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A Quick Note

I know there is a slough of related questions on stack exchange, but none of them really seem to answer my question. This post is the closest in relationship to my question, but the answer simply expresses a high level mathematical explanation, and not an example I can teach my kids. Growing up, my school always taught that division by zero was undefined or not allowed, but never really explained why, or how this was true.

The proposed duplicate has a very good answer, that I understand, however I'm not so sure my kids would understand that answer. The accepted answer will have to be understood by children under age 10 with a minimal working knowledge of multiplication and division.

Getting Started

The other day, I was working on a project at home in which I performed division by zero with a double precision floating point number in my code. This isn't always undefined in the computer world and can sometimes result in $\infty$. The reason for this is clearly explained in IEEE 754 and quite thoroughly in this Stackoverflow post:

Division by zero (an operation on finite operands gives an exact infinite result, e.g., $\frac{1}{0}$ or $\log{0}$) (returns ±$\infty$ by default).

Now, this got me thinking about basic arithmetic and how to prove each operation, and I created a mental inconsistency between multiplication and division.

Multiplication

As this is an important part of the thought process that lead me down this mental rabbit hole, I am including the elementary explanation of multiplication.

  • If I place $10$ marbles on my desk, $3$ times, I have placed $30$ marbles on my desk.
    • This is expressed as $10 \cdot 3 = 30$ and is true.
  • If I place $10$ marbles on my desk, $0$ times, I have placed $0$ marbles on my desk.
    • This is expressed as $10 \cdot 0 = 0$ and is true.

These two scenarios are true no matter what numbers are used.

Division

This is where things took an unexpected turn in my mind.

Let's say that I am a wandering saint and I have 50 apples. I want to help the hungry people of the world so I give my apples away freely. Now, let's handle two similar scenarios.

  • I come across $10$ people, and I want to give them all of my apples, I also want to ensure that each person receives the same number of apples. With $50$ apples to disperse across $10$ people, this means each person receives $5$ apples.
    • This is expressed as $\frac{50}{10} = 5$ and is true.

However, let's say I have the same $50$ apples, and I come across a town where no one is hungry, and no one wants my apples. Well, I have $50$ apples, and I have $0$ people to give them to, so I still have $50$ apples. I didn't disperse my apples evenly across any number of people, so it's still the same bag of $50$ apples.

I believe this may be my mind's way of bending the facts here, and that I've convinced myself that I'm dividing $50$ zero times, but in fact I may have divided $50$ one time (by me). But it has me thinking, if I divide a pizza into zero equal slices, well then I essentially didn't slice the pizza and thus still just have an entire pizza.

My Question

How can it be proved thoroughly, not just with math, but with an example explanation (understandable by children) that division by zero is truly undefined?

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    $\begingroup$ The explanations at this duplicate (linked to the above) give a thorough explanation, right? It has so many examples in the answers, too. This problem really has been explained so many times, with so many good answers. $\endgroup$
    – Dietrich Burde
    Apr 23, 2019 at 14:20
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    $\begingroup$ In your apple scenario, you are actually dividing 50 apples by 1 (giving them all to yourself). Here are some thoughts I typed up, which may be helpful: ee.usc.edu/stochastic-nets/docs/divide-by-zero.pdf You may like section F best , which talks intuitively about a photocopy machine shrinking/enlarging and the concept of invertible/non-invertible operations. $\endgroup$
    – Michael
    Apr 23, 2019 at 14:21
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    $\begingroup$ You have tricked yourself with your apples/pizza. If you were divvying up the $50$ apples to $10$ people, then you would have no apples left over. Similarly, if you came across $25$ people, you'd have none left over (and each other person would have $2$). The whole scenario ignores you, and assumes that you have none left over. If you take a fair share instead, then you are dividing it by a number one greater. To model division by zero in that case, even you yourself would disappear, and so would the remaining apples. Where did they go? $\endgroup$
    – Theo Bendit
    Apr 23, 2019 at 14:35
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    $\begingroup$ @Somos : Based on reading her post, I believe she wants the answer of $1/0$ to be 1, and $50/0$ to be 50, based on her pizza scenario, but she also knows that is not quite right. PerpetualJ : Before getting intuition about dividing 50 pizzas by 0 (which is not defined) you might try intuition on dividing 50 pizzas by $1/2$ (which is defined). $\endgroup$
    – Michael
    Apr 23, 2019 at 16:07
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    $\begingroup$ Re: Getting Started I don't understand why you would think a machine built by humans determines human conventions. I have never understood explaining to a student why $1/0$ is nonsense, and then they punch it into their calculator to confirm. What! I've never understood such muddleheadness. $\endgroup$
    – Allawonder
    Apr 23, 2019 at 17:28

5 Answers 5

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That division by zero is undefined cannot be proven without math, because it is a mathematical statement. It's like asking "How can you prove that pass interference is a foul without reference to sports?" If you have a definition of "division", then you can ask whether that definition can be applied to zero. For instance, if you define division such that $x\div y$ means "Give the number $z$ such that $y \cdot z =x$", there is no such number in the standard real number system for $y=0$. If we're required to have that $(x\div y) \cdot y=x$, then that doesn't work when $y$ is equal to zero. In computer languages where x/0 returns an object for which multiplication is defined, you do not have that (x\0)*0 == x. So we can can a class of objects in which we call one of the objects "zero", and have a class method such that "division" by "zero" is defined, but that class will not act exactly like the real numbers do.

Another definition of division is in terms of repeated subtraction. If you take 50 apples and give one apple each to 10 people, then keep doing that until you run out of apples, each person will end up with 5 apples. You're repeatedly subtracting 10 from 50, and you can do that 5 times. If you try to subtract 0 from 50 until you run out of apples, you'll be doing it an infinite number of times.

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    $\begingroup$ The example following glorified subtraction as is given with glorified addition for multiplication was perfect! Being able to explain why division by zero is undefined requires breaking the process down into terms they can understand. I can see their counter argument coming back as “well, you still have 50 apples” though. +1 $\endgroup$
    – Hazel へいぜる
    Apr 23, 2019 at 17:11
  • $\begingroup$ @PerpetualJ, I left a similar comment before I realized there was already an answer to this effect. :( In any case, I think you're hung up on the people and possession. If you give 10 people 50 apples, the apples still exist. Maybe think about baskets instead? "If I have 50 apples, I can put 1 apple into 10 baskets 5 times." I still have 50 apples; they're just in baskets now. "If I have 50 apples, I can put 1 apple into 0 baskets infinite times." Either way, I still have 50 apples. $\endgroup$
    – D. Patrick
    Apr 23, 2019 at 18:25
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Consider a problem where you have to divide a finite number like $5$ by zero. $5\div0$ is essentially a request for some number which when multiplied by zero gives you $5$:

$$5\div0=N\implies 0\cdot N=5.$$

Is there a number that when multiplied by zero gives you $5$? The answer is clearly no because any number times zero always gives you zero. Therefore, $5\div0$ is left undefined. "Undefined" here basically means that we can't explain what $5\div0$ really means.

What about the case $0\div0$? $$0\div0=N\implies 0\cdot N=0.$$

We know that any number times zero is zero. This means that $N$ can be any number at all. This kind of division problem gives you an infinite number of answers instead of just one as it should be. Because of this indeterminateness, $0\div0$ is also left undefined.

Here's another very simple example for good measure. You have $7$ pizzas and you want to divide them among zero people. How much pizza will each person get? Well, you have no people to give the pizzas to. You can pose that question and even write it mathematically as $7\div0$, but what could possibly be the answer to this question? Practically speaking, this is unanswerable. In other words, it's not clear what the statement $7\div0$ in this context means. In math-speak, we would say that this is undefined.

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  • $\begingroup$ Given the part of your answer that talks about $0\div 0$, shouldn't $0^0$ be undefined, too? I mean, for whatever value $k$ that you choose, $0^0=0^{k-k} = 0^k\div 0^k = 0\div 0$ so... $\endgroup$
    – Mr Pie
    Apr 29, 2019 at 23:48
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    $\begingroup$ How do you define $0^k$? If $k>0$, then $0^k=0$. In that case, you would be attempting to divide by zero ($0^k\div0^k$) which already has been left undefined. I don't think you can figure out what $0^0$ should be equal to this particular way. For a more thorough explanation, please consult the Wikipedia page that talks about raising zero to the zeroth power: en.wikipedia.org/wiki/Zero_to_the_power_of_zero $\endgroup$
    – Michael Rybkin
    Apr 30, 2019 at 0:00
  • $\begingroup$ A google search and my calculator both reveal that $0^0=1$, but yeah, I'll take a look at the page; I think it is a good idea to look into it a bit more. Thanks for that! $\endgroup$
    – Mr Pie
    Apr 30, 2019 at 0:17
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My understanding of division by zero goes back to the definition of rings. Let $R$ be a commutative ring and $a,b\in R$ with $b$ a unit in $R$. Then define the fraction $a/b$ as follows: $$\frac{a}{b} = a\cdot b^{-1}$$ i.e., division by $b$ is defined by multiplication with the inverse of $b$.

Since the zero element $0$ in a ring is absorbing (i.e., $a\cdot 0 = 0 = 0\cdot a$) and thus not a unit, division by $0$ is not defined.

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    $\begingroup$ The question clearly states that the explanation is targeted towards children. $\endgroup$
    – user1952500
    Apr 23, 2019 at 14:25
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    $\begingroup$ This is the first time I have seen someone explain division by zero by starting out "Let $R$ be a commutative ring..." $\endgroup$
    – Michael
    Apr 23, 2019 at 14:26
  • $\begingroup$ @Michael: In the algebraic sense, division is a derived operation as is subtraction. $\endgroup$
    – Wuestenfux
    Apr 24, 2019 at 12:11
  • $\begingroup$ @Wuestenfux : Your answer, and your comment to me, both seem misaligned with their intended audience. $\endgroup$
    – Michael
    Apr 24, 2019 at 19:35
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I explain it like this, ignoring the definition gap on purpose:

Have a look at the graph of $x/x$: it's a straight line at $y=1$. In this graph, we clearly see that $0/0=1$.

Then, look at $5x/x$: it's a straight line at $y=5$. We clearly see that $5*0/0=5$. Now this could be interpreted as $(5*0)/0 = 0/0 = 1$ (using the result of the graph before) or as $5*(0/0)=5$ (also using the result from before).

You can repeat this with other numbers as well, so the children can see that the result is arbitrary.

This should make it pretty clear that if we allow division by zero, other laws cannot hold. So it's better undefined.

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I think you've pretty much got it and others have gone into more detail on the various bits of maths related to this.

The simplest way to describe this for kids I can think of:

  1. Anything multiplied by zero is zero. Easy.
  2. Multiplying anything with two non-zero values gives a non-zero value. Shouldn't be a problem.
  3. Division of the result by one of those values gives you the other. We did the opposite of 2.

So 3 X 4 = 12, 12 / 4 gives you 3. 3 was the number you multiplied by 4 to get 12. We have a result 12, we ask "12 / 0 what was the other value multiplied by 0 to get 12"?

There is no such number because of statement 1 - hence divide by zero is undefined.

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  • $\begingroup$ Addendum: the basic mistake in the question is that how you get the answer - adding, subtracting or anything else - the process or algorithm of it - does not define the result. As different algorithms give different answers that shows the point. From a purely functional perspective where there are no algorithms this is the most sensible mapping for the inverse of the multiplication function. $\endgroup$
    – cyborg
    Apr 23, 2019 at 18:23

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