Show that the number is irrational.

Foundation

Mathematics Foundation

Properties of Real Number

Irrational Number

Rationalisation

Question

Show that the number $3 - \sqrt{5}$ is irrational.

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Let

3 - \sqrt{5}

be a rational equal to

\frac{a}{b}

Then,

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

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

\Rightarrow \sqrt{5} =, \frac{3 b - a}{b}

\Rightarrow \sqrt{5}

is rational.

This is a contradiction.

Hence,

3 - \sqrt{5}

is irrational.