common factor. But this contradicts the fact that a and b are coprime. This contradiction has arisen because of our incorrect assumption that is rational. So, we conclude that is irrational.
In Class IX, we mentioned that : the sum or difference of a rational and an irrational number is irrational and the product and quotient of a non-zero rational and irrational number is irrational. We prove some particular cases here. Example : Show that – is irrational. Solution : Let us assume, to the contrary, that – is rational.
That is, we can find coprime a and b ( b ) such that Therefore, Rearranging this equation, we get – a Since a and b are integers, we get – a is rational, and so is rational. But this contradicts the fact that is irrational. This contradiction has arisen because of our incorrect assumption that – is rational. So, we conclude that is irrational.
Example : Show that is irrational. Solution : Let us assume, to the contrary, that is rational. That is, we can find coprime a and b ( b ) such that Rearranging, we get Since , a and b are integers, b is rational, and so is rational. But this contradicts the fact that So, we conclude that is irrational.
EXERCISE . . Prove that is irrational. .
Prove that is irrational. . Prove that the following are irrationals : (i) (ii) (iii) . Summary In this chapter, you have studied the following points: .
The Fundamental Theorem of Arithmetic : Every composite number can be expressed (factorised) as a product of primes, and this factorisation is unique, apart from the order in which the prime factors occur. . If p is a prime and p divides a , then p divides a , where a is a positive integer. .