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.
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.
@@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.
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.
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.
@@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.
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.
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.
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".
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.
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.
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.
@@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
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'
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.
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.
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.
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.
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… :))))
@@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.
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 (?
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.
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.
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.
@@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.
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.
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
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.
This explains why mathematicians tend to be socially awkward: if you have 6 cookies and a friend they get all 6 cookies (6/1 = 6), however, if they're not such a good friend but you have 6 cookies to give, now you have to start baking more (6/0.25 = 24). The learning is that the worse friend you are the more cookies you get.
@@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
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.
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
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.
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?"
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.
@@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.
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.
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.
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.
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.
This might just be a restatement of the cookie example, or it might be a genuinely different way to think about it, but here is one explanation I like to use: 1. What is multiplication, really? In a sense, multiplication is just repeated addition. So, 6x1 is just a shorthand for 1+1+1+1+1+1 2. Division is just the same process in reverse. So, 6/1 is just another way of asking, "how many ones do we need to add up to get six?" 3. If we say "what is 6/0?", then we're asking, "how many times do we need to add 0+0+0... to get six?" and the answer is it can't be done. Right?
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.
@@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.
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
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
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 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.
you forgot the substraction definition of division where 6/2 is rephrased as "how often can you substract 2 from 6 until you reach 0: 3 times" which leads to 6/0 beeing "how often can you substract 0 from 6 until you reach 0" which also leads to the conclusion that its undefined. thats the one we learned in 1rst grade in germany
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.
I find it interesting how the level 5 answer kinda loops back to the level 3 answer, but more rigorously defined. Level 3 is knowing _the fact that_ 0•x=1 has no solution, but level 5 is knowing _why_
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.
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.
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 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.
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.
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.
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.
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.
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
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 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)
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
Dividing is also a way to find a proportion. 1/2= 0.5 = 1 is 0.5 ( half ) of 2 1/3 = 0.333 = 1 is 0.333 ( a third ) of 3 8/2 = 4 = 8 is 4 times bigger than 2 1/0 = infinite ( so 1 is infinitely bigger than zero )
in math history its like tradition to claim that something is not possible (negative numbers, irrational numbers, square roots of negative numbers) only to be disproved later
I do have some difficulty with the limit explanation. If memory serves a limit is a number to which a sequence tends. You can always find a number in your sequence which is closer to your limit than any number you chose. This doesn't seem to make much sense if your limit is infinity. The difference between a finite number and infinity can be anythng you like
Level 0: you can tell anyone you'd give everything you have to anyone as long as there's no one to give, the moment someone appears, how much you're willing to give quickly gets defined
I thought this video was gonna go even further into higher math. I propose an amendment: Level 6 should be projective geometry and how associating infinity and -infinity gives us a well-defined topological space where [x]/[0] is unambiguously well-defined.
I think of multiplying as an expedient way to add repeatedly. 3x2=2+2+2 Likewise, dividing is like subtracting the same number repeatedly: 6-2-2-2=0 I had to subtract 2 three times, so 6/2=3. If I try that with zero, nothing happens. 6-0-0-0-0-0…. I can do that forever, and I will still have 6.
Here is how I explain this. Multiplication is shorthand addition. 5 x 3 is the same as 3 + 3 + 3 + 3 + 3 (or 5 + 5 + 5). This can be read as, "What do i get if I add five threes (or three fives)?" Since division is the inverse of multiplication division is shorthand for subtraction. So if you look at 12 ÷ 4 you are saying how many times can I subtract 4 from 12? One gives me 8. Two give me for. And finally the third gives me 0 so the answer is three. The reason you cannot divide by zero is this. If you subtract 0 from any number you get the same number. You will never get any other number. Once you start you will die with the same number subtracting for infinity. This is why you cannot divide by zero.
I always sort of imaged that dividing by 0 always gave an Infinity. This might be due to how we learned to do division which was always the question of "How many time can the number fit inside of the other number" where in this case.. 0 can fit an infinite amount of times inside of X
I like remanbering the définition of division witch is "how many time can I remove a number to the initial quantity?" If you have a number and you remove 0 from it, it wont change. You can repeat this an infinite number of Time, you still would not have done anything. So the question : "how many time can I remove 0 to a number ?" Has no solution
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.
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
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 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."
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.
So why cant we devide by 0? First lets look at : 12/3. We know the awnser is 4. 12/3 = 4. But what happens? Well we can look at division as the following way: 12/3 : 12-3 = 9, 9-3=6, 6-3=3 and 3-3= 0. As you can see it took us 4 "steps" to get to 0. When it reach 0 it ends. So what about 12/0? We can do the same "trick" : 12-0=12 12-0=12 12-0=12 12-0=12 12-0=12 . . . U get the idee. We can never reach 0 and therefore we cant devide by 0. 12/0 = WtF
Suppose you project the y axis and values onto the circular sides of a cylinder. Then the limit as x goes to zero goes to infinity, which is neither positive nor negative since in this case the y axis is a circle, with one value for infinity. Maybe we have been too Euclidian in our thinking up until now.
The way I see it ×/0 is like saying "take x and divide it into no pieces. Basically, don't divide x" while ×/2 is "take x and divide it into 2 pieces". By my logic, ×/0=× because no division occurred
I dont like it when people say "the limit equals infinity." You may as well say the square is round or heat is cold. A limit is a boundary and infinity is limitless. No wonder people get confused.
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.
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
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....
My favorite explanation for why it's undefined is that division is just repeated subtraction. If we think of division as a word problem, the equation 8 ÷ 2 = 4 is like asking the question "how many times do I need to subtract 2 from 8 before it equals 0?" The answer is 4. Now apply that same logic to something like 8 ÷ 0. The question is "How many times do I need to subtract 0 from 8 before it equals 0?" Well, that question doesn't make any sense! You can subtract 0 from 8 as many times as you want and the 8 will never get any closer to being 0. Therefore division by 0 is undefined. One might want to think it's infinity, but even if you subtracted 0 from a number an infinite amount of times, you'd still never make any progress. It's kinda similar to the limit argument you made for calculus, just viewed from an arithmetic perspective.
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?)
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.
Literally everyone in the comments section: Reiman Spheres, economics, abstract algebra, geometry!?!?! Me: slope, it's simple, rise over run, when you have a slope of 6/0, you rise 6, you go over 0, snd you keep doing this indefinitely. I feel like this is where like the Differencial Calculus comes into play and makes some sort of derivitave or integral on both sides
Another way to explain it: I have 6. Divife by 2 = how many times can i substract 2 from 6 to reach 0? 6-2-2-2=0. -three times. Replace 2 with 0 and you cannot reach 0.
Couldn’t you use e^(i*pi) - 1 as a substitute for 0 when dividing? This gives us a genuine result, although imaginary, and can actually be used to solve problems such as 1^x = 2.
Well, I have questions. Does it matter? Do we use it for anything that works? If we do, what is it and how is it useful? MUST zero be used like this or not? Damn it. Lol.. I mean, eh ehm .. please and thank you.
0/0 is the only expression that will actually give you an answer (unless you're using hyperreals or surreals), but it's important that you must get this answer not from 0/0 itself, but from whatever gave you 0/0. This is only possible in calculus using limits. e.g. lim(x-->0) x/x = 1. So 0/0=1 right? Nope, because lim(x-->0) 2x/x = 2. So 0/0=2, and 1=2? No. This is what it means when we say something is "indeterminate." It can be *anything* depending on where it came from. The expression itself does not have a determinable value. You must know its source to know its value in a given instance.
If you think about it, zero is representation of nothing and you can't really fill anything with nothing, but on the other side 0/0 takes every single real number as an answer because it is possible to put nothing into nothing infinitely, so yes, it's still an exception
An interesting notion: If we try to define a division by zero the same way we define the square root of -1 and follow the logical consequences, we conclude that the neutral element of addition is the same of the neutral element of multiplication, in other words, 0 = 1. All numbers times 1 are themselves, and all numbers times 0 are 0, but we just concluded that 1 = 0, hence all numbers in this field are 0. That's a pretty useless field. I sometimes almost miss real analysis classes
I teach 9th graders who understand math at 5-6th grade level. That first explanation, unfortunately, seems to lose about half my class. I’ve found that it’s easier to explain division n/d as taking n objects and splitting them into groups of d size. You’re done when you used up all the objects. But if you try this with zero, you’ll never use them up, because all the groups are empty! When they get taught f(x)=1/x two years later, I hope that this explanation travels with them 😅
I'll take your calculus approach and expand. When solving with an approximation each side of your target, the answers should converge as you get closer. With 0, the answers diverge. That shows there is no possible solution
4:00 It kind of does make sense (to me) that giving six cookies to zero friends results in infinity because if you give six cookies to zero people then the cookies will last forever or am I being obtuse?
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.
@@soyoltoi 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.
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
“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.
@@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
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
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.
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?
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'
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
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.
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 :)
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 (?
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 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.
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.
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.
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
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.
This explains why mathematicians tend to be socially awkward: if you have 6 cookies and a friend they get all 6 cookies (6/1 = 6), however, if they're not such a good friend but you have 6 cookies to give, now you have to start baking more (6/0.25 = 24). The learning is that the worse friend you are the more cookies you get.
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
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.
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
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.
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
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?
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
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.
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.
“How do you split six cookies evenly among zero friends?”
I feel personally attacked.
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
This might just be a restatement of the cookie example, or it might be a genuinely different way to think about it, but here is one explanation I like to use:
1. What is multiplication, really? In a sense, multiplication is just repeated addition.
So, 6x1 is just a shorthand for 1+1+1+1+1+1
2. Division is just the same process in reverse. So, 6/1 is just another way of asking, "how many ones do we need to add up to get six?"
3. If we say "what is 6/0?", then we're asking, "how many times do we need to add 0+0+0... to get six?" and the answer is it can't be done.
Right?
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?
@@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.
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
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
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 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.
you forgot the substraction definition of division where 6/2 is rephrased as "how often can you substract 2 from 6 until you reach 0: 3 times"
which leads to 6/0 beeing "how often can you substract 0 from 6 until you reach 0" which also leads to the conclusion that its undefined.
thats the one we learned in 1rst grade in germany
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
I find it interesting how the level 5 answer kinda loops back to the level 3 answer, but more rigorously defined. Level 3 is knowing _the fact that_ 0•x=1 has no solution, but level 5 is knowing _why_
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.
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.
Excellent video, please keep it up!
Thanks so much, I'm glad you liked it!
Cardiologists HATE this one simple trick for dividing a pie into zero equal pieces.
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.
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.
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 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.
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
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 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)
@@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.
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
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
calc explanation made a ton of sense
Dividing is also a way to find a proportion.
1/2= 0.5 = 1 is 0.5 ( half ) of 2
1/3 = 0.333 = 1 is 0.333 ( a third ) of 3
8/2 = 4 = 8 is 4 times bigger than 2
1/0 = infinite ( so 1 is infinitely bigger than zero )
in math history its like tradition to claim that something is not possible (negative numbers, irrational numbers, square roots of negative numbers) only to be disproved later
I do have some difficulty with the limit explanation. If memory serves a limit is a number to which a sequence tends. You can always find a number in your sequence which is closer to your limit than any number you chose. This doesn't seem to make much sense if your limit is infinity. The difference between a finite number and infinity can be anythng you like
Level 0: you can tell anyone you'd give everything you have to anyone as long as there's no one to give, the moment someone appears, how much you're willing to give quickly gets defined
Was worried this was heading towards eigenvalue territory. Great video!
I thought this video was gonna go even further into higher math. I propose an amendment: Level 6 should be projective geometry and how associating infinity and -infinity gives us a well-defined topological space where [x]/[0] is unambiguously well-defined.
I think of multiplying as an expedient way to add repeatedly. 3x2=2+2+2
Likewise, dividing is like subtracting the same number repeatedly: 6-2-2-2=0 I had to subtract 2 three times, so 6/2=3.
If I try that with zero, nothing happens. 6-0-0-0-0-0…. I can do that forever, and I will still have 6.
Here is how I explain this.
Multiplication is shorthand addition. 5 x 3 is the same as 3 + 3 + 3 + 3 + 3 (or 5 + 5 + 5). This can be read as, "What do i get if I add five threes (or three fives)?" Since division is the inverse of multiplication division is shorthand for subtraction.
So if you look at 12 ÷ 4 you are saying how many times can I subtract 4 from 12? One gives me 8. Two give me for. And finally the third gives me 0 so the answer is three.
The reason you cannot divide by zero is this. If you subtract 0 from any number you get the same number. You will never get any other number. Once you start you will die with the same number subtracting for infinity.
This is why you cannot divide by zero.
Thank you for these explanations. I really enjoyed the video, it was so easy to understand. Thanks.
I always sort of imaged that dividing by 0 always gave an Infinity. This might be due to how we learned to do division which was always the question of "How many time can the number fit inside of the other number" where in this case.. 0 can fit an infinite amount of times inside of X
I like remanbering the définition of division witch is "how many time can I remove a number to the initial quantity?" If you have a number and you remove 0 from it, it wont change. You can repeat this an infinite number of Time, you still would not have done anything. So the question : "how many time can I remove 0 to a number ?" Has no solution
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.
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
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 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."
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.
So why cant we devide by 0? First lets look at : 12/3. We know the awnser is 4. 12/3 = 4. But what happens? Well we can look at division as the following way: 12/3 :
12-3 = 9,
9-3=6,
6-3=3 and
3-3= 0.
As you can see it took us 4 "steps" to get to 0. When it reach 0 it ends. So what about 12/0? We can do the same "trick" :
12-0=12
12-0=12
12-0=12
12-0=12
12-0=12
.
.
.
U get the idee. We can never reach 0 and therefore we cant devide by 0. 12/0 = WtF
Suppose you project the y axis and values onto the circular sides of a cylinder. Then the limit as x goes to zero goes to infinity, which is neither positive nor negative since in this case the y axis is a circle, with one value for infinity. Maybe we have been too Euclidian in our thinking up until now.
The way I see it ×/0 is like saying "take x and divide it into no pieces. Basically, don't divide x" while ×/2 is "take x and divide it into 2 pieces".
By my logic, ×/0=× because no division occurred
I dont like it when people say "the limit equals infinity." You may as well say the square is round or heat is cold. A limit is a boundary and infinity is limitless. No wonder people get confused.
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
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....
My favorite explanation for why it's undefined is that division is just repeated subtraction. If we think of division as a word problem, the equation 8 ÷ 2 = 4 is like asking the question "how many times do I need to subtract 2 from 8 before it equals 0?" The answer is 4. Now apply that same logic to something like 8 ÷ 0. The question is "How many times do I need to subtract 0 from 8 before it equals 0?" Well, that question doesn't make any sense! You can subtract 0 from 8 as many times as you want and the 8 will never get any closer to being 0. Therefore division by 0 is undefined. One might want to think it's infinity, but even if you subtracted 0 from a number an infinite amount of times, you'd still never make any progress.
It's kinda similar to the limit argument you made for calculus, just viewed from an arithmetic perspective.
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?)
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.
Literally everyone in the comments section:
Reiman Spheres, economics, abstract algebra, geometry!?!?!
Me: slope, it's simple, rise over run, when you have a slope of 6/0, you rise 6, you go over 0, snd you keep doing this indefinitely. I feel like this is where like the Differencial Calculus comes into play and makes some sort of derivitave or integral on both sides
Another way to explain it:
I have 6. Divife by 2 = how many times can i substract 2 from 6 to reach 0? 6-2-2-2=0. -three times.
Replace 2 with 0 and you cannot reach 0.
Couldn’t you use e^(i*pi) - 1 as a substitute for 0 when dividing? This gives us a genuine result, although imaginary, and can actually be used to solve problems such as 1^x = 2.
I saw this guy divide by zero irl at the U’s math lab. Absolute legend.
Well, I have questions.
Does it matter? Do we use it for anything that works? If we do, what is it and how is it useful?
MUST zero be used like this or not? Damn it. Lol.. I mean, eh ehm
.. please and thank you.
0/0 is the only expression that will actually give you an answer (unless you're using hyperreals or surreals), but it's important that you must get this answer not from 0/0 itself, but from whatever gave you 0/0. This is only possible in calculus using limits.
e.g. lim(x-->0) x/x = 1.
So 0/0=1 right?
Nope, because lim(x-->0) 2x/x = 2.
So 0/0=2, and 1=2?
No. This is what it means when we say something is "indeterminate." It can be *anything* depending on where it came from. The expression itself does not have a determinable value. You must know its source to know its value in a given instance.
If you think about it, zero is representation of nothing and you can't really fill anything with nothing, but on the other side 0/0 takes every single real number as an answer because it is possible to put nothing into nothing infinitely, so yes, it's still an exception
Apparently division by zero forms an algebraic structure called a wheel where addition is not always defined but multiplication and division are.
The example that always stuck with me was: 0/0 Zero divided by zero can equal 1 or 0 depending on the rule you follow, thus it's undefined.
An interesting notion:
If we try to define a division by zero the same way we define the square root of -1 and follow the logical consequences, we conclude that the neutral element of addition is the same of the neutral element of multiplication, in other words, 0 = 1. All numbers times 1 are themselves, and all numbers times 0 are 0, but we just concluded that 1 = 0, hence all numbers in this field are 0. That's a pretty useless field. I sometimes almost miss real analysis classes
I teach 9th graders who understand math at 5-6th grade level. That first explanation, unfortunately, seems to lose about half my class. I’ve found that it’s easier to explain division n/d as taking n objects and splitting them into groups of d size. You’re done when you used up all the objects. But if you try this with zero, you’ll never use them up, because all the groups are empty!
When they get taught f(x)=1/x two years later, I hope that this explanation travels with them 😅
I'm so delulu, I'm in grade 11 and I still think I can totally understand the math major answer😂
Edit: Wait no, I think I actually understood it yay
I'll take your calculus approach and expand.
When solving with an approximation each side of your target, the answers should converge as you get closer. With 0, the answers diverge. That shows there is no possible solution
Level 6 : Riemann sphere, projective geometry, Alexander compactification : yes please, divide by zero if you wish.
Why don't we represent dividing by zero with imaginary numbers, the same way we did with the square root of negative numbers?
Level 2 is my favorite; it makes you ask all the right questions.
4:00 It kind of does make sense (to me) that giving six cookies to zero friends results in infinity because if you give six cookies to zero people then the cookies will last forever or am I being obtuse?