Question

show that 5 minus root 3 is an irrational

Answer 1

Let us assume that 5 - root 3 is rational. Then it can be written in the form

5 - root3 = p/q

or 5 - p/q = root3

It implies root3 is a rational number [Since 5 - p/q are rationals]

But this contradicts to the fact that root 3 is irrational. Hence our supposition was wrong. Therefore 5 - root 3 is irrational.

Answer 2

Let if possible suppose , 5 - Root 3 is rational number
=>( 5 - Root 3 ) - 5 is also a rational number
[ since Rational - Rational = Rational ]
=> 5 - Root 3 - 5
=> - Root 3 is a rational number
=> ( - Root 3 ) / - 1
[ Rational / Rational = Rational ]
=> Root 3 is a rational number
=> This contradicts the fact that Root 3 is a rational number
=> Thus our assumption was wrong , hence 5 - Root 3 is an irrational number...