Prove that the following numbers are irrational (i) 3 + √5:

by

Prove that the following numbers are irrational:

(i) 3 + √5
(ii) 15 – 2√7
Rational and Irrational Numbers Class 9 ICSE ML Aggarwal img 48

Solution:

(i) If 3 + √5 is a rational number say x

Consider 3 + √5 = x
It can be written as
√5 = x – 3
Here x – 3 is a rational number
√5 is also a rational number.
Consider √5 = p/q where p and q are integers
q ˃ 0 and p and q have no common factor
By squaring both sides
5 = p2/q2
p2 = 5q2
We know that
5q2 is divisible by 5
p2 is divisible by 5
p is divisible 5
Consider p = 5k where k is an integer
By squaring on both sides
p2 = 25k2
So we get
5q2 = 25k2
q2 = 5k2
Here
5k2 is divisible by 5
q2 is divisible by 5
q is divisible by 5
Here p and q are divisible by 5
So our supposition is wrong
√5 is an irrational number
3 + √5 is also an irrational number.
Therefore, it is proved.

(ii) If 15 – 2√7 is a rational number say x

Consider 15 – 2√7 = x
It can be written as
2√7 = 15 – x
So we get
√7 = (15 – x)/ 2
Here
(15 – x)/ 2 is a rational number
√7 is a rational number
Consider √7 = p/q where p and q are integers
q ˃ 0 and p and q have no common factor
By squaring on both sides
7 = p2/q2
p2 = 7q2
Here
7q2 is divisible by 7
p2 is divisible by 7
p is divisible by 7
Consider p = 7k where k is an integer
By squaring on both sides
p2 = 49k2
It can be written as
7q2 = 49k2
q2 = 7k2
Here
7k2 is divisible by 7
q2 is divisible by 7
q is divisible by 7
Here p and q are divisible by 7
So our supposition is wrong
√7 is an irrational number
15 – 2√7 is also an irrational number.
Therefore, it is proved.

Rational and Irrational Numbers Class 9 ICSE ML Aggarwal img 48

Rational and Irrational Numbers Class 9 ICSE ML Aggarwal img 49
Here
¾ is a rational number and √5/4 is an irrational number
We know that
Sum of a rational and an irrational number is an irrational number.
Therefore, it is proved.

More Solutions:

Leave a Comment