Question:
Prove that $4-5 \sqrt{2}$ is an irrational number.
Solution:
Let us assume that $4-5 \sqrt{2}$ is rational . Then, there exist positive co primes $a$ and $b$ such that
$4-5 \sqrt{2}=\frac{a}{b}$
$5 \sqrt{2}=\frac{a}{b}-4$
$\sqrt{2}=\frac{\frac{a}{b}-4}{5}$
$\sqrt{2}=\frac{a-4 b}{5 b}$
This contradicts the fact that $\sqrt{2}$ is an irrational
Hence $4-5 \sqrt{2}$ is irrational
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