Prove That ?5 is Irrational Number Using Division Lemma Exercise 1.3 Chapter 1 Real Numbers Maths Class 10 NCERT

Question 1. Prove that √5 is irrational.

Answer:

Let take √5 as rational number
If a and b are two co prime number and b is not equal to 0.
We can write √5 = a/b
Multiply by b both side we get
b√5 = a
To remove root, Squaring on both sides, we get
5b² = a² ……………(1)
Therefore, 5 divides a² and according to theorem of rational number, for any prime number p which is divides a² then it will divide a also.
That means 5 will divide a. So we can write
a = 5c
and plug the value of a in equation (1) we get
5b² = (5c)²
5b² = 25c²
Divide by 25 we get
b²/5 = c²
again using same theorem we get that b will divide by 5
and we have already get that a is divide by 5
but a and b are co prime number. so it is contradicting .
Hence √5 is a non rational number