Charles Seif's Proof from his Book: a Biography of a Dangerous Idea

Let a = 1 and b = 1 such that b^2 = ab = 1.

Now let's start out with a simple equation.

a^2 = a^2

We will now subtract one from both sides and we will now have;

a^2 - 1 = a^2 - 1

Now let's substitute what was given to us, let one of the 1's be ab and the other 1 be b^2. We now have:

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

as you can see we have not changed the number at all. Now let's simplify.

a^2 - b^2 simplified = (a - b)(a + b)

a^2 - ab simplified = a(a - b)

so we now have

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

Now we can see that we have like terms and so we will divide by (a - b) from both sides:

We have:

a + b = a

Now subtract a from both sides we get

b = 0

Now b was given to us so we now have

1 = 0

This is very SCARY. What went wrong? We can never have 1 = 0.

If we go back up to when we divided (a - b) we are essentially dividing (1 - 1) which is zero and we cannot divide by zero.

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