Question

To prove √7 is an irrational number

Answer 1

The decimal expansion of √7 is 2.6457513110645907

As we can see that the decimal expansion of √7 is non terminating and non repeating hence it is an irrational number.

Hope it helps •_•

P.S. - To find decimal expansion of an under root number find its square root.

Answer 2

Hey buddy!

ltz pleasure to help you with the question :

"HOW TO PROVE ROOT 7 IS IRRATIONAL NUMBER."
________________________________

LETS SEE HOW:
_________________________________

HERE ITZ:

_______

Let us assume to contrary that root 7 is rational.

So we can find integers r and s ( no = to 0) such that
root 7=r/s

Suppose r and s have a common factor other than 1 Then divide both by the common factor to get root 7=a/b ,where a and b are co-prime.
So a=b*root 7

Squaring both sides ,we get a^2 = 7b^2
Thus 7 divides a^2 which implies that 7 divides a.(Theorem (3))

So we can write a=7c for some integer c

Substituting for a ,we get 7b^2=49c^2

i.e.,b^2=2c^2

This means 7 divides b^2

By theorem (3) ,7 divides b.

Therefore a and b have atleast 7 as common factor.But this contradicts the fact that a and b have no common factors other than 1.
So our assumption is wrong
So root seven is irrational
____________________
____________________

HOPE IT HELPZ..:-))