Prove that is an irrational number.

Foundation

Mathematics Foundation

Properties of Real Number

Irrational Number

Rationalisation

Question

Prove that $5 \sqrt{2} + 3$ is an irrational number.

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Assume,

5 \sqrt{2} + 3

is a rational number.

Then it should be represented in the form of

\frac{p}{q}

\Rightarrow 5 \sqrt{2} + 3 = \frac{p}{q}

5 \sqrt{2} = \frac{p}{q} - 3

\sqrt{2} = \frac{a - 3 b}{5 b}

\sqrt{2}

is irrational.

\frac{a - 3 b}{5 b}

is rational.

Irrational number can't be equated to rational number.

∴ 5 \sqrt{2} + 3

is an irrational number.