# How do you find the square root of 43?

Feb 14, 2017

See below

#### Explanation:

If you are looking at approximation methods that you can employ using pen, paper and some mental arithmetic, you can try a Binomial Expansion.

If you start at 36, a square number, so that you are looking for $\sqrt{36 + 7}$, you can now play with that a little:

$\sqrt{36 + 7} = \sqrt{36} \sqrt{1 + \frac{7}{36}} = 6 \sqrt{1 + \frac{7}{36}}$

You can then use the Binomial Expansion , ie:

(1+x)^alpha = 1 + alpha x + (alpha (alpha - 1))/(2!)x^2 + ...:

In this case:

$6 {\left(1 + \frac{7}{36}\right)}^{\frac{1}{2}}$

=6 (color(green)(1 +1/2 * 7/36) + 1/2 (-1/2)(1/(2!)) (7/36)^2 + ...)

Even just the first two terms give $\frac{79}{12} \approx 6.583$ and ${6.583}^{2} \approx 43.34$

We could get a little closer by using a different square number. If you start at 49, another square number, you are now looking at:

$\sqrt{49 - 6} = \sqrt{49} \sqrt{1 - \frac{6}{49}} = 7 \sqrt{1 - \frac{6}{49}}$

Using just the first 2 terms of the Binomial Expansion:

$= 7 \left(1 - \frac{1}{2} \cdot \frac{6}{49}\right) = \frac{46}{7} \approx 6.571$

And ${6.571}^{2} = 43.18$

Feb 17, 2017

$\sqrt{43} = \frac{13}{2} + \frac{\frac{3}{4}}{13 + \frac{\frac{3}{4}}{13 + \frac{\frac{3}{4}}{13 + \ldots}}}$

#### Explanation:

$43$ is a prime number, so its square root is irrational.

We can find approximations to it as follows...

Note that $43$ is roughly half way between $36 = {6}^{2}$ and $49 = {7}^{2}$

So a good first approximation for $\sqrt{43}$ would be $\frac{13}{2}$.

We find:

${\left(\frac{13}{2}\right)}^{2} = \frac{169}{4} = 42.25$

Given $n > 0$ and $0 < a < \sqrt{n}$ we can write $\sqrt{n}$ as a generalised continued fraction:

$\sqrt{n} = a + \frac{b}{2 a + \frac{b}{2 a + \frac{b}{2 a + \ldots}}}$

where $b = n - {a}^{2}$

So in our example:

$n = 43$, $a = \frac{13}{2}$ and $b = 43 - \frac{169}{4} = \frac{3}{4}$

So:

$\sqrt{43} = \frac{13}{2} + \frac{\frac{3}{4}}{13 + \frac{\frac{3}{4}}{13 + \frac{\frac{3}{4}}{13 + \ldots}}}$

We can truncate this to get rational approximations.

For example:

$\sqrt{43} \approx \frac{13}{2} + \frac{\frac{3}{4}}{13} = \frac{341}{52} \approx 6.5577$

$\sqrt{43} \approx \frac{13}{2} + \frac{\frac{3}{4}}{13 + \frac{\frac{3}{4}}{13}} = \frac{8905}{1358} \approx 6.557437$

A calculator tells me:

$\sqrt{43} \approx 6.5574385243$

See https://socratic.org/s/aCh3Xasm for another example and explanation of this method.