This week I presented you this proof, and asked you to find the flaw that allowed 2 to equal 1:

Take the ^ sign to mean ‘to the power of’, such that a^2 means ‘a to the power of two’ (or more commonly ‘a squared’).

Let a = b

a^2 = b^2

a^2 = ab

a^2 – b^2 = ab – b^2

(a + b)(a – b) = b(a – b)

a + b = b

b + b = b

2b = b

Which implies that 2 = 1.

Find out why this happens after the break!

Now obviously, 2 does not ever equal 1. So where’s the problem?

All the algebra is in fact correct, for those of you who maybe tried to nitpick my maths. If you follow it through and check every line, it works perfectly. Additionally, the idea is not to substitute numbers in – it’s an algebraic proof which means it will be true (and the same) for any values of a and b.

Instead you needed to look at what the algebra was actually doing. The problem comes between these two lines:

(a + b)(a – b) = b(a – b)

a + b = b

What are we doing here? Well, canceling the identical terms of course – we can cancel (a – b) to get rid of it from both sides. But what does that actually mean? What mathematical function are we using? The answer is division: we’re dividing both sides of the equation by the (a – b) term to simplify it.

Here’s the thing – a and b are the same number, so (a – b) = 0. And therefore when we divide by (a – b), we’re dividing by 0, which you simply can’t do for a myriad of reasons. * Not the least of which you’ll end up with strange results like 2 = 1.

Did you work it out?

Vel.

* Think about it with an example. For instance, when you divide 40 by 5, you’re asking ‘How many lots of 5 are there in 40?’. The answer is of course 8. So let’s divide 40 by 0. How many lots of 0 are there in 40? Um…

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January 8, 2010 at 10:42 am

Verily good, I might show that to my maths teacher 🙂