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Prove that √2 is irrational?

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Prove that √2 is irrational?

Posted by Swagat Mishra 7 years, 6 months ago

CBSE > Class 10 > Mathematics

2 answers

Shivani K 7 years, 6 months ago

√2 is irrational bcz √2 is non terminating and non repeating....

Susai Raj 7 years, 6 months ago

Supp0se √2 is rational.
Then √2 can be written as a fraction of integers in its simplest form.
So √2 = a/b , where a,b are integers, b is not equal to 0 and a, b are coprimes.
Squaring both sides we get
2=a^2/b^2
Then a^2 = 2×b^2 ......(1)
So 2 is a factor of a^2.
Since 2 is prime 2 is a factor of a ......(2)
So a = 2m for some integer m. Substitute this in (1),
4m^2 = 2b^2
=> 2m^2 = b^2
=> 2 is a factor of b^2
So 2 is a factor of b. ...(3)
From (2) and (3) 2 is a common factor of a and b.
Which is a contradiction to the fact that a and b are coprimes.
So our assumptio that √2 is rational is wrong .
So √2 is irrational.

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