Question

Example 1. Prove that V3 is not a rational number.​

Answer 1

To prove V3 is not a rational number that is V3 is irrational
so,
let us assume that V3 is rational

i.e V3 is rational =a/b
( rational no can written in this form)

where a and b are co-primes (b is not equal to 0)
hence V3 =a/b

V3 b=a
by squaring both sides
(V3b)*2 =a*2

3b*2=a*2

a*2/3=b
hence 3 divides a*2
by the theorem:if P is a prime no. and P divides a*2, then P divides where a is a positive no.

so 3 should divides a also
hence,
a/3=c where c is some integer
so,
a=3c
now we know that

3 b*2= a*2

putting a=3 c
3 b*2=(3 c )*2

3b*2=9c*2

b*2=9c*2/3

b*2=3c*2

b*2/3= c*2
hence 3 divides b
so 3 divides b also