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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., $frac10$ or $log0$) (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 $frac5010 = 5$ 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?

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 $xdiv y$ 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 $(xdiv y) cdot y=x$, 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 == 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.

Consider a problem where you have to divide a finite number like $5$ by zero. $5div0$ is essentially a request for some number which when multiplied by zero gives $5$:

$$5div0=Nimplies 0cdot N=5.$$

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, $5div0$ is left undefined. "Undefined" here basically means that we can't explain what $5div0$ really means.

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

We know that any number times zero is zero. This means $N$ can be any number. This kind of division problem gives you an infinite number of answers instead of just one as it should be. Because of this indeterminateness, $0div0$ 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 $7div0$, 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 $7div0$ in this context means. In math-speak, we would say that this is undefined.

My understanding of division by zero goes back to the definition of rings.
Let $R$ be a commutative ring and $a,bin R$ with $b$ a unit in $R$.
Then define the fraction $a/b$ as follows:
$$fracab = acdot b^-1$$
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., $acdot 0 = 0 = 0cdot a$) and thus not a unit, division by $0$ is not defined.

Numberphile shows how if you define division by 0 as infinity (a specific kind of infinity) you get a 1=2 proof. Division, is basically a loop subtraction with a counter variable.(jives well with multiplication as repeated addition.) Division by 0 then, makes an infinite loop. And the limits have 2 values.

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.

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.