# How do you find the square root of 7?

##### 1 Answer

#### Explanation:

Since

It is an irrational number, so cannot be exactly represented by

We can however find good rational *approximations* to

First note that:

#8^2 = 64 = 63+1 = 7*3^2 + 1#

This is in Pell's equation form:

#p^2 = n q^2 + 1#

with

This means that

#8/3 = 2 + 1/(1+1/(1+1/1))#

and hence we can deduce:

#sqrt(7) = [2;bar(1,1,1,4)] = 2 + 1/(1+1/(1+1/(1+1/(4+1/(1+1/(1+1/(1+1/(4+...))))))))#

The next economical approximation is given by truncating the continued fraction expansion just before the next

#sqrt(7) ~~ [2;1,1,1,4,1,1,1] = 2 + 1/(1+1/(1+1/(1+1/(4+1/(1+1/(1+1/1)))))) = 127/48 = 2.6458bar(3)#

This is also a solution of Pell's equation for

#127^2 = 16129 = 16128+1 = 7*48^2+1#

If you want more accuracy, truncate just before the next

By expanding the repeating part of the continued fraction for

#sqrt(7) = 21/8+(7/64)/(21/4+(7/64)/(21/4+(7/64)/(21/4+(7/64)/(21/4+...))))#

Using a calculator, we find:

#sqrt(7) ~~ 2.645751311#