PROVE THAT ROOT 7 is IRRATIONAL. - Bzziii.com

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Prove that `\sqrt{7}` is irrational.









Let us assume, to the contrary, that ,√7 is a rational number. Then, there exist co-prime positive integers a and b such that

`\sqrt{7}` = `"a"/"b"`    b ≠ 0

So, a `\sqrt{7}`b     .....(i)

Squaring both sides, we have 

`"a"^2` = 7`"b"^2`   = 7 divides a


So we can write

a = 7c,         (where c is any integer)


Putting the value of a = 7 c in (i), we have

49 `"c"^2` = 7`"b"^2`

7 `"c"^2` = `"b"^2`

It means 7 divides `"b"^2` and so  7 divides b.
So, 7 is a common factor of both a and b which is a contradiction.
So, our assumption that `\sqrt{7}` is a rational is wrong.
Hence, we conclude that `\sqrt{7}` is a irational number.

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