The Irrationality of the Square Root of 2.

You all heard that the square root of 2 is an irrational number. But has this fact ever been proven to you? If not, how can you be sure that it is indeed an irrational number? Today let us examine a surprisingly simple proof of this fact, a proof that dates back to the Antiquity.

The Proof:

Let A be the (positive) square root of 2, i.e. A 2 = 2. What we want to show is that A cannot be expressed as a ratio B/C, where B and C are two nonzero integers. This proof is an example of proof by contradiction.

First suppose that A is indeed a rational number such that A = B/C. Further, we can assume that B and C are relatively prime, i.e., they share no common factors other than 1. (Cancel out all such factors if necessary to get the final, irreducible, fraction of B/C.) Then we have:

2 = A2 = (B/C)2 = B2/C2.

Multiply through by C2, we have:

2C2 = A2C2 = B2.

Notice that the value on the left, 2C2, is an even number. So the value on the right of the equation, B2 , is even as well. What this says is that B has to be an even number itself, since the square of an odd number will again be an odd number.

Therefore, we can write B = 2D, where the integer D is equal B/2.

Replacing B = 2D in the above equation:

2C2 = A2C2 = (2D)2 = 4D2.

More importantly,

2C2 = 4D2, or

C2 = 2D2.

Using the same argument as before, we see that since 2D2 is always an even number, then C2 is also even, therefore so is C.

But then we have that both B and C are even numbers. That is, they share, at least, 2 as a common factor. This contradicts the original assumption that B and C have no common factors other than 1. However, there is nothing wrong with this assumption, since it can always be made true via appropriate cancellation(s). Therefore, the cause of this contradiction must be the first assumption - that the square root of 2 can be expressed as a ratio of two integers. Hence it must be true that it is an irrational number.

Next time: The Existence of Infinitely Many Prime Numbers