# Real Numbers

## NCERT Solution - Exercise - 1.3

Question 1 - Prove that is irrational.

Answer: Let us assume the contrary, i.e. is irrational.

Thus, there can be two integers a and b (bā‰ 0) and a and b are coprime so that;

Squaring on both sides, we get;

This means that a2 is divisible by 5 and hence a is also divisible by 5.

This contradicts our earlier assumption that a and b are coprime, because we have found 5 as at least one common factor of a and b.

This also contradicts our earlier assumption that is irrational.

Question 2 - Prove that is irrational.

Answer: Let us assume to the contrary, i.e. is irrational.

Thus, there can be two integers a and b (bā‰ 0) and a and b are coprime so that;

Since a and b are rational, so  is rational and hence, is rational.

But this contradicts the fact that is irrational.

This happened because of our faulty assumption.

Question 3 - Prove that following are irrationals:

Answer: Let us assume to the contrary, i.e. is rational.

Thus, there can be two integers a and b (bā‰ 0) and a and b are coprime so that;

Squaring on both sides, we get;

This means that b2 is divisible by 2 and hence a is also divisible by 2.

This contradicts our earlier assumption that a and b are co-prime, because 2 is at least one common factor of a and b.

This also contradicts our earlier assumption that is rational.

Hence, is irrational proved.

Answer: Let us assume to the contrary, i.e. is rational.

There can be two integers a and b (bā‰ 0) and a and b are coprime, so that;

Squaring on both sides, we get;

This means that a2 is divisible by 245; which means that a is also divisible by 245.

This contradicts our earlier assumption that a and b are coprime, because 245 is at least one common factor of a and b.

This happened because of our faulty assumption and hence, is irrational proved.

Answer: Let us assume to the contrary, i.e. is rational.

Thus, there can be two integers a and b (bā‰ 0) and a and b are coprime so that;

Since a and b are rational, so is rational and hence, is rational.

But this contradicts the fact that is irrational.

This happened because of our faulty assumption.

Hence, is irrational proved.

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