# How do you find the square root of 2000?

##### 1 Answer

#### Explanation:

If

So:

#sqrt(2000) = sqrt(400*5) = sqrt(400)*sqrt(5) = 20sqrt(5)#

Since

#sqrt(5) = [2;bar(4)] = 2 + 1/(4+1/(4+1/(4+1/(4+...))))#

According to how accurate an approximation we want we can terminate this continued fraction at more or fewer terms.

For example:

#sqrt(5) ~~ [2;4,4] = 2+1/(4+1/4) = 2 + 4/17 = 38/17#

So:

#sqrt(2000) = 20 sqrt(5) ~~ 20*38/17 ~~ 44.71#

Actually:

#sqrt(2000) ~~ 44.72135954999579392818#

As another way of calculating the successive approximations provided by the continued fraction, consider the sequence:

#0, 1, 4, 17, 72, 305,...#

where

This is similar to the Fibonacci sequence, except the rule is

This is strongly related to the continued fraction:

#[4;bar(4)] = 4+1/(4+1/(4+1/(4+1/(4+...))))#

The ratio between successive terms of the sequence tends to

For example, we can find an approximation for

#305/72 - 2 = 161/72#

Hence