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?
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asked Aug 15, 2018 at 10:33
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87$\begingroup$ How do you divide, say, $5$ apples between zero people? There is no meaningful way to distribute the apples. $\endgroup$– AlvinLAug 15, 2018 at 10:36
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15$\begingroup$ Division by zero should not be justified at all. $\endgroup$– PeterAug 15, 2018 at 11:39
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35$\begingroup$ Just ask Siri. It explains this very well. $\endgroup$– DonielFAug 15, 2018 at 15:32
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17$\begingroup$ This would also fit well on the Mathematics Educators site. $\endgroup$– gen-ℤ ready to perishAug 15, 2018 at 16:10
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38$\begingroup$ "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 $\endgroup$– jkdAug 15, 2018 at 19:12
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21 Answers
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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.”
answered Aug 15, 2018 at 11:42
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82$\begingroup$ A good enough answer till the little brother grows up and asks why it can't be $+\infty$ $\endgroup$ Aug 15, 2018 at 13:17
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48$\begingroup$ @PeeyushKushwaha, then you introduce them to a number system where 1/0 is $\infty$ $\endgroup$– ilkkachuAug 15, 2018 at 13:22
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57$\begingroup$ @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. $\endgroup$ Aug 15, 2018 at 15:14
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25$\begingroup$ @Kimball: I wanted to conjure up the image of OP talking to their younger brother, rather than of me addressing OP. $\endgroup$ Aug 16, 2018 at 8:57
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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.
answered Aug 15, 2018 at 13:32
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8$\begingroup$ 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. $\endgroup$ Aug 15, 2018 at 13:49
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40$\begingroup$ 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. $\endgroup$ Aug 15, 2018 at 13:54
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4$\begingroup$ 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. $\endgroup$ Aug 15, 2018 at 13:58
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4$\begingroup$ @theREALyumdub, the ring theorist will appreciate that every abelian group is a $\mathbb Z$-module in a natural way. $\endgroup$ Aug 15, 2018 at 17:55
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7$\begingroup$ @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. $\endgroup$ Aug 15, 2018 at 19:13
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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.
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5$\begingroup$ 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? $\endgroup$– nonuserAug 16, 2018 at 6:14
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8$\begingroup$ 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. $\endgroup$– NatAug 16, 2018 at 6:51
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3$\begingroup$ 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. $\endgroup$– RobAug 16, 2018 at 17:21
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4$\begingroup$ I really think we should be striving to get to a point where the 'older argument' is something a fifth grader can comprehend. $\endgroup$– Prince MAug 16, 2018 at 19:02
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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.
answered Aug 15, 2018 at 13:20
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3$\begingroup$ 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. $\endgroup$– SentinelAug 16, 2018 at 3:56
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3$\begingroup$ "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?" $\endgroup$ Aug 16, 2018 at 14:53
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1$\begingroup$ @MontyHarder: Good point; I'll change the wording. $\endgroup$ Aug 16, 2018 at 15:42
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6$\begingroup$ "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?" $\endgroup$ Aug 16, 2018 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.
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3$\begingroup$ 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." $\endgroup$ Aug 17, 2018 at 18:07
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How many nothings do you need to add together to get 12?
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4$\begingroup$ @Sentinel Ok, Donnie. I did that. I found all the nothings and added them, but I still have nothing. What now? $\endgroup$– SentinelAug 16, 2018 at 3:59
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$\begingroup$ 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. $\endgroup$ Aug 17, 2018 at 15:48
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$\begingroup$ @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 $[0,12]$ $\endgroup$ Aug 17, 2018 at 16:39
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1$\begingroup$ @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. $\endgroup$ Aug 17, 2018 at 22:08
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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.
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9$\begingroup$ If you include your imaginary friend, then the general form for dividing $X$ cookies is $-Xi$, because $(0+i) * (-Xi) = X$ $\endgroup$ Aug 15, 2018 at 22:43
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$\begingroup$ That is like saying that division by two is slicing evenly in half. Most other answers here think it is about groups of two. $\endgroup$– SentinelAug 17, 2018 at 21:18
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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.
answered Aug 16, 2018 at 14:50
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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!
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$\begingroup$ There are a lot of buzzwords for a 5th grader, but the explanation is good. $\endgroup$– AlvinLAug 16, 2018 at 7:28
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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/
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$\begingroup$ I believe your first sentence is circular logic $\endgroup$ Aug 15, 2018 at 15:59
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$\begingroup$ @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". $\endgroup$ Aug 17, 2018 at 15:45
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2$\begingroup$ 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. $\endgroup$ Aug 17, 2018 at 15:47
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2$\begingroup$ 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. $\endgroup$ Aug 17, 2018 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?
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$\begingroup$ 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. $\endgroup$ Aug 15, 2018 at 14:30
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$\begingroup$ 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. $\endgroup$ Aug 16, 2018 at 7:11
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$\begingroup$ 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". $\endgroup$ Aug 16, 2018 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.
answered Aug 15, 2018 at 17:53
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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!
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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."
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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!
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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.
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$\begingroup$ This is indeed similar with the 'division is sharing' concept in the answer of @Jason given here $\endgroup$ Aug 17, 2018 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!
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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.
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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 ...
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.
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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.
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2$\begingroup$ We can divide with $-1$ but can't cut pizza in $-1$ pieces... $\endgroup$– MeowOct 14, 2018 at 18:14
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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!!!
