Question

Find the square root of the following (correct to three decimal places).7

Answer 1

Answer:

√7≈2.645751311

Explanation:

Since 7 is a prime number, it has no square factors and its square root cannot be simplified.

It is an irrational number, so cannot be exactly represented by pq for any integers p,q.

We can however find good rational approximations to √7.

First note that:

82=64=63+1=7⋅32+1

This is in Pell's equation form:

p2=nq2+1

with n=7, p=8 and q=3.

This means that 83 is an economical approximation for √7 and it also means that we can use 83 to derive the continued fraction expansion of √7:

83=2+11+11+11

and hence we can deduce:

√7=[2;¯¯¯¯¯¯¯¯¯¯¯¯¯¯1,1,1,4]=2+11+11+11+14+11+11+11+14+...

The next economical approximation is given by truncating the continued fraction expansion just before the next 4, i.e.

√7≈[2;1,1,1,4,1,1,1]=2+11+11+11+14+11+11+11=12748=2.6458¯3

This is also a solution of Pell's equation for 7, since we find:

1272=16129=16128+1=7⋅482+1

If you want more accuracy, truncate just before the next 4 or the one after.

By expanding the repeating part of the continued fraction for √7 we can derive a generalised continued fraction:

√7=218+764214+764214+764214+764214+...

Using a calculator, we find:

√7≈2.645751311