Show that is an irrational number.

Foundation

Mathematics Foundation

Properties of Real Number

Irrational Number

Linear Equations in One Variable

Question

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

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Let

2 + \sqrt{5}

is a rational number such that

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

Where p and q are co-prime.

Therefore,

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

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

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

\sqrt{5}

is an irrational number and

\frac{p - 2 q}{q}

is a rational number.

∵ Rational ≠ Irrational.

Thus

2 + \sqrt{5}

is not a rational number.

Hence

2 + \sqrt{5}

is an irrational number.

Hence proved.