Question

Prove that 2+/5 is irrational. ​

Answer 1

$\huge\mathfrak\blue{Answer:}$

Given:

• We have been given a number 2+$\sqrt{5}$

To Prove:

• We have to prove that $2 + \sqrt{5}$ is an irrational number

Solution:

Let us assume that $2+\sqrt{5}$ be a rational number

Therefore it can be written in the form of a/b where a and B are non zero co-prime numbers.

$= > 2 + \sqrt{5} = \dfrac{a}{b}$

$=> \sqrt{5} = \dfrac{a - 2b}{b}$

We know that $\sqrt{5}$ is an irrational number

And an irrational number can never be equal to a rational number

Thus our assumption was wrong

Hence $2 + \sqrt{5}$ is an irrational number.

Hence proved !!

NOTE:

• This method of proving an irrational number is known as contradiction method
• In this method we first contradict a fact and then prove that our assumption was wrong

Answer 2

Answer:

Given:

We have been given a number 2+\sqrt{5}

5

To Prove:

We have to prove that 2 + \sqrt{5}2+

5

is an irrational number

Solution:

Let us assume that 2+\sqrt{5}2+

5

be a rational number

Therefore it can be written in the form of a/b where a and B are non zero co-prime numbers.

= > 2 + \sqrt{5} = \dfrac{a}{b}=>2+

5

=

b

a

=> \sqrt{5} = \dfrac{a - 2b}{b}=>

5

=

b

a−2b

We know that \sqrt{5}

5

is an irrational number

And an irrational number can never be equal to a rational number

Thus our assumption was wrong

Hence 2 + \sqrt{5}2+

5

is an irrational number.

Hence proved !!

NOTE:

This method of proving an irrational number is known as contradiction method

In this method we first contradict a fact and then prove that our assumption was wrong