origin. Now every line has a defined slope and every pair of lines has an intersection point (even parallel lines).
But this is at the cost of making a whole bunch of other operations undefined. For anyone curious about the details, look up the wikipedia page for protectively extended real line and Riemann sphere.
@@davyx_max831 Its infinity as -infinity is the same in pga
I should at least be able to divide 0 by 0
@@tt3925from what i gather, it would cause paradoxes.
one explanation (algebra): for a*b = c*d and therefore a/d = c/b if c=b=0 then a/d = 0/0 when a,d = R (C,Q etc.). we simply cannot tell what 0/0 equals that's why it is indeterminate. it could be 8/11, -1113/π+i³ and that would be equal to 0/0
0/0 would actually make sense, but it's just not useful to math I guess. Division is just repeated subtraction, so how many times do you need to subtract 0 from 0 to get 0? 0 times. The logic works, so the answer should be 0/0=0 but since 0 is the term that gets rid of other terms, I guess it gets really weird when you actually try to start doing anything meaningful with that fact. See: proofs that 1= 0 or 1=2 when division by 0 is allowed to happen. I don't remember exactly why, but it breaks math for 0/0 to be defined
This is the first of your videos I watched and I have been following you ever since. Great work you’re doing, explaining several fascinating concepts and often even showing a graspable proof. I think this channel has great potential and I can’t wait to see where this goes. Keep going!
The best explanation I've come across is considering the function f(x)=0•x
If the reciprocal of 0 exists, then we can find the inverse function of f(x), which is f-¹(x), and it would undo the multiplication by 0. And for a function to have an inverse it has to be bijective. Yet we can clearly see that our function isn't an injection in the first place as we have f(5)=0=f(4) for example yet 5≠4. Which means that f-¹ doesn't exist, and so is the reciprocal of 0.
It is worth noting that there are a few areas in math where dividing by 0 makes sense, like Wheel Theory.
As for another way to think about it within just the Real Numbers, let's do some Analysis. Suppose that we can divide by 0. Then, 1/0 exists. Take the sequence {1/(1/n)}, with n being a natural number. On one hand, as n approaches infinity, 1/(1/n) tends towards 1/0, which exists by our assumption. However, 1/(1/n) = n, which the limit of that sequence doesn't exist. As a sequence cannot both converge and diverge, we get a contradiction. Thus, 1/0 doesn't exist, so we couldn't divide by 0 in the first place.
This is brilliant!
But can't it just mean 1/0 is infinity?
@@CallmeSamyNo. The concept of the limit was explained in the video a bit, but this is still not the same as analytic evaluation of infinity. Infinity is not a scalar. It is the cardinality of a set, in this case the "size" of the set of Reals.
Dividing by a single scalar, regardless of which, will not and cannot yield the size of the entire set of all elements within it.
Even when people try to argue that a sum, product, or some other higher order "hyperpower" equals infinity, what they really mean is that there exists a series divergence.
@@TheLethalDomain i mean there are a lot of infinities out there, and we use infinities in lims too, and i said 1/0 is infinity as 1/infinity is always taken as 0, so it would be just one way otherwise which is kinda wierd for Real numbers
@samudrarajofficial1254 But then 1/0 would be equal to infinity and negative infinity at the same time.
Also Infinity is not a Number. It's just a concept.
I would explain it using localisation of a ring. Suppose I have a ring R. If I choose a multiplicative set S (that includes 1), I can form the localisation S^{-1}R, such that it is now possible to divide by the elements in S in this new ring. Formally, we have a ring homomorphism R -> S^{-1}R, mapping r to r/1. One can check S^{-1}R consists of all elements r/s, where r is in R and s in S, such that a/b=c/d iff s(ad-bc)=0 for some element in S.
We can try to put 0 in S to form S^{-1}R. What happens is that now a/b=c/d for all elements, so you make the ring of 1 element (should one be allowed to consider it as a ring). In fact, 0*0=0=1 since 0 and 1 are actually the same element.
what
And after explaining all this, you ask "Do you understand, sweetie?" and she says "Yes daddy, can I go play with my dolls now?" 😊
Hahahahahaha. You are insane! And I loved it.
@@RealLeBronJamesFitnessOfficial localisation is a generalization of the field of fractions over integral domains.
@@samueldeandrade8535 Give me like 4 years and I'll get it, I'm 12
Dr. Sean, you asked for some alternative views.
We do a lot of funny things with 0 that dont make intuitive sense: e.g. exponents, factorials. The real value of the symbol 0 is to more easily handle places (making addition and multiplication easier) as well as represent nothing. A blank space (as at first used) is now different from 0. A blank space represents missing data. While the symbol 0 represents that some effort was made to count something and there was nothing: i.e. 0.
Its when we do things algebraically with 0 do we get all fuddled up.
We are confusing symbols with values. 12.45 is 5 symbols represent something. 0 is a symbol for representing nothing. I've concluded that 0 is not really a number or value, just as the infinity symbol. Infinity represents some abstract number that can be further multiplied by 10 ad infinitum. Likewise, 0 is an abstract symbol representing some number divided by 10, ad infinitum. This introduces us to the realm of transfinite and infinitesimal numbers ... and we can claim that 1 divided by infinity is equal to 0. And algebraically, 1 divided by zero is infinity. ... thus making division by zero a useless operation since infinity is a make believe useless value in the world of solving problems.
Thus, don't divide by nothing.
But we can still make some funny claims that 0 factorial = 1 = 1 factorial. And x raised to 0 = 1, when in fact they are limits (not actual values) and the value 1 is never never never ever reached. Gets real, real close ... but (mathematically) no cigar.
Someone already mentioned calculus ... which deals with sin 0 / 0 = 1
That's just my view ... in short ... dont bother dividing by nothing ... and if you do, just realize the solution (infinity) has no specific value and is thus not a useful answer.
Awesome video. As a math and physics major, the final level scratched the itch I wanted with this video, but the rest were all still really well done especially when viewed from the lens of someone in the target group.
You sir have earned a subscriber. I can’t wait to see where this channel goes, I would love to see more vids of this type, maybe something like different levels of square root of a negative, and go from impossible to complex analysis.
Thanks! I'm glad you liked it. I added that to my list for future videos
Suppose we have a unital ring and Z:= 0^-1 is defined such that 0*Z=1. Let x be an arbitrary element of the ring. We then know:
x= x * 1 = x * (0 * Z) = (x * 0) * Z = 0 * Z = 1
So every element must be equal and we are left with the Ring with 1 element. Which is therefore the only ring where division by 0 is possible.
very nice
How do you know that 0 * x=0 for all x? And if you know that then you can immediately say x=Z and so 0=1
@@omaduck5583 i guess what he means is if 0x=0, 1x=1 for all x and 0z=1 for some z then 0=1.
@@omaduck5583 if we take x = Z, then x * 0 = 1 so i dont think it's true for all x
@@omaduck5583 (0+0)=0
(0+0)x=0x
0x+0x=0x
0x =0x-0x=0
This follows from the ring axioms and is therefore true in every ring
i like to think of it like this...
multiplication is iterative addition, so 6x3 = 6 + 6 + 6
that must mean that division is iterative subtraction, where we have the quotient is equal to the number of times you can subtract the divisor from the dividend (call the dividend, d), before reaching an integer, n, where 0
👍
Actually, multiplication is 'iterated' (repeated) addition ... 'iterative' is normally reserved for computational mathematics, for instance an algorithm that generates a sequence of values that successively approach a target value defined by a relation.
But if you iterate it an infinite number of times you still never progress past the first step, so if your conclusion is that infinity is the correct answer you must also conclude that 1 is the correct answer also (or any other positive integer, for that matter)
The way I was taught calculus, we did the approaching 0 by hand technique by trying to get the slope of a formula at a given point using the rise over run thing. The teacher showed us that you could use a variable to represent the divisor that would normally be zero and cancel it out, effectively giving you the slope at that point in the line. The concept broke my brain so hard that I'm still trying to fully wrap my head around it.
Okay, this made me look up a few structures where division by 0 is technically allowed
- Wheel theory where division is defined for every element including 0
- The zero ring ({0}, +, ·) where 0 is both the additive and multiplicative identity
- Real projective line RP1
- Floating-point arithmetic, except ∞/∞ and 0/0 (unless you count NaN as an element?)
As an example in a graduate math class, I noted that 0 cannot be the denominator of a fraction because the denominator is the number of pieces the whole is divided into. Saying that it is divided into 0 pieces is refusing to follow the steps in forming a fraction.
Which graduate math class?
I think he means hes in 1st grade math (graduated kindergarten math) @@williamwilliam4944
@@williamwilliam4944 Number theory
Great video! And yes, I would say there is a "level 6" that was not spoken about: there are situations where you CAN divide by zero. For example, let R denote the real numbers. Then, the one point compactification of R is the set R U {∞}, where the usual rules of arithmetic on R apply and we also have (for nonzero real numbers r):
1.) r / ∞ = 0,
2.) r / 0 = ∞.
The symbol "∞" is the "point at infinity." You might realize this structure is homeomorphic to the unit circle! The key isn't asserting "you can't divide by zero" as a blanket statement, it is asking "does dividing by zero make sense in my mathematical structure?"
Of course. Also, trivially, the zero ring allows for division by zero
I’m currently a calc student, and honestly looking at the explanation between all the different levels is super intriguing! While I don’t plan to pursue math as a major in college hearing the different explanations, especially in the levels above me is super fascinating beyond just limit notation
Projective spaces can allow for division by zero in more meaningful ways, and more generally wheel algebras encapsulates this notion in an abstract way, providing formal setting for some number systems inside programming language designs, where including well-defined behavior of NaNs might be useful.
Excellent video, please keep it up!
Thanks so much, I'm glad you liked it!
Even as a kid i pondered in algebra class Trying to mathematically reach 0 through decimals and called it the Bowtie shape.
Later on in Calculus did i find that shape Was the function on a graph of lim and -lim.
smart kid
The modular arithmetic example makes things pretty clear.
On a clock, 3 times some number makes intuitive sense. It also happens that 3 times 12 equals zero in modular arithmetic. And this holds up for all whole numbers, because any whole number times 12 is obviously divisible by 12. That means that the input is lost in that calculation, because all whole number inputs map to the same output, which means that the inverse operation does not have a defined output.
So long as division is defined solely as the inverse of multiplication, it will remain undefined. And any change in that definition of division means you're no longer describing whatever numerical set/etc you were describing before.
Another interesting approach (for lower abstraction levels) is viewing division as repeated subtraction.
How do you share 6 cookies among 3 friends ?
You give one to each friend, leaving you with 6 - 3 cookies.
You give a second one to each friend, leaving you with 6 - 3 - 3 = 0 cookies.
The number of cookies each person has is equivalent to the number of times we substracted 3 from 6 before getting to zero, which is in this case 2.
In that sense, if we want to know what 6 divided by zero is, we would have to subtract 0 repeatedly from 6 until we get to 0.
This is where we see an issue : subtracting zero doesn't change anything, meaning the process never terminates.
This could either indicate that dividing by 0 is undefined, since the process doesn't end, or it could indicate that the answer is infinity in a certain non rigorous sense.
This gives the intuition that there is something to do with infinity without needing to introduce calculus or limits.
i think the 4th one is the best as it provides a real answer, despite being a bit abstract
Important to note though that it doesn't answer 6/0. It only provides the limit, which at no point actually divides by zero.
Disagree. There are infinitely many examples of a limit not existing at x0, but a function existing at x0.
I believe it is the worst because it provides an answer, which in this context really goes against what you are trying to teach. Also the theorical considerations imply continuity and limits that are way less intuitive imo
@@swenji9113 whether its intuitive or not dosent matter its math
@@9308323 ok by that logic when you integrate to find the area under a curve at no point do you truly find the area its just an approximation
Both wheel theory and projection onto the Riemann Sphere permit division by zero giving ⊥ and ∞ respectively. IEEE 754 floating point numbers have elements of both, with distinct representations of ±0 and all numbers understood to implicitly carry some error bound. ±0/±0 then evaluates to NaN (Not A Number: equivalent to ⊥) and any other number divided by zero evaluates to ±∞ depending on the signs of the operands, because there are an infinity of values *very close* to 0 that are represented as 0
One of my teachers in high school explained it like this: division is really a shortcut for subtraction. Like your 6 cookies and 3 people, you take a cookie away, give it to one person, you have 5 left (the remainder), take one away, give to second, you have 4 left, give it to third then start over. By going through twice, you're out of cookies and every one has 2, with no remainder, which is the key concept here. He went on to ask about the same problem with 0 cookies to any random number of people, e.g., the same 3 people. Take 0 away and give to each person, remainder is 6 with everyone having 0 cookies. You can repeat that as many times as you wish, you still have everyone with 0 and a remainder of 6. So, 6 / 0 means you can take away 0 40 times, still same as you started, then take away 47 more times, then take away 693 times, etc. That means 6 / 0 is simultaneously 40, 87, 780, etc. That means you cannot do it since it's undefined and a logical contradiction. RIP to that teacher, he died too young so that he couldn't teach even more people.
Your logic is impeccable, with the exception of one thing...
NEVER read the comments. As a content creator, you open yourself to mental health breaking abuse by reading them.
PS - Great video
Imagine I have 6 cookies and I want to divide them between 0 people, then I still have 6 cookies, haven't I
The question is how many cookies each person has once you've gone through all of your cookies. But you never go through all of your cookies, so the question makes no sense.
In the case of 0 divided by 0, that's like sharing 0 cookies among 0 people. In this case, it's slightly different; you're already out of cookies from the get-go, so it would would seem like you're done and the answer is 0, since nobody got any cookies. However, what we're specifically asking about is how many cookies each person has. You could just as well say, "Each person has 37 cookies," which would be vacuously true, as there are no people in the group.
Yes, in my opinion, you do. The only person to see my point. See my earlier replies to this video!
That's dividing 6 cookies among 1 person, since you;re a person as well (at least I hope so)
I am obviously not dividing, but keeping all to myself. Nevertheless: the thing I am making fun of is the very impractical description for the case: the video talks about having X Cookies and diving them among Y friends, where in one case Y approaches 0. But does not clearly state, that the result in question is: how many cookies does each and everyone of these Y get. It is supposed to talk about maths and fails at the simplest case already on precision.
I have never understood why some people are so uncomfortable with just giving the answer: because it is not defined. The wonderful thing about mathematics is that we have clear definitions of everything. Yes, it makes sense to talk about the reasons for why we have made a definition a certain way, but as mathematicians we don't have to worry about this. We simply just need to be clear about what the definition is. I have cleared up much confusion with my students in exactly that way. There is no need to worry about the philosophical aspects of "why can't we divide by zero".
@@ldov6373But that is kinda the point. The definition tells you what something is. It tells you why you can't divide by zero (because it isn't defined). You might not *feel* like that is satisfying. A part of maturing as a mathematician is exactly getting used to this. There is a big difference between dropping formulas from the sky and appealing to definitions that *define* what something is. A formula will have a proof that we can and should go through. Why is the theorem true? Because "the proof".
I absolutely agree tho I believe it is very useless and not a good way to teach at all if you stop there.
If you say "because it's undefined" you should continue and then explain how we define divisions by nonzero real numbers (for example) while making sure that your explanation showcases well that you use a number that is not 0.
This should answer the question and give a better insight on what is maths at the same time imo.
The thing is, if someone asks why you can't divide by 0, you should tell them that it's not a sensible question but still answer the underlying question, because you know very well that them expecting to be able to divide by 0 comes from a misunderstanding of how maths work. You should therefore answer the questions " Why can you divide by 2, by π, by -4.3566, etc?"
@@swenji9113 I didn't say to stop there. But it isn't useless. It is a common problem that students struggle to "prove things" and it is quite an eye opener when they realize that for many things you simply have to look at the definition. If you want to prove that something is a group, then we are not asking for some intuitive explanation on how it makes sense. We are asking for *a proof*. And there you need the definition (as you go through this initially at least). And that is the important part. The definition is very important. It is literally what defines what a thing is. The right answer to why we can't divide by zero is that it isn't defined. It is no more no less. Sure, you can try to explain how this emotionally makes sense. But there is a lot of value in being able to just approach things abstractly without having to rely on some intuitive or emotional understanding. On the other hand, if you don't point people to the definition, you run in to the problem that people can't actually work with whatever you are working with. I see this all the time in calculus. Students have this nearly philosophical issues with the concept of infinity. But we have a clear definition of what, say, limits are when we talk about infinity. Yes, you can *illustrate* this with examples and pictures. But at higher levels of mathematics you are not going to survive if you rely on that.
@@swenji9113I think I disagree. Asking "Why can't I divide by zero?" is a fine a very sensible question. But it also has a very clear and simple answer.
There are at least three cases where division by 0 is perfectly fine. The first is where a removable discontinuity would arise rather than an asymptote, the second is where L'Hopital's Rule can be used, and the third is where the squeeze theorem can be used. E.g. lim x-> 1 of y = (x^2 - 1) / (x - 1), lim x-> 0 of y = x / x, and lim x -> 0 of y = x^2 * sin(1 / x), respectively.
Love how he has an image of a black hole in the background. As that's a visual example of dividing by zero. Gravity over infinitely small distance. Dividing by 0 squared. And why string theory was so prominent, as it evades dividing by zero because nothing is smaller than a string.
You forgot Level 6: Computer Programmer
In the standard for floating point arithmetic IEEE 754, 1/0 is defined to equal Infinity and 1/-0 is -Infinity.
isn't it :
underflow => inf
because an underflow can occur during an computation because of finite register size.
Additionally, I'm looking to see if meadow theory will be adopted by software developers. Fields do not admit for a purely equational axiomatization. Meadows do, and division by zero may be definable in some meadows.
Wait wait, mathematicians have friends?
other Mathematicians 😂
@@ijhhcfionlkgs That would make them a group, and not a ring, because I am fairly sure no multiplication is happening
People they have defined as friends
@@leow.2162 That seems more like an indeterminate form.... and L'H is most likely not going to help.
No that's undefined
I always theorized that while dividing most numbers by zero is not possible, zero divided by zero is. I forget what property of math says this, but if x/y=z then zy=x. That explains why most numbers don't work. 6/0=x but if we flip it, 0x=6. What multiplied by zero equals six? We don't know that. But, what about 0/0? Well, the starting equation is 0/0=x and flip it: 0x=0. Any number multiplied by zero equals zero, (as far as I know with my 9th grade math knowledge) so zero divided by zero has an infinate number of solutions. That's just what I have though. The topic has interested me for a while now, but whenever I ask someone why, they just reply with "Because you can't divide by zero."
mapping spherical coordinates to the x-y plane, you get a hole when projecting the uppermost point of the sphere. this corresponds to the limit calculation in the sense that x and y coordinates can be between inf and -inf
My approach to this problem is to think of nonzero number in fractions as 1 dimensional (length) and zero as 0 dimensional (a point with zero length) . And the notion of division as asking the question how many of the denominator can fit into the numerator . For example 9/5 can be thought of as how many lines of length 5 can fit into a line of length 9 . 1.8 lines of 5 would fit into a line of 9. Now considering 4/0 , the question could be thought of as how many points can fit inside of a line of length 4 . There exists infinitely amount of points that can fit inside the line of length 4. The bridge between dimensions is infinity.
It was quite a while after I'd taken Physics and learned Ohm's Law (V=IR) when I figured out the mathematical explanation of how a short circuit works and why it's damaging and dangerous. Solve Ohm's Law for I (current) and you get I = V/R. In a short circuit, resistance approaches 0, so for a given voltage, the current goes to infinity. Of course, a protected circuit would blow a fuse or trip a circuit breaker, and an unprotected circuit would be destroyed when the current overloads the the circuit at a value FAR lower than infinity.
So basically it is either fractions, limits, equations, calculus and group theory that explains the idea of something divided by zero. Great stuff.
Well, ring theory, but yes
Absolute legend. Im a math major and that fact this was explained so well in 6mins is really inpressive. Subbed
Thanks!
As a physics teacher I'm always trying to apply math to real-world applications. I usually hold a meter stick or some long thing and have people imagine if I divide by 4, 3, 2, 1. Just like his cookie example. But when we get to zero pieces, there would be no stick. This breaks the law of conservation of mass. Dividing by zero would literally make atoms vanish. That is not possible.
to stay in the abstract algebra, we could just complete a field K with the multiplicative inverse of zero and add as many rules as we can. we can notably keep commutativity, but not multiplicative associativity, nor multiplication's distributivity over addition. and just like 0 is aborbing on K, we can also have 0^-1 to be absorbing on (K+{0,0^-1})\{0}. and this, even though it means 0^-1 = -0^-1, hence 0^-1 + 0^-1 = 0, even though 2*0^-1 = 0^-1 =/= 0... yeah, multiplication gets weird. but ay, if it works, it works.
1-4 makes perfect sense, and its things I've heard before. My favorite is the "oops we suddenly got infinite cookies for our 6 friends somehow", feels broken. Then level 5 really breaks things. Great video.
Note that there kind of is a field where dividing by zero is defined: {0}, or the zero ring, which contains a single element, usually denoted 0, where 0 is both the additive and multiplicative identity. Since x*0=x, x/0=x (or 0/0=0, since 0 is the only element). The thing is, although the zero ring technically meets the definition of a field, it's not considered a field because of how trivial it is, just like how 1 is not considered a prime number.
Nice video! For the last example, it's probably worth mentioning why 0 =/= 1, since the 0 ring (not a field, I know) is a commutative unital ring with 0 = 1. Namely, if 0 = 1, then for all x in the field (ring), x = 1x = 0x = 0, so the ring is the zero ring. Hence, when working with something more "intuitive" or "basic" like Z or R, 0 has a multiplicative inverse exactly when 0 = 1 exactly when we are in the zero ring, which contradicts the fact that we are working with Z or R.
I enjoyed this video quite a bit! Would you consider doing a video on Nash equilibrium at some point in the future? I don't have a background in math but I watched A Beautiful Mind not long ago and it made me curious to know more from someone that can break it down really well.
Treat it as repeated subtraction and counting the iterations until you hit or pass zero. This can be the basis for programming a computer to do division if it lacks an instruction to do so. Some early calculators actually did implement division like this, and would end up in an infinite loop if you tried to divide by zero, requiring a restart to get them out of it.
When I was in elementary school, they explained division in a few ways. One being how many (or what portion) of the divisor you can fit into the dividend. I remember thinking about division by zero under that logic during recess and I came to the conclusion that you could fit infinite zeros into any number. Then the conclusion I came to was that division by zero is not okay because "infinity is not a number".
My teacher told me that I was wrong and just that division by zero just doesn't work and not to look into it any further.
I think it would have been a good lead-in for my teacher to explain the basic idea of limits to me since I would definitely have been able to grasp that smaller and smaller divisors would "fit into" a number more and more times. When I learned calculus a decade later I thought it was really cool that I had already thought about a lot of the more basic topics in elementary school.
The example with 12 hour clock is really mind opening.
It's called modular arithmetic, a modulo 12 operation is basically the remainder after division by 12 (if you call the 12 on the clock 0 instead, so 30 hours after 3 o'clock = (30 + 3) mod 12 = 9 o'clock). What's interesting is that computers represent numbers this way, a byte for instance is 8 bits so 1-byte numbers are modulo 2⁸ = 256 and the largest number that fits in 1 byte is 256 - 1 = 255. Larger numbers can be represented in more bits/bytes (2-byte numbers are 16 bits so the largest number is 2¹⁶ - 1 = 65535, etc.)
@@JerehmiaBoaz Oh yes, I'm pretty aware of the usage of modular arithmetic in computer science. Mostly under the terms of modulo division and integer overflow. Never really thought of it in terms of abstract algebra. Even though I was taught about groups, rings etc. the only examples I recall were real numbers and their vectors.
I'm glad you liked that example! I think sometimes groups are taught so abstractly that they aren't connected with the groups that we are already familiar with
Good video. I used to teach math at a university and often thought about banning division from my classes and insisting on strict adherence to multiplication by inverses. I'm not sure it would have ultimately solved the divide by zero errors I saw, but it would have circumvented many problems stemming from non-commutative operations that are at least as annoying.
If you have six cookies and zero friends, you'll eat all the cookies yourself in a desperate attempt to fill the loneliness.
I had first heard that we cant divide by zero because depending on where you approach from it can either be positive or negative infinity, so i think we could fix that parficular issue by superimposing -0 on top of 0 on the number line. 1/0=infinity and 1/-0=negative infinity. For virtually all other purposes, there wouldn't be a need to differentiate, but its one of those things that feels like it needs an answer whether its useful or not, math does that sometimes anyway
I am not a math guy. In my profession as sw dev, depending on the compiler version/standard, divide by 0 yields either INF or NAN, or undefined result. For some target processor it may also trigger exception and leads to a reset. That’s why it’s always a good ideal to check 0 for variable operand in run time before executing a divisional statement
Thank you for this video! I never took math above precalculus, but I was able to understand you when you got to Level 4 and Level 5 despite that. In your "clock" example, you never used the following word, but the term "modulo" was used in textbooks in courses where I briefly studied that type of math. That was how I understood your example of how 11 + 2 could equal 1.
I'm a tutor and whenever one of my students (who are way way way too old and should know better) say 6/0 = 0, I've used explanation 2. I will try explanation one next time! This video did exactly what I needed for it for my professional life and it was also nice to revisit abstract algebra!
I'm glad you liked the video! I'd be interested to hear if it helps your students 😊
Good way of explaining groups and fields. As a math major that isn't a math major, I approve.
Level 2 is my favorite; it makes you ask all the right questions.
It is also worth noticing the fact that a division by zero approaches infinity ONLY when you consider positive infinitesimals, but it goes to NEGATIVE infinity if you consider approaching from the left. + Inf and - Inf are entirely opposite numbers. As opposite as numbers can get.
Now, this is significant because engineers might, and sometime they do, mistakenly think that it makes perfect sense to "define" div by zero as the biggest number a computer can represent. But that's because they are looking at the limit from the right only.
Having said that, I have seen systems which define div by zero that way.
Sure, but neither negative nor positive infinity are possible, so why not just call them equivalent?
@@CBlarghright, that's what sometimes done. The problem here is that they are in the opposite ends of the representation scale. So what should you pick, the largest positive or the largest negative? which one you choose will completely change any subsequent calculations.
The usual reasoning for picking the largest number (in representation for infinity) is that the calculation that led to zero could have resulted in an infinitesimal instead. For example, because of a rounding effect known as catastrophic cancellation where you subtracted two numbers which were not really the same by close enough.
The problem is that the reasoning is biased toward a positive infinitesimal, hence a positive infinite (well, largest number in practice). But as you can see, the order of the operands in the subtraction, for example, would determine whether the zero is positive or negative (or rather, collapsed from the right or the left). So the choice of positive over negative infinity can be completely wrong.
NOTE: Interestingly, computers DO have both positive and negative zeros (depending on how they reach it). In fact they also have a representation for both positive and negative infinite. The problem is that (almost all) programming languages don't know about them. You CAN use all of them if you use a low level language such as C or C++ but only through special functions.
@@fernandocacciola126 Don't get me started on trinary logic... Infinity isn't really a question of applied science. You can either divide by the number or you can't. If you want to represent infinity, choose whatever symbol you want. It doesn't really matter if it's negative or positive if it doesn't compute.
for level 4 it should be noted that since the limits on the right and left side of zero do not agree, the definitive limit as x approaches zero is also undefined
0 has a different behavior depending on what set we are in and what division means in that set. In the first one, we are talking about the natural numbers, then the rational numbers, and finally the real numbers. It gets interesting in calculus because we can approximate using the notion of limits to conclude that the limit doesn't lie on the line. I would use measure theory, using the right assumptions there's a right answer, it's positive infinity all along
There's a relation-algebraic *converse* to multiplication by zero: the relation that takes zero to all numbers (in your ring, ...). A fun visual introduction to this is the Graphical Linear Algebra blog by Sobocinski et al.
Fantastic explanations! Best video on this topic by far! Thanks👍👍
1/0 (or the point at infinity) makes more sense when you use complex numbers, and map the Argand Plane onto a globe (with 0 at the South pole and infinity at the north pole).
It doesn't matter whether you approach infinity (using 1/z as z tends to 0) along the positive real, negative real or positive/negative imaginary axis - the answer will always approach the 'point at infinity' (I.e. the North Pole)
I've had the most luck communicating division by zero through looking at operations as repeated simpler operations. The simplest is counting up or down, addition and subtraction are repeated counting. Then multiplication is repeated addition, and most people make the leap to division as repeated subtraction. So I do an example, say 12/3, so 12-3 is 9, 9-3 is 6, 6-3 is 3, and 3-3 is 0. This took 4 subtractions of 3 from 12 to get to zero, so 12/3=4. Then it's a small step to see how no matter how many times you subtract zero you still get the same number back, and this is a good opportunity to reinforce the idea of zero as the additive identity.
I actually came up with a concept of visualizing asymptotes on a sphere on my own, only to realize that was already a thing called a riemann sphere. I always loved the idea. Instead of this extreme incongruity of infinite negatives immediately flipping to infinite positives, you just see the approach to negative infinity wrap around the sphere to join onto the approach of positive infinity. It's so interesting to imagine infinity as a counterpart to zero that behaves basically the same on a graph, except on the opposite side of the sphere... Of course it doesn't behave the same. Unlike zero, the coordinate that would map to infinity on the sphere is still a single point of incongruity, while zero is congruent as normal. But an incongruity at a single point on a sphere is easier to ignore and thus more aesthetically pleasing than an incongruity that spans an entire axis.
“You have 6 cookies to divide among 0 friends.” This hurt a lot more than I expected
A variant of level 4 is my preferred approach, graphing y = 1/x as x approaches 0, it's intuitive that x will never actually reach 0, so division by 0 is not only undefined, it doesn't even exist.
Actually it does but not in the real number plane. Division by 0 is the same as tan(90) or sin(90)/cos(90). It is orthogonal to the +x-axis and is coplanar to the +y-axis. If we take any real number x along the x-axis and rotate it by 90 degrees or multiply it by +/- i. It exist in the complex plane on the unit circle defined in terms of the sine and cosine function as in (cos, sin). Here the real part becomes 0 and the imaginary part is all that is left over. Many don't consider the imaginary or the complex numbers when looking at division by 0, vertical slope, or the tangent of 90 degrees... It does exist. It's just not a real number. It's perpendicular to the real numbers, it's orthogonal to them. The real numbers are along the horizontal and division by 0 has no relation to the horizontal as it is purely in the vertical.
If you walk across a flat road, we say we have no slope... this is inclination or angle of 0. And within the slope formula (y2-y1)/(x2-x1) or dy/dx or sin(t)/cos(t) or tan(t) the numerator is 0 and the denominator is 1. sin(0)/cos(0) = 0/1 = 0.
When both the sine and cosine are equal or we have an even amount of translation or displacement in both the horizontal and the vertical, the slope is 1 and this is equivalent to a 45 degree or PI/4 radian angle. sin(45)/cos(45) = sqrt(2)/2 / sqrt(2)/2 = 1.
When there is no displacement in the horizontal plane (real numbers) but there is only displacement in the vertical we have vertical slope which is perpendicular - orthogonal to no slope. sin(90)/cos(90) = 1/0 = infinite slope.
The sine and cosine functions are basically the same exact wave function with the same range, domain, properties and limits as each other except for one major difference. That difference is their initial y value or f(x) at x when x = 0. The sine starts at 0 and the cosine starts at 1. This is the only difference between these two functions. Other than that, they are horizontal linear transformations of each other. In fact they are at a 90 degree or PI/4 radian horizontal translation from each other.
Both functions are continuous smooth curves that are rotational, oscillatory, periodic, transcendental wave functions. Their domains are at minimal the set of all reals and their range is [-1,1]. Their limits exist for all points on their graphed curves even when they are within the context of the tangent function. Just because they are used within another function doesn't mean their independent limits cease to exist. Those limits persist.
Taking the above information we can easily see that 0 and 1 are opposites of each other. Why are they opposites? Simply because they are 90 degrees from each other. They are orthogonal to each other. They are perpendicular to each other. Zero has No Value, and One has All Value. Without this orthogonal - perpendicular relationship, binary arithmetic, log2 mathematics, logic, etc. wouldn't be possible. Division by 0 does exist and it isn't undefined. It is infinite repeated non terminating subtraction that tends towards infinity and the result of the operation is ambiguous. Stating that it is "undefined" is a copout of laziness because people don't want to deal with its complexity. If division by 0 didn't exist or is "undefined". Your clock would break at midnight but it doesn't. The wheels, tires and axles on your vehicle would break. The electricity going through the wires to your house wouldn't work. You computer wouldn't work as there would be no internal clock or oscillator. The planets, stars and galaxies would spin or spiral, and Life wouldn't exist. Division by 0 is real, although it doesn't yield a real number. It is ambiguous, not undefined. It is also not indeterminate either. It's only indeterminate if the numerator is either +/- infinity itself or 0. These 3 cases: +inf/0, -inf/0, and 0/0 are indeterminate forms, all other cases are ambiguous as they tend towards +/-inf.
For level 1 intuition you can even add in that 0! Is 1 because even though there are zero people to receive cookies, there is 1 way to represent (arrange) that no one receives cookies.
"How do you split 6 cookies among zero friends?" That hit hard.
I like to imagine the number system as three ranges, 0, ARN (all real numbers) and infinity. Each of these fields has similar properties (0*2=0, ARN*2=ARN, inf*2=inf)and equal ranges. When dividing a real number by 0, it makes infinity (which we'll assume is the opposite of 0). We can return to 0 by dividing by inf. Now what about 0 * x = 1. Well, when multiplying 0 and inf (or 0/0, inf/inf etc.) it makes ARN, because 0 and inf both have infinitely long ranges (stay with me here). This can also be proven because 0 * any number = 0, so divide those, and any number * 0/0 = 0/0 (or 0 * ARN/0 = 0/0, which is also the same) This equates to ARN^2 = ARN, which is true. You can also see this graphically. As a goes closer to 0 in the graph (y = a/x), the graph looks closer and closer to two joined graphs y = 0 and x = 0. (The true graph of 0/x is just y = 0 with a hole at x=0). Though if this is the case, it would not be a function, which causes a lot of philosophical problems regarding mathematics. Sorry if this reasoning sucks I thought of most of this when I was like in Algrebra I. There's definitely a plenty details missing but I'm tired so yeah....
Infinity. Mostly because as soon as I understood the concept of division using numbers smaller than 0, it makes perfect sense for it to work this way in any practical context, and everyone shouted at me angrily for pointing it out.
Negative infinity
There are some contexts where dividing by zero makes some sense. On the extended complex plane (which can be visualized via the Riemman sphere) we say that any non-zero complex number divided by zero is infinity, and any finite complex number divided by infinity is zero. It's not exactly complete, though, because it leaves 0/0, ∞/∞, and 0 × ∞ undefined.
Best visual definition of modular multiplicative inverse so far !
The way I see it is: when you place the function, [f(x) = 1/x] into a graphing calculator, you see that when “x” approaches 0, the function moves towards both negative and positive infinity. The only extra thing I’m adding is that the reason is is undefined, is that both hemispheres of the graph are trying to reach separate but equal points. Since these points go on countably forever, there is no way to distinguish the farthest point (it is infinitely far ahead). Therefore the reason for “undefined” or “syntax error” is because the function reaches a point where it no longer passes the vertical line test nor remains a function. However, since there is no way to be 100% sure of this and because the answer varies by definition of infinity, the function remains a function and the error remains somewhat of an error. We humans simply cannot compute everything.
Research Wheel Theory. It's a variation of fields that are specifically designed to allow for division by zero. For instance, a wheel of integer fractions works exactly like the rational numbers, except that the wheel allows the denominator to be zero. This results in two more values in the set: one represents 1/0 (usually called ∞), and the other represents 0/0 (usually called ⊥). As an example, the rule for addition of two fractions is a/b+c/d=(ad+bc)/(bd). So ∞+1 is 1/0+1/1=(1·1+0·1)/(0·1), which simplifies to (1+0)/0, or 1/0. So ∞+1=∞.
you can divide by zero in a meadow, where most things are like a field but multiplicative inverses are instead defined by
(x^-1)^-1 = x
x * (x * x^-1) = x
in which case, 0^-1 = 0, because (0^-1)^-1 = 0^-1 = 0, and 0 * (0 * 0^-1) = 0 * 0 = 0
so then 1/0 = 0
I would add the continuity and divergence/convergence concepts when talking about limits approaching from opposite directions. Like in order to have a solution the limits must converge in a single value and in the case of 0 not only the solutions go to two different locations (infinite and negative infinite) but also there’s no continuity as the function breaks.
the calculus one gives me the impression that a number divided by 0 is both more than infinity and less than negative infinity at the same time which is completely mind breaking.
Thank you! I enjoyed this. Zero and infinity feel like they share some common ground of being edge cases where things start to collapse because you're mixing unlike elements. Like zero is an infinitely small value between things which are negative and things which are positive. Neither positive nor negative and infinitely not anything, as infinity is infinitely something.
Thanks so much! I'm glad you liked it.
To be honest, I start wondering what if Zero didn't exist. because I think all its uses in operations are "useless" for example: 5 + 0 = 5; 1 - 0 = 1; 0 - 1 = -1 (no effect). 6 × 0 = 0 (just gives itself so if it exists just to give itself then why it exists in the first place?) 12 ÷ 0 (undefined) 0 ÷ 15 = 0 (also just gives itself) maybe you have right zero should be just like infinity, a concept but not a number. already logically, a number is a value but zero has no value. maybe zero should be banned in algebra and only exists in limits as infinity.
Or just forget, I'm mad
You could add a level 6, the transreals, structure that breaks only a few points of being a field (it is not, then), but works almost as if. It is actually easily definable but not so easy to construct. Basically 1/0 is infinity if i recall and 0/0 is nulity, a point that absorbs anythinf it operates with. It is basically the closed line on infinity with an extra point that breaks a bit of division and associative, but makes dividing by zero possible in a sense.
Fun thing: Calculus stuff remain functional.