Question

Prove that (2+√3) is irrational no.. Given that √3 is irrational

Answer 1

Step-by-step explanation:

let, 2+√3 is a rational no.

then, 2+√3= a/b

√3=a/b-2

a-2b/b=√3

here, LHS is rational but √3 is irrational

this is only because our assumption is wrong

so,2+√3 is irrational

Answer 2

GIVEN :-

$\rm{ a \: irrational \: number \: 2 + \sqrt{3} }$

TO PROVE :-

$\rm{ \: that \: 2 + \sqrt{3} \: \: is \: irrational}$

SOLUTION :-

Let us assume, to the contrary, that

$2 + \sqrt{3}$ is rational. Then, there exist co-prime positive integers a and b such that

$\implies \rm{ \: 2 + \sqrt{3} = \dfrac{a}{b} }$

$\implies \rm{ \: \sqrt{3} = \dfrac{a}{b} \: - 2 }$

$\implies \rm{ \ \sqrt{3} = \dfrac{ \: a \: - 2b \: }{b} }$

$\implies \: \rm{ \sqrt{3 \: } \: \: \: is \: \: \: rational}$

[ 2,a and b are integers ∴

$\rm{\dfrac{a - 2b}{b} }$ is a rational number]

This contradicts the fact that $\sqrt{3}$ is irrational. So, our assumption is not correct. Hence, $\rm{ \: 2 + \sqrt{3} }$ is an irrational number.

OTHER INFORMATION :-

IRRATIONAL :

• A number is called rational if it cannot be expressed in the form p/q where p and q are integers ( q> 0) Example : √3,√2,√5,p etc.

• Pythagoras Theorem: In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. Using this theorem we can represent the irrational numbers on the number line.

• They have non-terminating and non-repeating decimal expression

• The sum, difference, multiplication and division of irrational numbers are not always irrational. Irrational numbers do not satisfy the closure property under addition, subtraction, multiplication and division