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.

@@soyoltoi Only if you take the engineering concept that 1/infinty = 0 (which, of course it is not)

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.

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.

7 year old's existential crisis... trust me, it only gets worse from here kid!

😂

Simple. I get all the cookies.

@@Koyasi78 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

It's sad to have zero friends

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.

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

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

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?

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.

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

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

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 (?

Awww

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.

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.

Thanks, that is a new and refreshing approach! 😊

ah, you make a very valid argument, and give an excellent alternative!

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?

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 :)

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.

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

This is brilliant!

But can't it just mean 1/0 is infinity?

​​@@SamudrarajOfficialNo. 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.

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

👍

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)

@@robo3007 The right answer will "terminate" the loop.

I think of addition and subtraction as just a higher form of counting. In other words, it's impossible to not quantify your subjects.

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

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

Of course. Also, trivially, the zero ring allows for division by zero

Which graduate math class?

I think he means hes in 1st grade math (graduated kindergarten math) @@williamwilliam4944

@@williamwilliam4944 Number theory

​​@@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

@@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.

smart kid

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

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.

Thanks so much, I'm glad you liked it!

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.

Thanks! I'm glad you liked it. I added that to my list for future videos

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.

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

Well, ring theory, but yes

sqrt(-1) is a bit of abuse of notation. Technically i is defined to be a solution to the polynomial x^2 + 1 = 0. Even more precisely, if R is the field of real numbers, and R[x] is the ring of all polynomials in x with real coefficients, then define f:R[x] -> R[x]/(x^2+1) by p(x) -> p(x) (mod (x^2+1)). Then we define i to be f(x). Not trying to be "that guy" or anything, I'm just pointing out that mathematicians can't just define _anything_ . There has to be a solid motivation and many rigorous reasons why we are able to do it. Without understanding this, it's easy to fall into precisely this trap that we can simply define ourselves a division by zero. In fact it can happen, but only under strict circumstances like Wheel algebras or the zero ring.

But 0/0 is not 1. It's still undefined. Nothing divided by nothing is not something. Nice try, though. :)

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.

Thanks!