Given that is an irrational number, prove that is an irrational number.

Foundation

Mathematics Foundation

Properties of Real Number

Irrational Number

Rational Numbers

Question

Given that $\sqrt{3}$ is an irrational number, prove that $(2 + \sqrt{3})$ is an irrational number.

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Given

\sqrt{3}

is irrational number

Let

2 + \sqrt{3}

is rational number.

then

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

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

Since

\sqrt{3}

is irrational number but

\frac{p - 2 q}{q}

is rational it mean

irrational = rational

which contradict,

∴ our assumption is wrong and

2 + \sqrt{3}

is irrational number.