Prove that 5 + \sqrt{} 2 is an irrational number,

Foundation

Mathematics Foundation

Properties of Real Number

Irrational Number

Rational Numbers

Question

Prove that

5 + \sqrt 2

is an irrational number,

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To prove

5 + \sqrt{2}

is irrational,

Let us assume

\sqrt{2}

is rational

\Rightarrow \sqrt{2} = \frac{a}{b}

(a

and

b

has no common factors except

1

)

\Rightarrow

2 = \frac{a^{2}}{b^{2}}

\Rightarrow

2 b^{2} = a^{2}

Hence

a

is even ...(1)

Let

a = 2 k

\Rightarrow 2 b^{2} = {(2 k)}^{2}

\Rightarrow b^{2} = 2 k^{2}

Hence

b

is also even ...(2)

This contradicts our assumptions as both

a

and

b

are even because they have

2

as the common factor.

Hence,

\sqrt{2}

is irrational.

Hence, proved.