Prove that 4 minus root 3 is an irrational
Answers
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
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