Division by Zero

Note – The information contained in this article is not examinable in your GCSEs. However hopefully you will find it interesting and helpful nonetheless.

Hopefully by now you will know that division by zero is impossible!

Going back to primary mathematics, let’s say you have 5 apples, and want to share them between zero people, how would you go about this? You give zero people an apple, then you give zero people another apple, then you give another zero people another apple… You would go on for ever.

As mathematicians, we say that a number divided by zero is undefined, basically it cannot be done! Below is an example illustrating one of the paradoxes that occurs when you divide by zero.

Let’s say you have two numbers, a and b, and it is known that:

a = b

Now we’ll multiply both sides by a, giving

a^2 = ab

Looks ok so far, now let’s minus b^2 from both sides.

a^2 - b^2 = ab - b^2

Now we can factorise both sides of this equation. The left hand side is the difference of two squares, and on the right both terms contain a b. This now gives us,

(a + b)(a - b) = b(a - b)

Now both sides of our equation have a common factor of (a - b), so we can divide through by this.

(a + b) = b

From our first equation we know that a = b, so we can replace the a in our first equation with b.

(b + b) = b, simplifying to give us 2b = b.

Now both sides have a common factor of b, so dividing both sides by b should give us our equation in the lowest possible form.

2 = 1

Eh? Can you spot where we went wrong?

The problem occurs at this stage of the working:

(a + b) (a - b) = b (a - b).

The red highlighted brackets are essentially zero, and by cancelling these off we have broken the rule of never dividing by zero! Obviously we know that 2 = 1 can never be true.

I hope this article has helped illustrate how dividing by zero is never a good idea!