# Prove that of the Numbers 2 -3 Sqrt(5) is Irrational: - Mathematics

Prove that of the numbers 2 -3 sqrt(5) is irrational:

#### Solution

Let 2 -3 sqrt(5)  be rational.
Hence 2 and 2 -3 sqrt(5) are rational.
∴ 2 – ( 2 -3 sqrt(5) ) = 2 – 2  + 3 sqrt(5) =  3 sqrt(5) = rational    [∵Difference of two rational is rational]
∴ 1/3 × 3sqrt(5) = sqrt(5) = rational   [∵Product of two rational is rational]
This contradicts the fact that sqrt(5) is irrational.
The contradiction arises by assuming 2 -3 sqrt(5) is rational.
Hence, 2 -3 sqrt(5)  is irrational.

Concept: Concept of Irrational Numbers
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