Question

prove that 2+√5 is irrarational

Answer 1

ANSWER:

• 2+√5 is an Irrational number.

GIVEN:

• Number = 2+√5

TO PROVE:

• 2+√5 is an irrational number.

SOLUTION:

Let 2+√5 be a rational number which can be expressed in the form of p/q where p and q have no other common factor than 1.

$\implies \: 2 + \sqrt{5} = \dfrac{p}{q} \\ \\ \implies \: \sqrt{5} = \dfrac{p}{q} - 2 \\ \\ \implies \: \sqrt{5} = \dfrac{p - 2q}{q}$

Here:

• (p-2q)/q is rational but √5 is Irrational.
• Thus our contradiction is Wrong.
• 2+√5 is an Irrational number.

NOTE:

• This method of proving an Irrational number is called contradiction method.
• In this method we first contradict a fact then we prove that our supposition was worng.

Answer 2

$\huge\mathfrak{Answer:}$

Rational numbers:

• Rational numbers are the numbers that can be written in the form of p/q where p and q are integers and q is not equal to zero.
• Example: 2/3, 4/5, 6/7 etc.
• All whole numbers, integers are rational numbers.

Given:

• We have been given a number 2+√5.

To Prove:

• We need to prove that 2+√5 is irrational.

Solution:

Let us assume that 2+√5 is a rational number.

Therefore, 2+√5 can be written in the form of a/b, where a and b are coprime.

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

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

But, √5 is an irrational number.

An irrational number can never be equal to a rational number.

Therefore, our assumption was wrong.

Hence, 2 + √5 is an irrational number.

Hence proved!