Learn Exercise 1.3 with Free Lessons & Tips

Let is a rational number. Therefore. we can find two integers a, b (b  0)
such that   =
Let a and b have a common factor other than 1. Then we can divide
them by the common factor. and assume that a and b are co-prime.
a =
=
Therefore. is divisible by 5 and it can be said that a is divisible by 5. Let a
= 5k. where k is an integer
=
= This mans that is divisible by 5 and hence, b is divisible
by 5.
This implies that a and b have 5 as a common factor. And this is a
contradiction to the fact that a and b are co-prime.
Hence, cannot be expressed as  and it can be said that  is irrational.

To prove : 3+2√5 is rational number.

Ans: 3 is rational number.

2 is rational number.

2√5 is irrational number (product of rational and irrational is an irrational number)

3+2√5 is an irrational number (sum of rational and irrational number is irrational)

Or 

Let P = 3+2√5 is rational number ( we are assuming )

P= 3+2√5

P-3 =2√5

(P-3)÷2=√5

L.H.S. we assume P is a rational number .

P-3 is also rational number substraction of rational number always rational number .

(P-3)÷2 rational number (P-3) divide by rational number 2 hence (P-3)÷2 is rational number .

L.H.S. is rational While R.H.S. √5 is irrational L.H.S ≠R.H.S.

Our assumption is wrong so we can say 3+2√5 is an irrational number.