06 September 2007

Proving that Pi Equals 3

...proves to be less of a pain than you might originally think. In fact, it's no problem at all to unravel millennia of careful mathematical postulation with just one simple trick. Can you spot it?

Given: a=b

(1) pi*a^2 = pi*a*b (multiplying pi*a)
(2) 3*a*b = 3*b^2 (multiplying by 3*b)
(3) pi*a^2 - 3*a*b = pi*a*b - 3*b^2 (subtracting (2) from (1))
(4) pi*a^2 - pi*a*b = 3*a*b - 3*b^2 (adding 3*a*b and subtracting pi*a*b on both sides)
(5) pi*a^2 - pi*a*b + a*b - b^2 = 4*a*b - 4*b^2 (adding a*b and subtracting b^2 on both sides)
(6) pi*a*(a-b) + b*(a-b) = 4b*(a-b) (factoring out common terms)
(7) pi*a + b = 4*b (removing common terms)
(8) pi*a = 3*b (subtracting b)
(9) pi*b = 3*b (substituting a for b since a = b)
(10) pi = 3 (removing common terms)

Therefore pi=3.

Spoiler ahead! If you want to figure it out yourself, then don't read on just yet!

The trick is actually the same employed over and over to prove all kinds of things in math, including that 1 = -1 and 1 = 2, among the more famous. Take a look at step 7. This is where the error is. In order to remove the terms, we must divide, right? Well, if a = b, then a - b = 0, thus we are dividing by zero, which is a big math no-no. Aside from being a fun trick to throw at your math students, it is a great example of why we can't divide by zero. Indeed, if we allowed it, then we can prove all kinds of false statements in our system!

Try your own hand at it for a minute. What other "convenient" conclusions can you come to? Can you prove that the world is like this math t-shirt? Maybe prove that e = 2 or that sqrt(2) = 1.5 or, if you're really adventurous, prove that pi = e!

Now, consider branching out into other illegal areas of math. :) How about negative square roots? What kind of false statements can you arrive at using that loop hole?

Just for fun, here is another fake proof. See if you can find the error:

(1) John = Time * Money (or anyone that you want to stick it to...)
(2) Time = Money (Time is Money)
(3) John = Money * Money (substitution)
(4) Evil = sqrt(Money) (Money is the Root of Evil)
(5) John = Evil

Of course, here the problem is in (4) because Paul said that the love of money was the root of all evil...har-de-har-har...oh, well, it was funny to me...