Prove that √5 is irrational

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Prove that √5 is irrational

Posted by Veena Jha 6 years, 5 months ago

CBSE > Class 08 > Mathematics

1 answers

Sia ? 6 years, 5 months ago

Let take √5 as a rational number
If a and b are two co-prime number and b is not equal to 0.
We can write √5 = a/b
Multiply by b both side we get
b√5 = a
To remove root, Squaring on both sides, we get
5b² = a² ……………(1)
Therefore, 5 divides a² and according to a theorem of rational number, for any prime number p which is divided 'a²' then it will divide 'a' also.
That means 5 will divide 'a'. So we can write
a = 5c
and putting the value of a in equation (1) we get
5b² = (5c)²
5b² = 25c²
Divide by 25 we get
b^2/5 = c²
again using the same theorem we get that b will divide by 5
and we have already get that a is divided by 5
but a and b are co-prime number. so it is contradicting.
Hence √5 is an irrational number

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CBSE > Class 08 > Mathematics