Prove that is an irrational number.

Foundation

Mathematics Foundation

Properties of Real Number

Irrational Number

Rational Numbers

Question

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

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Solution:

If possible, let us suppose

3 + \sqrt{5}

be a rational number.

⟹ 3 + \sqrt{5} = \frac{p}{q}

where, p, q ∈ Z, q ≠ 0.

⟹ \sqrt{5} = \frac{p}{q} - 3

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

which is rational.

This leads to contradiction because

\sqrt{5}

is irrational.

∴

3 + \sqrt{5}

is an irrational number.