Prove that √5 is irrational.
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let √5 be a rational number. So √5 = a/b where b or a is not equal to 0 and a & b are co prime with no common factor other than 1. So
√5 = a/b
(√5)^2 = (a/b)^2
5 = a^2/b^2
5b^2=a^2
5 divides a^2
5 divides a
let a^2=5c
5b^2= 25c^2
b^2= 5c^2
2 divides b^2
2 divides b
this shows that a and b have 5 as common factor. but a and b are co prime numbers with only 1 as common factor. hence this contradicts our assumption.
√5 = a/b
(√5)^2 = (a/b)^2
5 = a^2/b^2
5b^2=a^2
5 divides a^2
5 divides a
let a^2=5c
5b^2= 25c^2
b^2= 5c^2
2 divides b^2
2 divides b
this shows that a and b have 5 as common factor. but a and b are co prime numbers with only 1 as common factor. hence this contradicts our assumption.
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