# Irrational Fraction

Pro Problems > Math > Logic > Proofs > Indirect Proofs## Irrational Fraction

Let a and b be positive integers, with b *not *a perfect square. Show that the following expression is irrational, using an indirect proof.

*Hint: Remember that an irrational number is a number which cannot be written as the quotient of two integers.*

## Solution

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### X, Y, and Z proof

Prove by contradiction: for any integers x, y, and z with z > 1, x^{2} - yz^{2} ≠ z.

Hint: This is a more generalized version of the problem found here: X Squared Proof

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### Radical Two Proof

*Hint: An irrational number is a number which cannot be written as the quotient of two relatively prime integers.*

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