Prove that is an irrational number.

Foundation

Mathematics Foundation

Properties of Real Number

Irrational Number

Linear Equations in One Variable

Question

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

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Let us assume that

√ 5

is a rational number.

we know that the rational numbers are in the form of p/q form where p,q are coprime numbers.

so,

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

p = \sqrt{(} 5) q

we know that 'p' is a rational number. so

\sqrt{(} 5)

q must be rational since it equals to p

but it doesnt occurs with

\sqrt{(} 5) q

since its not an integer

therefore, p is not equal to

\sqrt{(} 5) q

this contradicts the fact that

\sqrt{(} 5)

is an irrational number

hence our assumption is wrong and

\sqrt{(} 5)

is an irrational number