# How do you find the square root of 20?

Aug 31, 2016

Find approximation:

$\sqrt{20} \approx \frac{2889}{646} \approx 4.472136$

#### Explanation:

The prime factorisation is:

$20 = {2}^{2} \cdot 5$

Hence:

$\sqrt{20} = 2 \sqrt{5}$

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 $\frac{9}{2}$ - halfway between $4$ and $5$.

In fact, we find:

${9}^{2} = 81 = 80 + 1 = 20 \cdot {2}^{2} + 1$

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 $\frac{9}{2}$...

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:

$\sqrt{20} \approx 4.472135955$