Question

prove that -5 + 2/√5 - √5 is an irrational number.​

Answer 1

Answer:

Let us assume that √5 is a rational number.

we know that the rational numbers are in the form of p/q form where p,q are integers.

so, √5 = p/q

    p = √5q

we know that 'p' is a rational number. so √5 q must be rational since it equals to p

but it doesn't occur with √5 since its not an integer

this contradicts the fact that √5 is an irrational number

∴ p ≠ √5q

hence our assumption is wrong and √5 is an irrational number.

Then,

Let us assume that -5+2/√5-√5 is a rational number.

so, -5+2/√5-√5= p/q

p=-√5-√5q/5+2

we know that 'p' is a rational number. so -√5-√5q/5+2 must be rational since it equals to p

but it doesn't occur with -√5-√5q/5+2 since its not an integer

this contradicts the fact that -√5-√5q/5+2 is an irrational number

∴ p≠-√5-√5q/5+2

hence our assumption is wrong and -√5-√5q/5+2 is an irrational number.