is a prime. One of the theorems, we use in our proof, is the Fundamental Theorem of Arithmetic. Recall, a number ‘ s ’ is called irrational if it cannot be written in the form p q where p and q are integers and q ¹ . Some examples of irrational numbers, with which you are already familiar, are : , , , .10110111011110 .
. . , etc. Before we prove that is irrational, we need the following theorem, whose proof is based on the Fundamental Theorem of Arithmetic.
Theorem . : Let p be a prime number. If p divides a , then p divides a, where a is a positive integer. *Proof : Let the prime factorisation of a be as follows : a = p p .
., p n are primes, not necessarily distinct. Therefore, a = ( p p . . .
. . p n . Now, we are given that p divides a .
Therefore, from the Fundamental Theorem of Arithmetic, it follows that p is one of the prime factors of a . However, using the uniqueness part of the Fundamental Theorem of Arithmetic, we realise that the only prime factors of a are p , p , . . ., p n .
So p is one of p , p , . . ., p n . Now, since a = p p .
. . p n , p divides a . We are now ready to give a proof that The proof is based on a technique called ‘proof by contradiction’.
(This technique is discussed in some detail in Appendix ). Theorem . : Proof : Let us assume, to the contrary, that is rational. So, we can find integers r