Question

1. Prove that root 5 - root 3
is not a rational number.​

Answer 1

Answer:

Hope helps....................................

You can use very simple method to prove this by putting the value of √5 and √3 as follows:

√3+√5 = (1.7320….) + ( 2.2360….) > this is non-terminating and non-recurring. Therefore, this sum is irrational number(By definition of irrational number).

2nd method:- Let, √3 + √5 is a rational number(Just saying and assuming that it is rational and is equal to Q). If this is satisfied then it is rational number otherwise assumption is false and hence it is irrational number.

Let Q= √3 +√5……..(1), squaring both sides we get, Q^2 = (√3+√5)^2 = 8 + 2×√3 ×√5. Or we can write, (Q^2 -8)/2 = √15…..(2)

Equation( 2) says, left hand side is a rational number because Q^2- 8 is rational. But right hand side is irrational number. So,we concluded that our assumption was false. Therefore, √3 +√5 is irrational number.