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Real Numbers Solutions
Prove that √5 is irrational
Proof: To prove that √5 is an irrational number
Let us assume that √5 is rational. Then √5 =
Since a and b are co primes with only 1 common factor and b≠0.
⇒ √5 =
⇒ a = √5b
On squaring both sides, we get
⇒ a² = 5b² ------- (i)
⇒ b² = [By theorem if p divides q then p can also divide q²]
So, 5 can also divide a. ……….. (ii)
⇒ a = 5c
On squaring on both sides, we get
⇒ a² = 25c² ……… (iii)
From equations (i) and (iii)
⇒ 5b² = 25c²
⇒ b² = 5c²
⇒ c² = [By theorem if p divides q then p can also divide q²]
So, 5 can also divide b. ……….. (iv)
We know that a and b are co-primes having only 1 common factor but from (ii) and (iv) we can see a and b are not co-primes.
This contradiction arises because we assumed that √5 is a rational number.
∴ our assumption is wrong.
∴ √5 is an irrational number.
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