NCERT Solutions 10th Maths Chapter 1 Real Number Exercise 1.3

NCERT Solutions For Class 10 Maths Chapter 1 Exercise 1.3 we will restart our exploration of the world of real numbers. We will study Euclid’s division algorithm and the Fundamental Theorem of Arithmetic. Also, we will see decimal representation of real numbers.

Exercise 1.3
1. Prove that v5 is irrational.
Answer
Let take v5 as rational number
If a and b are two co prime number and b is not equal to 0.
We can write v5 = a/b
Multiply by b both side we get
bv5 = a
To remove root, Squaring on both sides, we get
5b2 = a2 … (i)
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
Putting value of a in equation (i) we get
5b2 = (5c)2
5b2 = 25c2
Divide by 25 we get
b2/5 = c2
Similarly, 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 contradicts.
Hence v5 is not a rational number, it is irrational.
2. Prove that 3 + 2v5 is irrational.
Answer
Let take that 3 + 2v5 is a rational number.
So we can write this number as
3 + 2v5 = a/b
Here a and b are two co prime number and b is not equal to 0
Subtract 3 both sides we get
2v5 = a/b – 3
2v5 = (a-3b)/b
Now divide by 2, we get
v5 = (a-3b)/2b
Here a and b are integer so (a-3b)/2b is a rational number so v5 should be a rational number But v5 is a irrational number so it contradicts.
Hence, 3 + 2v5 is a irrational number.
3. Prove that the following are irrationals:
(i) 1/v2 (ii) 7v5 (iii) 6 + v2
Answer
(i) Let take that 1/v2 is a rational number.
So we can write this number as
1/v2 = a/b
Here a and b are two co prime number and b is not equal to 0
Multiply by v2 both sides we get
1 = (av2)/b
Now multiply by b
b = av2
divide by a we get
b/a = v2
Here a and b are integer so b/a is a rational number so v2 should be a rational number But v2 is a irrational number so it contradicts.
Hence, 1/v2 is a irrational number
(ii) Let take that 7v5 is a rational number.
So we can write this number as
7v5 = a/b
Here a and b are two co prime number and b is not equal to 0
Divide by 7 we get
v5 = a/(7b)
Here a and b are integer so a/7b is a rational number so v5 should be a rational number but v5 is a irrational number so it contradicts.
Hence, 7v5 is a irrational number.
(iii) Let take that 6 + v2 is a rational number.
So we can write this number as
6 + v2 = a/b
Here a and b are two co prime number and b is not equal to 0
Subtract 6 both side we get
v2 = a/b – 6
v2 = (a-6b)/b
Here a and b are integer so (a-6b)/b is a rational number so v2 should be a rational number.
But v2 is a irrational number so it contradicts.
Hence, 6 + v2 is a irrational number.