# Prove that (4 - 5Sqrt(2) ) is Irrational. - Mathematics

Prove that (4 - 5sqrt(2) ) is irrational.

#### Solution

Let x = 4 - 5sqrt(2) be a rational number.
x = 4 - 5sqrt(2)
⇒x^2 = (4 - 5sqrt(2)  )2
⇒ x^2 = 4^2 + (5sqrt(2) ) 2 – 2(4) (5 sqrt(2) )
⇒ x^2 = 16 + 50 – 40sqrt(2)
⇒ x^2 – 66 = – 40 sqrt(2)
⇒( 66− x^2)/40 =sqrt(2)
Since x is a rational number, x^2 is also a rational number.
⇒ 66 -x^2 is a rational number
⇒ (66− x^2)/40 is a rational number
⇒sqrt(2) is a rational number
But sqrt(2) is an irrational number, which is a contradiction.
Hence, our assumption is wrong.
Thus, (4 - 5sqrt(2) ) is an irrational number.

Concept: Concept of Irrational Numbers
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