State whether the following statement is true or false.The following number is irrational

Foundation

Mathematics Foundation

Properties of Real Number

Irrational Number

Rational Numbers

Question

State whether the following statement is true or false.

The following number is irrational

6 + \sqrt{2}

True

False

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An Irrational Number is a real number that cannot be written as a simple fraction.

Let us assume

6 + \sqrt{2}

is rational.

Hence,

6 + \sqrt{2}

can be written in form

\frac{a}{b}

Where, a and b (n ≠ 0) are co-prime.

Hence,

6 + \sqrt{2} = \frac{a}{b}

\Rightarrow

\sqrt{2} = \frac{a}{b} - 6

\Rightarrow

\sqrt{2} = \frac{a - 6 b}{b}

Here,

\frac{a - 6 b}{b}

is a rational number, but

\sqrt{2}

is irrational.

Since, Rational ≠ Irrational

This is a contriadition

∴ Our assumption is incorrect.

∴

6 + \sqrt{2}

is irrational number.