Question

Show that the following numbers are irrational.

`1/sqrt(5)`

Answer 1

Given: `1/sqrt(5)`

Step-wise calculation:

1. Show `sqrt(5)` is irrational.

Assume `sqrt(5) = m/n` in lowest terms (m, n integers, gcd = 1). 

Squaring: 5n² = m². Then 5 divides m², so 5 divides m, say m = 5k. 

Substituting gives 5n² = 25k² ⇒ n² = 5k², so 5 divides n. 

Thus m and n share factor 5, contradicting gcd(m, n) = 1. 

Hence `sqrt(5)` is irrational.

2. Write `1/sqrt(5) = sqrt(5)/5`. Since `sqrt(5)` is irrational and `1/5` is a nonzero rational, a nonzero rational multiple of an irrational number is irrational. 

Therefore `sqrt(5)/5` `("i.e." 1/sqrt(5))` is irrational.