Why Can't the Denominator Be Zero?

We know that division can be written in another way.

6 ÷ 2 = 6 × 1/2

Here, 1/2 is the multiplicative inverse of 2.

The multiplicative inverse of a number is a number which, when multiplied by it, gives 1.

2 × 1/2 = 1

So:

1/2 = multiplicative inverse of 2

Now suppose we want to divide a number by zero.

6 ÷ 0

If division is multiplication by the multiplicative inverse, we would need the multiplicative inverse of 0.

So we would need to find a number that satisfies:

0 × ? = 1

But one of the fundamental rules of mathematics is:

Any number × 0 = 0

Therefore, there is no number that can make:

0 × ? = 1

Zero has no multiplicative inverse.

Therefore, division by zero is not defined.

A Rational Number is written as:

p/q

Since q represents the divisor, it cannot be zero.

q ≠ 0

That's why we write:

p/q, where q ≠ 0