Question

Prove that (3-
$\sqrt{5}$
3) is irrational.​

Answer 1

QUESTION:

$\red {prove \: that \: 3 - \sqrt{5} is \: irrational}$

ANSWER:

Let,

3-root 5 is a rational number in the form of a by b where a and b are Co prime.

$\blue {According \: to \: statement}$

$3 - \sqrt{5} = \frac{a}{b}$

$- \sqrt{5} = \frac{a}{b} - 3$

$\orange {(taking \: lcm \: in \: rhs \: side)}$

$- \sqrt{5} = \frac{a - 3b}{b}$

$\sqrt{5} = \frac{ - (a - 3b)}{b}$

Hence, here contradiction arises as we know that root 5 is a irrational number but we let 3-root 5 as a rational number.

Hence proved.

Additional information :

☆Rational numbers :

Numbers that can be expressed in the form of p by q where p and q are integers and q is not equal to zero.

eg. 4/3; 2/4 etc.

☆Irrational numbers :

Real numbers which non terminating and non repeating

eg, value of pie.