Question

Prove that the number is irrational:
√8​

Answer 1

Heya user!!

Here we begin

you would go with a proof by contradiction:

suppose √8 = a/b with integers a, b

and gcd(a,b) = 1 (meaning the ratio is simplified)

then 8 = a²/b²

and 8b² = a²

this implies 8 divides a² which also means 8 divides a.

so there exists a p within the integers such that:

a = 8p

and thus,

√8 = 8p/b

which implies

8 = 64p²/b²

which is:

1/8 = p²/b²

or:

b²/p² = 8

which implies

b² = 8p²

which implies 8 divides b² which means 8 divides b.

8 divides a, and 8 divides b, which is a contradiction because gcd (a, b) = 1

therefore, the square root of 8 is irrational.

hope it helps