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I wish to explain my younger brother: he is interested and curious, but he cannot grasp the concepts of limits and integration just yet. What is the best mathematical way to justify not allowing division by zero?

SQB
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Shubh Khandelwal
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    How do you divide, say, $5$ apples between zero people? There is no meaningful way to distribute the apples. – AlvinL Aug 15 '18 at 10:36
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    Division by zero should not be justified at all. – Peter Aug 15 '18 at 11:39
  • Related: https://math.stackexchange.com/questions/26445/division-by-0 – Hans Lundmark Aug 15 '18 at 12:40
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    Just ask Siri. It explains this very well. – DonielF Aug 15 '18 at 15:32
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    This would also fit well on the Mathematics Educators site. – gen-ℤ ready to perish Aug 15 '18 at 16:10
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    @AlvinLepik Would have been a perfect answer as-is – Lightness Races in Orbit Aug 15 '18 at 18:03
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    "Imagine that you have zero cookies and you split them evenly among zero friends. How many cookies does each person get? See? It doesn't make sense. And Cookie Monster is sad that there are no cookies, and you are sad that you have no friends." - Siri – jkd Aug 15 '18 at 19:12
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    You divide 5 cookies among zero people by keeping them all yourself. Makes perfect sense! – Laurence Payne Aug 15 '18 at 19:13
  • @AlvinLepik [Alvin, Simon, Theodore....](https://www.youtube.com/watch?v=OxD6mmJaYqk) – Facebook Aug 15 '18 at 19:43
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    The first thing he did was asked Google Assistant which led to a Wikipedia page full of limits... – Shubh Khandelwal Aug 16 '18 at 01:06
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    @LaurencePayne: no, you can’t divide five cookies among zero people that way. Because that doesn’t work If you aren’t there. And if you’re there, that’s not zero people. – WGroleau Aug 16 '18 at 03:04
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    I'm pretty sure that my father explained $n/0=∞$ using limits to me at that age. (1 / ½ = 2, 1 / ⅓ = 3, as we make the divisor closer to zero, the result gets larger. It's easy enough to chose a divisor which makes the result larger than any given value => 1/0 is infinitely large.) – Martin Bonner supports Monica Aug 16 '18 at 10:45
  • Then you go on to show that as the divisor approaches zero from below, the result tends to $-∞$, and you end up saying "it is meaningless". – Martin Bonner supports Monica Aug 16 '18 at 10:47
  • The divisor does not get closer to zero, only in some absolute sense, otherwise the process is bound to never reach zero so proximity is meaningless. – Sentinel Aug 16 '18 at 11:49
  • You shouldn't try to do that. Instead make counter question. "What should it be, then?" and let them think about it. – mathreadler Aug 16 '18 at 14:49
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    Divide 10 by 2 by subtracting 2 repeatedly. Then try it again, dividing 10 by 0. There's nothing really mysterious about the fact that you could try for hours and make no progress at all. – Daniel R Hicks Aug 17 '18 at 01:24
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    The same reason we don't print zero-dollar bills. – Wyck Aug 17 '18 at 05:09
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    The problem 5th graders may have with this may be that there is something in mathematics that has no solution. This may be the first time they encounter this phenomenon. Maybe just present them with more problems that do not have a solution. For example: You have two apples and eat three of them. What is the result? ... In the end failing division by zero will not seem so odd any more. – Trilarion Aug 17 '18 at 07:33
  • Teach your child one-point compactification of the reals, and then when they don't understand say: "see its easier if we just pretend 1/0 doesn't exist." That's how I first came to terms with it. – tox123 Aug 17 '18 at 14:36
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    ? What's wrong with teaching 1/0 = infinity. Why happens when you ask him what happens to the result if the numerator stays the same and demonimator halves ... and halves. – Randy Zeitman Aug 18 '18 at 00:43
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    @RandyZeitman Infinity is not a number, and 1/0 does not have a value, and 1/0 = infinity is, depending on the details of what you mean by its parts, meaningless and/or valueless and/or false and/or undefined. But it's not true for division & equality of reals. – philipxy Aug 18 '18 at 01:22
  • @philipxy You're welcome to address the point I made - not points I did not make. – Randy Zeitman Aug 18 '18 at 21:39
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    @AlvinLepik The same argument (cannot divide between 0 people) can be made for $\pi$ people, or even 0.5 people; it is not convincing. A lot of things that were labeled "impossible" earlier in life are all of a sudden part of the curriculum this year ;-). – Peter - Reinstate Monica Aug 19 '18 at 03:18
  • @PeterA.Schneider it'll be ok for a few years. I'll think of something better if little brother presses the issue :D – AlvinL Aug 19 '18 at 06:41
  • I've always thought of it this way: You have 100 legos and 5 bins. How do you divide your legos amongst five bins evenly? 20 legos per bin. What about 4 bins? 25 legos per bin. 3 bins? Well, you'll have to break one into thirds, but 33.3 legos per bin. 2 bins? 50 legos. 1 bin? 100 legos. Zero bins? Uh . . . can't really do anything with zero bins. Of course, we could continue and say what about -1 bins? And this is where it gets weird, because now the real world example still doesn't work . . . but it's mathematically valid. -100 legos. But how do you explain -100 legos? – Jonathan Kuhl Aug 20 '18 at 00:53
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    @RandyZeitman he addressed it perfectly. what is wrong with $1/0=\infty$ is that it is not true. – The Count Aug 20 '18 at 15:57
  • @jkd Math doesn't make sense at first. That's just going to cause problems later. –  Aug 20 '18 at 21:15
  • I am speaking to the OPs question - a request to help explain why division by zero is meaningless, not a mathematical technicality. The answer is that it's not meaningless ... it's a limit ... and you explain that by showing what happens as denominator approaches zero. – Randy Zeitman Aug 21 '18 at 02:44
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    @Wyck erm.... your explanation would fail in the Eurozone :-) https://www.banknoteworld.com/zero-euro/ – Radovan Garabík Aug 21 '18 at 13:36
  • "but he cannot grasp the concepts of limits and integration just yet." You sure? – ibuprofen Sep 13 '18 at 04:17

21 Answers21

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“One of the ways to look at division is as how many of the smaller number you need to make up the bigger number, right? So 20/4 means: how many groups of 4 do you need to make 20? If you want 20 apples, how many bags of 4 apples do you need to buy?

So for dividing by 0, how many bags of 0 apples would make up 20 apples in total? It’s impossible — however many bags of 0 apples you buy, you’ll never get any apples — you’ll certainly never get to 20 apples! So there’s no possible answer, when you try to divide 20 by 0.”

Peter LeFanu Lumsdaine
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    A good enough answer till the little brother grows up and asks why it can't be $+\infty$ – Peeyush Kushwaha Aug 15 '18 at 13:17
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    @PeeyushKushwaha, then you introduce them to a number system where 1/0 _is_ $\infty$ – ilkkachu Aug 15 '18 at 13:22
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    The point is, division doesn't mean what the kid probably thinks it means - division is defined to be the inverse of multiplication, it's not about chopping up apple pies. And if that's too abstract for him to wrap his head around for now, then honestly he may as well just say $6/0 = 6$ or whatever seems reasonable to him. – Jack M Aug 15 '18 at 13:36
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    @PeeyushKushwaha: Then you explain that you can introduce such a symbol and define 1/0 that way if you want, but that it doesn't behave like a number and whether it's useful for modelling anything will depend highly on the situation. – R.. GitHub STOP HELPING ICE Aug 15 '18 at 14:13
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    Well, this can really be understood by even my brother now! Thanks for the good explanation. There are tons of videos on YouTube which are good, but just too difficult for kids to grasp. And this is the time when their basic concepts are built. So, a BIG Thank you from my side. – Shubh Khandelwal Aug 15 '18 at 14:18
  • @PeeyushKushwaha Division by zero can mean anything you find convenient. It is not necessarily infinity. – Dúthomhas Aug 15 '18 at 14:30
  • To understand the problem intuitively...Division as the reverse of multiplication says that 6/3=2 because 2x3=6. The quirk is that because 0 x anything = 0, the division doesn't work anymore. 6x0=0, meaning that you're essentially saying that by equivalency 0x0=6. or indeed any other number you choose to put in the equation. This is probably nothing like the underlying reality of the math, but it's probably a fair explanation for a 5th grader. The rules break and you can't divide by 0 and get meaningful results. **Edit** just realised Jack M already explained it this way :P derp. – Ruadhan2300 Aug 15 '18 at 14:38
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    @JackM: If you look at my answer carefully, you’ll see I really am describing division as the inverse of multiplication (specifically, in ℕ), and arguing that 0 has no multiplicative inverse. I’m just presenting it concretely, because to just about anyone short of a mathematically fairly mature undergraduate, that’s clearer and more convincing than a formal algebraic proof. This isn’t “dumbing down” either — the real-world understanding of division has just as much claim to being the “real thing”, and mathematics was being done well for centuries without modern notions of definition and proof. – Peter LeFanu Lumsdaine Aug 15 '18 at 15:14
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    @PeterLeFanuLumsdaine This is why it's a pretty good answer. I would say you should explicitly bring up the fact that division is *by definition* the inverse of multiplication (I always like to say that division is just the name we give to the process of answering a certain question about multiplication), but that might be over an 8 year old's head. – Jack M Aug 15 '18 at 15:16
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    From looking at the comments, I see that it is not just your kid brother, that does not get it. @ilkkachu 1/0 = ∞ is false. It may seem that way as you approach from positive y=1/x as x→0, when x starts positive, then y →0, but what about when x starts negative or if x starts as a multiple of $\sqrt(-1)$ or … – ctrl-alt-delor Aug 15 '18 at 15:17
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    @ctrl-alt-delor, that's why I said to pick a system where it works: https://en.wikipedia.org/wiki/Riemann_sphere – ilkkachu Aug 15 '18 at 15:36
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    @ctrl-alt-delor or IEEE 754 floating point: 1/+0 = +$\infty$, 1/-0 = -$\infty$. – user71659 Aug 15 '18 at 15:40
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    "What's with the quotation marks?" – Kimball Aug 15 '18 at 22:24
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    @PeeyushKushwaha But, $\infty$ doesn't seem to break this analogy! If you have infinitely many empty bags, an apple (or $20$) will not simply materialise! – Theo Bendit Aug 16 '18 at 00:56
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    @Kimball: I wanted to conjure up the image of OP talking to their younger brother, rather than of me addressing OP. – Peter LeFanu Lumsdaine Aug 16 '18 at 08:57
  • I like how this answer demonstrates that a number divided by zero is not infinite despite how zero can go into 20 infinite times, but that there just is no numerical answer. I mean, infinite zeros added together is still zero. If you can turn a pie into eight pieces (1/8) can you turn a pie into zero pieces (1/0)? (That's rhetorical, but you can't do it without eliminating the pie altogether.) – Brōtsyorfuzthrāx Aug 16 '18 at 10:57
  • _"20/4 means: how many groups of 4 do you need to make 20? If you want 20 apples, how many bags of 4 apples do you need to buy?"_ These are **not** the same questions (though they have the same answer since multiplication is commutative in N), and might disturb. – YSC Aug 16 '18 at 13:30
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    This argument struggles a bit when you consider 0/0. – nurdyguy Aug 16 '18 at 21:32
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    @nurdyguy I think this argument is actually pretty elegant for 0/0. If you can steer the learner to the idea that 1 bag, 0 bags, and a million bags of zero apples are all reasonable answers, you can then point out that it's unreasonable for a question to have pretty much unlimited answers, so we have to say that we just can't give an answer. It's mathematically founded while also letting you subtly introduce the concept of "well-defined". – Erdős-Bacon Aug 16 '18 at 22:06
  • @Erdős-Bacon Actually, that is exactly what I do when teaching College Algebra. I'm not saying the explanation here is bad but rather that it stops short of being complete. It really just needs to add exactly what you described to correct this. – nurdyguy Aug 16 '18 at 22:19
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    I like this answer as it beats the “you’ll open a black hole every time you do” answer I was given as a kid. – Eric McCormick Aug 17 '18 at 11:53
  • @Theo Bendit - Are you sure? You should try the experiment and see what happens instead of just declaring what the result will be! ;-) – Paul Sinclair Aug 17 '18 at 15:27
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    As someone who once taught 5th grade math & science I would definitely recommend this method with one addition that is not explicitly stated (but may be implied): use a physical object. Asking a student to "split these 20 apples into groups of 0" and letting them work through on their own with guidance ("well I have 20 left" *but that means you have 1 group of 20 not 20 groups of 0*) gets them engaged and is easier to grasp than abstract concepts for most – LinkBerest Aug 17 '18 at 20:15
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    Your entire answer is in a pair of quote marks. What is the source you're quoting? – jpmc26 Aug 17 '18 at 23:04
  • You comment that you are presenting division as inverse of multiplication, and so presenting is good, but the actual text in your question does not do that. Doing that would be, is there a number (or numbers) we can multiply by zero to get that & the answer is no. Eg your actual text begs the "infinity" response, although, of course, infinity is not a number--and again the explanation is, there's no number that multiplied by 0 etc. Ultimately we must correctly justify our axioms/model. Eg adding the point at infinity does give a workable system, but a *different* system. – philipxy Aug 18 '18 at 00:05
  • +1 You answered a question I was wondering about my whole life! – Artur Aug 18 '18 at 00:34
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    @jpmc26 read the comments before trying to be pedantic – Darren H Aug 18 '18 at 07:42
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    @DarrenH I see nothing in the comments that answers jpmc26's question. If there is an easy answer, consider giving it instead of throwing shade on the person asking the question. – Yakk Aug 20 '18 at 15:42
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    @Yakk see the answerers comment from Aug 16 at 8:57. I'll quote it here for you. "I wanted to conjure up the image of OP talking to their younger brother, rather than of me addressing OP". That comment was in direct reply to someone who previously asked exactly the same question as jpmc26 and answers it exactly – Darren H Aug 20 '18 at 15:51
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    @DarrenH Thanks, spotted it now. – Yakk Aug 20 '18 at 16:05
  • Thus, the colloquial name "gozinta"s for division, as in "4 gozinta 20 5 times." – Steven B. Segletes Aug 20 '18 at 19:45
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When we first start teaching multiplication, we use successive additions. So,

3 x 4 = 3               | 3
          + 3           | 6
               + 3      | 9
                    + 3 | 12
=12

Division can be taught as successive subtractions. So 12 / 3 becomes,

12 - 3 -> 9 (1)
9 - 3 -> 6 (2)
6 - 3 -> 3 (3)
3 - 3 -> 0 (4)

Now apply the second algorithm with zero as a divisor. Tell your brother to get back to you when he's done.

While this algorithmic approach is not rigorous, I think it is probably a good way of developing an intuitive understanding of the concept.

Chris Cudmore
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    I should comment that in some schools of thought, teaching young students that multiplication is repeated addition can be hindering. I've certainly seen kids who only count on their fingers - they can't actually multiply, just multiple add. So there are arguments from Ring Theory to education where you may be wrong, although it is perfectly reasonable to say in the integers. – theREALyumdub Aug 15 '18 at 13:49
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    I can't comment on the first part, but for the second part; We're talking about a 9 year old. One of the problems with mathematical pedagogy is we prioritize rigour over understanding. I suggest we provide the understanding first, then make it rigourous. After all, we have thousands of years of mathematical development before the hard rigour came in in the 19th century. – Chris Cudmore Aug 15 '18 at 13:54
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    I agree completely, and I have hit walls with mathematics involving rigour and a lack of understanding. I was more making the point that this sounds much like a computerized argument, and I have blindly heard of education arguments against this method of approach for young children - it's more or less my best understanding of division, but it can be made more abstract and perhaps more practical for education. – theREALyumdub Aug 15 '18 at 13:58
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    @theREALyumdub, the ring theorist will appreciate that every abelian group is a $\mathbb Z$-module in a natural way. – Carsten S Aug 15 '18 at 17:55
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    @ChrisCudmore Seeing as most college educated people *never* get a rigorous definition of almost any of the mathematical concepts they were taught, I don't think an over-emphasis on rigor is the problem. You seem to be confusing calculation with rigor. Frankly, I don't think the wave of formalization had almost any impact on how math was/is taught, at least pre-college. I strongly suspect that there is even *more* of an emphasis of intuitive understanding in grade school classrooms now than in the 1810s say. – Derek Elkins left SE Aug 15 '18 at 19:13
  • @theREALyumdub, that’s true if you stop too soon. A good next step is to show that the method is impractical for many situations and thus we have this more powerful technique. – WGroleau Aug 16 '18 at 03:06
  • @theREALyumdub It's not like I multiply numbers. If you ask me what 9*6 is, I'll just ask my [System 1](https://en.wikipedia.org/wiki/Thinking,_Fast_and_Slow). I'm not sure how I would consciously do the multiplication, other than repeated addition. – Acccumulation Aug 16 '18 at 20:43
  • @Accumulation Yes, it appears I held to the opposite view and the smart people here do not like what I say. I think that you expressed the other view well. I prefer mind over matter in a world of machines. But the comments are not for extended discussion - talk with me in chat if you wish to say something, publically or otherwise. – theREALyumdub Aug 17 '18 at 21:56
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    @Accumulation your system 1 knows 9*6=54 but if it didn't would you really add up 9 six times? You wouldn't do 10*6=60 then subtract a 6? Consider if the posed question was 92*99. Your system 1 probably doesn't know the solution instinctively, but surely you aren't going to add 92 to itself 99 times. No, you'll say 92*100=9200 then subtract 92 to give 9108 – Darren H Aug 18 '18 at 07:50
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New story

Suppose that we can divide numbers with $0$. So if I would divide $1$ with zero i would get some new number name it $a$. Now what can we say about this number $a$?

Remember:

If I divide say $21$ with $3$ we get $7$. Why? Because $3\cdot 7 = 21$.

And similiary if I divide $36$ with $9$ we get $4$. Why? Because $9\cdot 4 = 36$.

So if I divide $1$ with $0$ and we get $a$ then we have $a\cdot 0 =1$ which is clearly nonsense since $a\cdot 0 =0$.


Old explanation:

Suppose that ${1\over 0}$ is some number $a$. So $${1\over 0} =a.$$ Remember that $$\boxed{{b\over c} = d\iff b = c\cdot d}$$ So we get $$1= a\cdot 0=0$$ a contradiction. So ${1\over 0}$ doesn't exist.

nonuser
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    I see a lot of people doesn't like this answer although it is perfectly correct. I agree it is perhaps to advance for 5th grader but then again how else could I do it? There is a nice way Peter gave, but what if brother asks what is 5/4. How many bags with 4 apples do we need to get 5 apples? – nonuser Aug 16 '18 at 06:14
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    The logic proposed in this answer seems constructive, but a plain-English explanation would probably be more helpful. Separately, it's probably best to avoid saying that $\frac{1}{0}$ "_doesn't exist_"; it'd seem a bit better to say that $\frac{1}{0}$ doesn't cleanly match up with a number. – Nat Aug 16 '18 at 06:51
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    You have to be careful when declaring contradictions. If you arrive at a "contradiction", it's actually a consequence of asserting the truth of the statements used. If you arrive at $12 = 0$ as a result, it really is just asserting a mod 12 system. if you accept $\frac{1}{0} = a$ as a statement, then you are in a mod 1 system as a consequence. Suppose that you assert a statement that only after millions of manipulations asserts that $0 = 2^{32}$. Not only is it not a "contradiction", but it leads to the most general solution. On a computer, you have to deal with this frequently. – Rob Aug 16 '18 at 17:21
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    I really think we should be striving to get to a point where the 'older argument' is something a fifth grader can comprehend. – Prince M Aug 16 '18 at 19:02
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    This is the only correct and valid answer: I do not understand how the "distribution of apples" is any easier than just applying the definition. – gented Aug 17 '18 at 12:13
  • Your explanations are the same. Merge them together to create a single one that discusses the issue in both words and equations. – jpmc26 Aug 17 '18 at 23:07
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    @LightnessRacesinOrbit I think it's accessible for a bright 5th grader (I ran a math club last year at my son's school, grades 1-6). School "math" is often pretty dumb and at the same time "mechanically" hard to do; "real" math like this is often the opposite. I think kids are often underestimated -- the brain is all there, it's experience which is lacking. There is no principle hurdle to logical reasoning at any age but it needs some fun with mental puzzles. – Peter - Reinstate Monica Aug 19 '18 at 03:27
  • I must know some exceptionally smart fifth graders because this would work with the ones I know. Maybe not a third grader, but that's when we first teach long division around here. By fifth grade they're learning stuff like absolute value and the beginnings of Algebra. I would expect most 5th graders to understand that subtraction is the inverse operation of addition and that division is the inverse of multiplication, and that $1/a$ is the same as dividing by $a$. I was taught in second grade that division is the inverse of multiplication. How else would you introduce it? – Todd Wilcox Aug 19 '18 at 17:16
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An explanation that might make sense to a fifth grader is one that gets to the heart of why we have invented these operations in the first place.

Multiplication is a trick we use to add similar things to form a sum. When we say 5 x 3, what we really mean is take five things of size three each and add them all together. We invented this trick because we are frequently in the situation where we have many of a similar thing, and we wish to know their sum.

Division is the same trick but the other way. When we say 15 / 3, we are asking the question "how many times would we have to add a thing of size three starting from nothing to make a thing of size fifteen?" We'd have to add five things of size three together to make a thing of size fifteen. Again, division is just a trick we use to answer questions about sums.

Now it becomes clear why division by zero is not defined. There is no number of times you can add zero to itself to get a non-zero sum.

A sophisticated fifth grader would then note that 0 / 0 is by this definition defined as zero. Going into why 0 / 0 is not defined would require more work!

For non-zero divided by zero, there is no number at all of times that you can add zero to itself to get non-zero. For zero divided by zero, every number of times you add zero to itself, you get zero, so the solution is not unique. We like our mathematical questions to have unique answers where possible and so we by convention say that 0 / 0 is also not defined.

Eric Lippert
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    Actually I have never thought of division like that. 15/2 has always meant 'how big is each half when you cut it in two' to me. – Sentinel Aug 16 '18 at 03:56
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    "how many times would we have to add a thing of size three to itself to make a thing of size fifteen?" 3+3=6; 6+3=9; 9+3=12; 12+3=15. I count four additions. Your question is therefore not worded correctly. It's not adding three to itself, it's "Starting with zero (nothing), how many 3s do you need to add to get 15?" – Monty Harder Aug 16 '18 at 14:53
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    @MontyHarder: Good point; I'll change the wording. – Eric Lippert Aug 16 '18 at 15:42
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    "A sophisticated fifth grader would then note that 0 / 0 is by this definition defined as zero. Going into why 0 / 0 is not defined would require more work!" Simple enough. Grab a plate and set it on the table and say "there are 0 cookies on this plate". Grab another and set it down. "there are 0 cookies on each plate, for a total of 0 cookies. If you leave the room and all you know is that there are 0 total cookies, how can you know how many empty plates I put on the table (or picked back up) when you were gone?" – Monty Harder Aug 16 '18 at 16:31
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The Wikipedia article Division by zero lists the usual arguments why there is no good choice for the result of such an operation.

I prefer the algebraic argument, that there is no multiplicative inverse of $0$, this would need you to explain a bit about algebra.

The argument from calculus, looking at limits of $1/x$, I find also useful, but perphaps harder to explain.

mvw
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    To me this is the answer, though I would combine it with "let the 5th grader try to come up with some ideas, and help them see why they fail." Personally, I like this approach because then, when they come across the sqrt(-1), they're going to be more comfortable when we say "actually, there *is* a good choice for how to deal with this." – Cort Ammon Aug 17 '18 at 18:07
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How many nothings do you need to add together to get 12?

Sentinel
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    Add all the nothings! – Don Branson Aug 15 '18 at 19:54
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    @Sentinel Ok, Donnie. I did that. I found all the nothings and added them, but I still have nothing. What now? – Sentinel Aug 16 '18 at 03:59
  • Not in 5th grade, maybe, but by 7th or 8th I think I might have asked "what if you added an infinite number of nothings?" Running into the same issue as some other explainations. – Ian D. Scott Aug 17 '18 at 15:48
  • @DonBranson Of course. Mathematicians have done for 4 centuries already... Infinitesimals.. :) Imagine integrating constant function with value 12 over $[0,1]$ or value 1 over $[0,12]$ – mathreadler Aug 17 '18 at 16:39
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    @IanD.Scott infinitesimals and calculus. That is why you should not tell kids that/why you can't divide by 0. Playing with the idea of multiplying "almost 0" with infinity is so fruitful it is more valuable if they wrestle around a bit with it for themselves. It is not inconceivable that a kid could come up with some variant of infinitesimal calculus based on adding infinite many very small slices. – mathreadler Aug 17 '18 at 22:08
  • I don't agree. Adding a an *actual* infinity of infinitesimals vs a *potential * infinity of *potential* infinitesimal outcomes is mathematical philosophy still even today contentious. – Sentinel Aug 20 '18 at 20:22
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Ask Siri.

Imagine that you have zero cookies and you split them evenly among zero friends. How many cookies does each person get? See? It doesn't make sense. And Cookie Monster is sad that there are no cookies, and you are sad that you have no friends.

Evorlor
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    If you include your imaginary friend, then the general form for dividing $X$ cookies is $-Xi$, because $(0+i) * (-Xi) = X$ – Chronocidal Aug 15 '18 at 22:43
  • That is like saying that division by two is slicing evenly in half. Most other answers here think it is about groups of two. – Sentinel Aug 17 '18 at 21:18
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@Jack M and @greedoid probably highlight a good point: division does not exist. It's only the inverse operation of multiplication.
You could explain your brother the complete truth: dividing 20 by 5 is about finding the only answer (if it exists) to this question: what number can be multiplied by 5 to give 20?. The unique answer is easy: 4 times 5 is 20.
And the division is only another phrasing to say the exact same thing: 20 divided by 5 is 4.
Can you always find one and only one answer? Yup, almost always...
There's only one exception...

What number, multiplied by 0, gives 20? There's none.
So "division" by 0 has no meaning, since we cannot find any number that satisfies our definition.

You could even draw his attention by mentioning that most grown-ups don't know there's no such thing as "division", and that's the first step to learn about "E-vector spaces", "rings" and other funny-named artefacts when he's in college... or before that!

Note: what if he raises a question about "0/0"?
OK, let's try: "what number, multiplied by 0, gives 0?" All of them! We cannot find one and only one answer, so, it's still impossible to divide 0 by 0!

Evariste
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You shouldn't try to do that. Instead make counter question.

"What should it be, then?" and let them think about it.


(Lengthy) justification: There are many important concepts in math you can come up with if you start experimenting with multiplication. Take for example area of a rectangle. You multiply the sides. Area of a curve? You take the integral. What is an integral? Well Riemann imagined thin thin slices, almost infinitely thin, actually. The idea that we can calculate area of these slices where one side is so tiny it almost is 0. If we disqualify limits, or the idea of multiplying something "almost 0" to be 0 then we would have a tougher time coming up with an excuse to investigate integrals, which have been veeery important to the development of modern technology.

Any kid who could come up with some new interpretation of this could be very valuable.

mathreadler
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Division by zero is meaningless because that's what we decided division means. All you can do is explain why such a convention is a useful one for ordinary arithmetic.

It might even help to demonstrate some other context (e.g. arithmetic in the projectively extended number line) where it can be useful to define division by zero, so that the student is able to compare and contrast the reasons why we might or might not like to define something.

Your question might be better placed on https://matheducators.stackexchange.com/

  • I believe your first sentence is circular logic – Ruadhan2300 Aug 15 '18 at 15:59
  • @Ruadhan2300 No, it's not circular. I think Hurkyl's point is that all mathematical concepts are just definitions that we decided on. You could define division differently, as $1/0 = 37$ and still develop all of modern mathematics; it would only be less convenient, not less "correct". – 6005 Aug 17 '18 at 15:45
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    That being said, I think this meta-explanation may be a bit too difficult to grasp for a 5th grader. At that stage, most students think of definitions as immutable truth. – 6005 Aug 17 '18 at 15:47
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    Much like most of science, all mathematics is rooted in modelling reality. The rules and concepts we produce exist entirely because we found they apply to real situations and remain consistently effective. I maintain that "because that's the definition we gave it" is circular and unhelpful. – Ruadhan2300 Aug 17 '18 at 16:05
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I don't have kids (my wife says one 3-year-old in the house is enough for her) and it's been a while since I was in the 5th grade (although at work sometimes...), but I'll give it a go.

I know you're too old to play with blocks, but lets start with 12 blocks.

Let's start with $12/6$ - that's $2$, right? Take $6$ at a time and there are two "sets". There are $2$ sets of $6$ in $12$.

Then $12/4$ is $3$ - $3$ sets of $4$ in $12$.

Then $12/3$ is $4$ - $4$ sets of $3$ in $12$ (commutation of the last case).

Then $12/2$ is $6$ - $2$ sets of $6$ in $12$ (commutation of first case).

Then $12/1$ is $12$ - $1$ set of $12$ in $12$ (degenerate case).

Notice the size of the result set is getting bigger as the denominator (the number on the bottom) gets smaller.

Before we go to $0$ let's try something between $1$ and $0$ - $1/2$ or $0.5$. Think of just splitting each block into two (take a hatchet to the wooden blocks blocks, or just imagine it if mom doesn't want you handling a hatchet).

$12/0.5$ is $24$ - $24$ sets of $0.5$ (half-pieces) in $12$

$12/0.25$ is $48 - 48$ sets of $0.25$ (quarter-pieces) in $12$

$12/0.125$ is $96 - 96$ sets of $0.125$ (pieces of eight**) in $12$

$12/0.0625$ is $192 - 192$ sets of $0.0625$ (pieces of 16) in $12$

The close you get to zero, the larger the set you get gets.

$12/0.000000001$ (a billionth) is $12$ billion sets of a billionth of a block (aka, sawdust)

The as you approach zero, the resulting set size is too large to represent (not enough paper in this room, not enough memory on this computer) and the size of the pieces approach zero.

A cheat for "Too large to represent" is "infinity".


** pirate reference - do 5th graders still like pirates these days?

JayTuma
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Nufosmatic
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  • The problem I think is a conceptual one, As you describe, it's actually talking about sets, and while you can have a single set containing everything, you can't logically have less sets than you started with. "I want you to put all of these objects into a box too small for any of them" would be a similar task. The only appropriate response is to take Exception at the impossible task. – Ruadhan2300 Aug 15 '18 at 14:30
  • This is how my grandma explained it to me when I was like 5. I got it immediately and it stuck with me ever since. – htmlcoderexe Aug 16 '18 at 07:11
  • A "piece of eight" (or, in what is perhaps a more "piraty" version, "piece o' eight") was a Spanish coin that was worth eight Spanish reales. So the piece of eight was the whole, while the real was the eighth. Another term for a real was a "bit". This survives in a bit being one-eighth of a byte, and in "shave and a hair-cut, two bits [i.e. 25 cents]". So you could say "half-piece, quarter-piece, bit-piece". – Acccumulation Aug 16 '18 at 20:49
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One would need to first explain what we mean by division. That is, what does $/$ mean in the expression $a/b,$ where $a$ and $b$ are integers?

Well, whatever it is, it is a way of combining two numbers. Now recall that every time we defined an operation (say addition), we always had a unique result as the product of the combination, so that we would like this to continue to hold. What else? We define $/$ indirectly, by looking at what we want $a/b$ to mean. Well, we want it to stand for the number $c$ which when multiplied together with $b$ recovers $a.$ (Recall how we similarly defined subtraction as the inverse operation of $+.$)

Therefore, in summary, if we let $a/b=c,$ then by definition this equality is equivalent to $c×b=a.$ Also, we want $c$ to be unique for all possible integers $a$ and $b.$

Now consider the expression $a/0.$ First let us take $a\ne0.$ Then if we let $a/0=c,$ it follows by definition that $c×0=a.$ But with the way we defined multiplication (remind him of this), we required that $0$ must make any number vanish, so that there simply is no such $c$ as we seek. If now we let $a=0,$ then we want a unique $c$ such that $c×0=0.$ But again, by the property $r×0=0\,\,\,\forall r$ which we've previously allowed in defining $×,$ we have infinitely many candidates for $c$ and there is no other condition we can impose to select one uniquely. We therefore do not allow ourselves to divide by $0$ in any case, in order to avoid all that mess.

Allawonder
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Division is sharing:

1 / 10:

10 boys in a class grab at a toy -- they rip the toy to tiny bits!

1 / 2:

2 boys fight for a toy -- they rip the toy in half!

1 / 0:

A different toy is alone -- he is a special boy!

Jason
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The way I taught it, even to junior college students who were taking elementary mathematics courses, was with a calculator.

I would show them that 1/1 = 1, 1/0.1 = 10, 1/0.01 = 100, and so on. I would ask them if they saw how the numbers kept getting bigger as we divided by smaller and smaller numbers. Then I would ask them what they thought would happen when we hit zero. "We would get the biggest possible number that exists, right? But there is no biggest number. So dividing by zero gives you a number that doesn't exist. Does that make any sense? No. So we say that dividing by zero is undefined."

user2303321
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Number of marbles : Number of boxes = Number of marbles in each box.

20 marbles : 4 boxes = 5 marbles per box

0 marbles : 4 boxes = 0 marbles per box

20 marbles : 0 boxes = "how many marbles in each box while no box?" ---> undefined!

Friendly Ghost
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Because before you think about dividing something, it is more important to consider if you have someone to divide it for (he/she/it must be present, exist, etc). If you do not have anyone who can 'benefit' from the division, no point in dividing. Non rigorous, pragmatic, heuristic approach. It might pave the way for more reasoned proofs and demonstrations.

XavierStuvw
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  • This is indeed similar with the ['division is sharing'](https://math.stackexchange.com/a/2885271/446004) concept in the answer of @Jason given here – XavierStuvw Aug 17 '18 at 15:06
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To divide means to subtract many times. So, how many times can we subtract $0$ from a given number?

It might be a duplicated answer and I apologize, in case. But, according to my experience as a teacher, this worked well.

The point, as others had observed, is what does "to divide" mean. This sometimes looked obscure to the students, whereas the concept of subtraction was more clear.

Thus, once you convey the message that "to divide" means "to subtract many times", everything becomes more clear.

How many times can we subtract $3$ from $10$? Well, usually my students got this.

How many times can we subtract $0$ from $10$? Well, how many times we want!

So there is not a precise answer, because any answer is good. This made more clear the sense of "not defined", at least to my students.

Hope it helps!

0

Explain him the problems, don't enforce him as an "official view".

Explain him, what are the problems of the division by zero.

Let him to think about a possible solution.

You might also explain, that also the negative numbers don't have a suqare root, but this problem had a solution, the imaginary numbers. Let him try to think about a similar solution for the division by zero.

peterh
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The following explanation in terms of division as the inverse of multiplication may help, as modern fifth graders should have been introduced to the idea of division as something that undoes multiplication.

6/2 = 3. Why? Because 3 * 2 = 6 and division means find the number (3) that multiplies the dividing number (3) to give the number being divided (6). To divide 6 by 2, we ask what number, when multiplied by 2, gives 6.

Ask your brother to do this exercise for 6 and 0. What number, when multiplied by 0, will give 6? He should see the problem here, because, no matter what number we try, when we multiply it by 0, we get the same answer 0.

A diagram might help to bring the problem into sharper sight. What you're doing in the following is conveying the lack of bijectivity of $x\mapsto 0 \times x$, in age appropriate words, of course ...

enter image description here

The left hand diagram shows the mapping $x\mapsto 2 \times x$; encourage your brother to think of multiplication as a stretch or shrink induced on the number line. The crucial property to note here is that every arrow on the diagram is reversible, meaning that you can find one and only one number that 2 multiplies to get the answer. Every answer is 2 times a unique something. Multiplication by 2 is reversible - use this word - in the sense that we do not lose the knowledge of what has been multiplied by 2 to get the answer.

The same kind of situation holds for every nonzero multiplier - the real line is stretched or shrunken, and sometimes flipped in orientation as well, but we can always work out what was multiplied originally to arrive at the end of any given arrow.

Now have your brother look at the diagram for $x\mapsto 0 \times x$. Everything goes awry because all the arrows wind up at the image 0. Given only our answer (0), we have no idea what we multiplied by 0 to get the answer, because it could have been any real number. Multiplication by 0 destroys the knowledge of what was multipled.

Later on, your brother might like to come back to this idea to understand the pole of $z\mapsto 1/z$ at 0 in a bit more detail: multiplication by a very small number $\epsilon$ corresponds to a very severe shrink, but, as long as the number is not nought, the arrows do not quite merge and the shrink can be undone.

0 as a multiplier is a destroyer of information: no other real number is like this and this property is why we can't invert the multiplication. One boy in my daughter's class whom I explained this to (I help out with numeracy at my daughter's school) has a particular love and encyclopoedic knowledge of Greek, Hindu and other gods (I think he may know every pantheon conceived!). He was most chuffed to learn that $0$ was the "Shiva" number.

Selene Routley
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Try to make him realize himself that there's no solution.

Take a (imaginary) pizza.

Ask him to cut the pizza into one piece.
Ask him to cut the pizza into two pieces.
Ask him to cut the pizza into three pieces.
Ask him to cut the pizza into zero pieces.

corvus_192
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Just give him some questions e.g 2/0 ,5/0 ,6/0 and tell him to divide just using simple division tell him to keep on dividing till he reaches a satisfactory.Let him try for some time.And that satisfactory won't come how much me try.

Now you tell him that you will never come to a satisfactory result. Hence it's answer will be meaningless!!!

jame samajoe
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