Question

Prove that 1/2 - root 5/3 is an irrational number

Answer 1

Answer:If possible let (1/2 - √5/3) be a rational number x

1/2-√5/3 => x

making denominator equal

(3-2√5)/6 => x

3-2√5 = 6x

3-6x = 2√5

2(1-2x)/2 = √5

1-2x = √5

since , x is rational number (by our supposition)

2x is also rational

1 is rational

and we know that , difference of 2 rationals is rational number

so, this tells us that √5 is also rational

but , √5 is irrational

thus , our supposition is wrong

so x is not rational ,

and x = 1/2 -√5/3

so, 1/2 - √5/3 is irrational

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