[Math is Fun] Why 0 cannot be divisor?

I remember my secondary teacher told me that 1 cannot be divided by 0.

Why 0 cannot be divisor?

“Because it is just not permitted in arithmetic”, he said.

Well. It is not permitted because it is not permitted. A kid shouldn’t ask much…

Throughout my N years of learning and teaching, I know that 0 cannot be a divisor because if it does, something will go wrong in mathematics. But how things can go wrong other than the calculator showing some error messages?

Here is a simple example.

Let say

  1 / 0 = SOMETHING,

is valid.

If so, I can multiply both left hand and right hand sides by 0.  And, it becomes

1 = SOMETHING * 0
1 = 0.

Oops, something is wrong with the arithmetic. That’s why we can’t have 0 as the divisor.

Well. Perhaps you are not yet convinced and ask “how about 0 / 0 ? Since  a number divide itself always gives 1”.

Well, assuming

0 / 0 = 1,

is valid.

So, we can have a statement

2 = 2
2 = 2 x 1
2 = 2 x 0 / 0
2 = (2 x 0) / 0
2 = 0 / 0
2 = 1.

Wait. Something is wrong again. We can’t have 0/0 =1 then.

* The above examples are taken from Professor Steward’s Cabinet of Mathematical Curiosities by Ian Steward.