# What is the square root of 30?

Sep 14, 2015

You can't get an exact answer by hand, but you can estimate it using some tricks.

$\sqrt{25} = 5$
$\sqrt{36} = 6$

so it must be between $5$ and $6$.

$30$ is slightly less than halfway between $5$ and $6$.

What you can begin with is assuming that because:

$30 - 25 = 5$
$36 - 25 = 11$
5/11 = 45.45%

... the square root of $30$ is probably near $0.45 + 5 = 5.45$. What you are assuming then is:

$x + 0.45 \cdot \left[\left(x + 1\right) - x\right] \approx {x}^{2} + 0.45 \cdot \left[{\left(x + 1\right)}^{2} - {x}^{2}\right]$

or more specifically,

$5 + 0.45 \cdot \left(6 - 5\right) \approx 25 + 0.45 \cdot \left(36 - 25\right)$

Actually, it's not a bad bet. The actual square root is about $5.477$.

Sep 14, 2015

$30 = 2 \cdot 3 \cdot 5$ has no square factors, so it is not possible to simplify $\sqrt{30}$.

You can calculate an approximation by hand as shown below...

#### Explanation:

I explained my favourite method (a sort of Newton Raphson method) for approximating square roots of integers in an answer to the following question:

Given an integer $n$, choose integers ${p}_{0}$ and ${q}_{0}$ so ${p}_{0} / {q}_{0}$ is a reasonable first approximation to $\sqrt{n}$.

Then iterate using the formulas:

${p}_{i + 1} = {p}_{i}^{2} + n {q}_{i}^{2}$

${q}_{i + 1} = 2 {p}_{i} {q}_{i}$

If the resulting values of ${p}_{i + 1}$ and ${q}_{i + 1}$ have a common factor, then divide both by that factor before the next iteration.

The successive pairs ${p}_{i}$, ${q}_{i}$ provide a sequence of rational approximations ${p}_{i} / {q}_{i}$ to $\sqrt{n}$ that converge quite rapidly.

For our example, let $n = 30$, ${p}_{0} = 11$, ${q}_{0} = 2$ (using an initial approximation of $5.5$ since $30$ is about halfway between ${5}^{2} = 25$ and ${6}^{2} = 36$).

${p}_{1} = {p}_{0}^{2} + n {q}_{0}^{2} = {11}^{2} + 30 \cdot {2}^{2} = 121 + 120 = 241$

${q}_{1} = 2 {p}_{0} {q}_{0} = 2 \cdot 11 \cdot 2 = 44$

This would give $\sqrt{30} \approx \frac{241}{44} = 5.47 \dot{7} \dot{2}$

${p}_{2} = {p}_{1}^{2} + n {q}_{1}^{2} = {241}^{2} + 30 \cdot {44}^{2} = 58081 + 58080 = 116161$

${q}_{2} = 2 {p}_{1} {q}_{1} = 2 \cdot 241 \cdot 44 = 21208$

This gives $\sqrt{30} \approx \frac{116161}{21208} \approx 5.477225575$