Can 1 equal 2?

I don't think that anyone doubts the power of mathematics, but even so, it can be used to perform some nasty tricks for the unsuspecting observer or indeed learner. Some of my favorites are those that deal with infinities and zero; it seems that many strange and mysterious things can happen when numbers become very large or very small.

I'm sure many have seen this before, but it's always worth a look ... and what is it? Well, let's prove, algebraically that 1=2! Sound impossible? Not so ...

Let's say that there are two non-zero values a and b such that

a = b

seems pretty harmless doesn't it, nothing really strange here yet ... ok, now multiply both sides of this expression by b

Step 1: ab = b²

Now that's fair since we did the same thing to both sides ... now subtract a² from both sides

Step 2: ab - a² = b² - a²

still ok, since the same quantity has been subtracted from both sides ... now we factorize both sides

Step 3: a(b - a) = (b + a)(b - a)

Now in case you don't believe me with the right hand side, if we multiply it out

bb - ab + ab - aa = b² - a²

this is actually called the difference of two squares, so it appears to be fine too.

Now if we look at step three, we see a common factor of (b-a) so we can divide both sides by this factor ...

Step 4: a = b + a

At the beginning we also made the statement that a = b, so we can replace the b on the right hand side with a ...

Step 5: a = a + a

Simplify step 5;

Step 6: a = 2a

Last step, once again there is a common factor of a, so we can divide this out

1 = 2

Well who would have guessed ... 1 = 2!

Now we all know that 1 cannot equal 2, well that's what we've been told. Yet this little proof is perfectly correct algebraically ... so what happened?

Let me know ... I've actually given you a little hint earlier on. Enjoy!