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Prove that is irrational.
Asked by Reap Last Modified
Answer 
Let is a rational number. Therefore. we can find two integers a, b (b
0)
such that =
Let a and b have a common factor other than 1. Then we can divide
them by the common factor. and assume that a and b are co-prime.
a = =
Therefore. is divisible by 5 and it can be said that a is divisible by 5. Let a
= 5k. where k is an integer =
=
This mans that
is divisible by 5 and hence, b is divisible
by 5.
This implies that a and b have 5 as a common factor. And this is a
contradiction to the fact that a and b are co-prime.
Hence, cannot be expressed as and it can be said that
is irrational.
Palak Kanti Maitra
I have a degree in Mechanical Engineering I have conducted over 12000 online sessions on Math. durin
Let√5 is a rational number.So √5=a/b, squaring both sides 5=a^2/b^2 and a^2=5b^2,So a^2 is divisible by 5.
Hence a is also divisible by 5. Let a =5c where c is an integer. Then a^2=5b^2=25c^2
b^2=5c^2. So b^2 is divisible by 5. Hence b is also divisible by 5. So and b have a common factor5.But for a rational number a/b ,common factor between a and b cannot be other than 1.So this contradicts √5 is a rational number. So √5 is irrational.
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