Proofs Of Irrationality

 

A number‘s’ is called irrational if it cannot be written in the form, p q where p and q are integers and q ≠ 0.

Some examples of irrational numbers are √2, √3, √15,p, -√2  / √3 , etc.

Theorem  : √2 is irrational.

Proof :

Let us assume, to the contrary, that √2 is rational.

So, we can find integers r and s (≠ 0) such that √2 = r/ s . Suppose r and s have a common factor other than 1.

Then, we divide by the common factor to get √2 =  a / b , where a and b are coprime.

So, b√ 2 = a.

Squaring on both sides and rearranging, we get 2b^2 = a^2 . Therefore, 2 divides a^2 .

Now, Fundamental Theorem of Arithmetic , it follows that 2 divides a.

So, we can write a = 2c for some integer c.

Substituting for a, we get 2b^2 = 4c^2 , that is, b^2 = 2c^2 . This means that 2 divides b^2 , and so 2 divides b (again using Fundamental Theorem of Arithmetic  with p = 2). Therefore, a and b have at least 2 as a common factor. But this contradicts the fact that a and b have no common factors other than 1.

This contradiction has arisen because of our incorrect assumption that √2 is rational. So, we conclude that √2 is irrational.

 

Example:

Show that 3√2 is irrational.

Solution:

Let us assume, to the contrary, that 3√2 is rational.

That is, we can find coprime a and b (b ≠ 0) such that 3√2 = a/ b. Rearranging, we get √2 = a/3 b Since 3, a and b are integers, a/3 b is rational, and so√ 2 is rational. But this contradicts the fact that √2 is irrational. So, we conclude that 3√2 is irrational.