Number of Type Rational r plus s Root 2 is Irrational

Theorem

Let $r, s \in \Q$ be rational numbers.

Then $r + s \sqrt 2$ is irrational.

Proof

Aiming for a contradiction, suppose $t = r + s \sqrt 2$ be rational.

Then:

$\sqrt 2 = \dfrac {t - r} s$ is also rational.

This contradicts the fact that Square Root of 2 is Irrational.

Hence the result by Proof by Contradiction.

$\blacksquare$

Sources

• 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 1$: Real Numbers: Exercise $\S 1.20 \ (4)$