Given that √5 is irrational, prove that 2√5-3 is an irrational number
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Given that √5 is irrational , prove that 2 √5 – 3 is an irrational number.
We have to prove that 2√5 - 3 is an irrational number.
Let us assume the contrary the 2√5 - 3 is a rational number.
So we can represent 2√5 - 3 in the form of where p and q are co-prime integers and q ≠ 0.
2√5 - 3 =
⇒ 2√5 = 3 +
⇒ 2√5 =
⇒ √5 =
⇒ √5 =
Here 3 , 2 , p and q are integers so is a rational number.
But we are given that √5 is irrational number so our assumption that 2√5 - 3 is an rational number was wrong.
So we conclude that 2√5 - 3 is an irrational number.
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