There are a number of ways of explaining why you cannot divide by zero. Here are my two favourites.

Any Set of Numbers Collapses to a Single Number

How old are you? Zero years old.

How tall are you? Zero metres old.

How many teeth do you have? Zero.

How many Superbowls has Tom Brady won? Zero

Yep, if you allow division by zero you only end up with one number to measure everything with.

In any number system we like we have to include zero and one. They are the identities for addition and multiplication; where a is any number we assume that zero and one have the properties:

a+0=a and a\times 1=a.

There are a number of things that we assume about any number system. One of them is the distributive law, that for any three numbers a,b,c:

a\times(b+c)=a\times b+a\times c.

As 0+0=0, for any number b, we have, using the distributive law:

b\times0=b\times (0+0),

\Rightarrow b\times 0=b\times 0+b\times 0.

Now, taking b\times 0 away from both sides (see here), we end up with (as 0\times b=b\times 0 by commutativity)

0=0\times b           (Theorem 1)

Zero times anything is equal to zero.

Now consider the inverse of multiplying (more on this below). Suppose that you have a number b and you multiply it by another number a:

b\times a.

We assume that there is another number, a^{-1} (spoken a-inverse), such that if we multiply by a^{-1}, it will undo the action of multiplying by a:

\displaystyle b\overset{\times a}{\longrightarrow} b\times a\overset{\times a^{-1}}{\longrightarrow} b,

i.e. b\times a\times a^{-1}=b.

This means that multiplying by a… and then by a^{-1}… is the same as multiplying by one:

b\times \underbrace{a\times a^{-1}}_{=1}=b\times 1=b,

so that, at the moment, for any number a, there exists a number a^{-1}, which, when multiplied by a, gives one. This number is also denoted by \frac{1}{a} and in fact is nothing but division by a:

\displaystyle \times a^{-1}=\times\frac{1}{a}=\div a;

and of course we have the following:

\displaystyle a\times \frac{1}{a}=1,      (Axiom 1)

a\div a=1.

So let us put Theorem 1 and Axiom 1 together:

0\times b=0  and a\times \frac{1}{a}=1.

These are supposed to be true of all numbers a and b. If we can divide by zero: that means that \frac10 exists as:

\displaystyle \div 0=\times 0^{-1}=\times \frac10.

Consider Theorem 1 and Axiom 1 with b=\frac10 and a=0:

\displaystyle 0\times \frac10=0,   by Axiom 1, but

\displaystyle 0\times \frac10=1,   by Theorem 1.

Of course no matter how we calculate 0\times\frac10 we must get the same answer. Some people stop here and say that one and zero are different and so we have a contradiction: we cannot allow the existence of \frac10 (i.e. we cannot divide by zero).

We can go a little further.  This result above means that one and zero must be the same number… is it possible that there is a new number system where one and zero are the same? There is! If 1=0 then we have for any number a:

a=a\times 1\underset{1=0}{=} a\times0\underset{\text{Th. 1}}{=}0,

that is every number is equal to zero. There is only one number and whenever we calculate anything we always get the same answer: zero… not much good for applications… although you can divide by zero! For more, see here.

The, Ah!! I Can’t Get Back Approach

This is the more traditional approach… with pictures.

You can think of multiplication (and indeed any other operation) as somehow dynamically acting on a numberline. For example, starting with b and multiplying by a, we end up at ba:

graph1

Now division is simply the inverse of multiplying. So if you first multiply by a, and get to ba, dividing by a will simply bring you back to b. This is done by multiplying by a^{-1} (see above):

graph2

Now  imagine starting at b and multiplying by 0. You end up at zero:

graph3

To divide by zero you must get back to b by multiplying by 0^{-1}:

graph4

However this is impossible, because, from zero you cannot multiply by anything to get you back to b. As we saw above, zero times anything is zero.

So you cannot divide by zero because you cannot undo the action of multiplying by zero. This is connected to the fact that, if you think of zero as a destination after a multiplication by zero, there are many, many numbers you could have come from:

graph5

How can you send back to all these places simultaneously? This is versus the situation where, for example, when multiplying by two, and you arrive at six, there is only one place you could have come from:

graph6

In other words multiplying by zero is not invertible. The function m_a:\mathbb{R}\rightarrow \mathbb{R}, given by m_a(x)=a\times x, has an inverse:

\displaystyle m_a^{-1}(x)=\frac{1}{a}x,

only when a=0. When a=0, the inverse image is the whole of \mathbb{R}.

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