How do you find the square root of 20?

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Answer 1 · 2016-08-31T20:59:55

Find approximation:

$sqrt(20) ~~ 2889/646 ~~ 4.472136$

The prime factorisation is:

$20 = 2^2*5$

Hence:

$sqrt(20) = 2sqrt(5)$

This is an irrational number between $4$ and $5$ since:

$4^2 = 16 < 20 < 25 = 5^2$

It is not expressible as an exact fraction, but we can find rational approximations...

Since $20$ is roughly halfway between $4^2$ and $5^2$ , its square root is approximately $9/2$ - halfway between $4$ and $5$ .

In fact, we find:

$9^2 = 81 = 80+1 = 20*2^2 + 1$

which is in Pell's equation form:

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

with $n = 20$ , $p = 9$ and $q = 2$

That means that we can deduce the continued fraction for $sqrt(20)$ from the continued fraction for $9/2$ ...

$9/2 = 4+1/2 = [4;2]$

Hence:

$sqrt(20) = [4;bar(2,8)] = 4+1/(2+1/(8+1/(2+1/(8+1/(2+1/(8+...))))))$

To get a good approximation for $sqrt(20)$ truncate this continued fraction early, just before an ' $8$ '...

For example:

$sqrt(20) ~~ [4;2,8,2,8,2] = 4+1/(2+1/(8+1/(2+1/(8+1/2)))) = 2889/646 ~~ 4.472136$

From a calculator:

$sqrt(20) ~~ 4.472135955$