Question

Prove that 4- √3 is irrational

Answer 1

First let us look at the laws:

Irrational+rational (and vice versa) = irrational

Rational + rational = rational

Let us assume √3 to be rational.

==> √3 =p/q. (Where q#0 , gcd of p and q is 1)

==> 3 = P²/q²

==> 3q² = p². ————1

Therefore p² is a multiple of 3.

Hence p is also a multiple of 3. ———2

Let p = 3x (for some x)

==>(3x)² =p²

By 1

(3x)² = 3q²

==>3x= q²

Therefore q² is a multiple of 3.

Hence q is also a multiple of 3. ———3

By the notes 2 and 3 we get a contradictory statement as both p,q are divisible by 3. This means GCD of p and q #1.

Therefore √3 is irrational.

‘#’ means ‘not equal”

So now we know√3 is irrational.

Now let us assume 4-√3 is rational.

Let 4-√3 = y ( where y is a rational number)

==>(4-√3)² = y²

==> 16+3+9√3 = y² ( y² is also rational)

==> (y² -19) ÷9= √3

(y²-19) ÷9 is rational if y is rational. By our assumption we know y is rational.

A rational # irrational

Therefore the assumption was wrong. So y is irrational.

y = 4-√3

Therefore 4-√3 is irrational

Answer 2

Any rational number being subtracted from an irrational always gives an irrational .