Given that \sqrt{3} \textrm{ } a n d \textrm{ } \sqrt{5} are irrational numbers, prove that \sqrt{3} + \sqrt{5} is an irrational number.

Foundation

Mathematics Foundation

Properties of Real Number

Irrational Number

Rational Numbers

Question

Given that

\sqrt{3} a n d \sqrt{5}

are irrational numbers, prove that

\sqrt{3} + \sqrt{5}

is an irrational number.

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\sqrt{3}

\sqrt{5}

are irrational [given]

To prove:

\sqrt{3} + \sqrt{5}

is irrational

Let us assume

\sqrt{3} + \sqrt{5}

is rational

Now

\sqrt{3} + \sqrt{5} = \frac{a}{b},

where a & b are coprime

integers as

b \neq o

\sqrt{3} + \sqrt{5} = \frac{a}{b}

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

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

is a form of

\frac{a}{b}

then

it is rational number

\frac{a - \sqrt{5} b}{b}

is a rational number

then

\sqrt{3}

is also rational numbers

which is not true

∴

our assumption is false

\sqrt{3} + \sqrt{5}

is an irrational number