# How do you find the square root of 4783?

Jan 31, 2017

$\sqrt{4783} \approx 69.159$

#### Explanation:

As $4783$ is not a perfect square, to find the square root of $4783$, we should do a special long division, where we pair, the numbers in two, starting from decimal point in either direction.

Here first pair is $47$ and the number whose square is just less than it is $6$, so we write $36$ below $47$. The difference is $11$ and now we bring down next two digits $83$. As a divisor we first write double of $6$ i.e. $12$ and then find a number $x$ so that $12 x$ (here $x$ stands for single digit in units place) multiplied by $x$ is just less than the number, here $1183$. We find for $x = 9$, we have $129 \times 9 = 1161$ and get the difference as $22$.

Now as we still have a remainder of $22$, we bring $00$ after decimal point. Also put decimal after $69$ and this makes it $6900$.

Now recall we had brought as divisor $6 \times 2 = 12$, but this time we have $69$ so we make the divisor as $138 x$ and identify an $x$ so that $138 x$ multiplied by $x$ is just less than $2200$. This number is just $1$, as making it $2$ will make the product $1382 \times 2 = 2764 > 2200$.

We continue to do this till we have desired accuracy.

$\textcolor{w h i t e}{\times} 6 \textcolor{w h i t e}{\times} 9 \textcolor{w h i t e}{\times} .1 \textcolor{w h i t e}{\times} 5 \textcolor{w h i t e}{\times} 9$
$\underline{6} | \overline{47} \textcolor{w h i t e}{.} \overline{83} . \textcolor{w h i t e}{.} \overline{00} \textcolor{w h i t e}{.} \overline{00} \textcolor{w h i t e}{.} \overline{00} \textcolor{w h i t e}{.} \overline{00}$
$\textcolor{w h i t e}{X} \underline{36} \textcolor{w h i t e}{X} \downarrow$
$\textcolor{red}{12} 9 | \textcolor{w h i t e}{X} 11 \textcolor{w h i t e}{.} 83$
$\textcolor{w h i t e}{\times \times x} \underline{11 \textcolor{w h i t e}{.} 61}$
$\textcolor{w h i t e}{x} \textcolor{red}{138} 1 | \textcolor{w h i t e}{X} 22 \textcolor{w h i t e}{.} 00$
$\textcolor{w h i t e}{\times \times \times x} \underline{13 \textcolor{w h i t e}{.} 81}$
$\textcolor{w h i t e}{\times} \textcolor{red}{1382} 5 | \textcolor{w h i t e}{.} 8 \textcolor{w h i t e}{.} 19 \textcolor{w h i t e}{.} 00$
$\textcolor{w h i t e}{\times \times \times \times} \underline{6 \textcolor{w h i t e}{.} 91 \textcolor{w h i t e}{.} 25}$
$\textcolor{w h i t e}{\times} \textcolor{red}{13830} 9 | \textcolor{w h i t e}{} 1 \textcolor{w h i t e}{.} 27 \textcolor{w h i t e}{.} 75 \textcolor{w h i t e}{.} 00$
$\textcolor{w h i t e}{\times \times \times \times} \underline{1 \textcolor{w h i t e}{.} 24 \textcolor{w h i t e}{.} 47 \textcolor{w h i t e}{.} 81}$
$\textcolor{w h i t e}{\times \times \times \times x} \textcolor{w h i t e}{.} 32 \textcolor{w h i t e}{.} 71 \textcolor{w h i t e}{.} 9$

Hence $\sqrt{4783} \approx 69.159$

Feb 5, 2017

$\sqrt{4783} = 69 + \frac{22}{138 + \frac{22}{138 + \frac{22}{138 + \ldots}}}$

#### Explanation:

Here's a method using generalised continued fractions.

First the theory...

Suppose we can find numbers $a$ and $b$ such that:

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

Then we have:

$a + \frac{b}{a + \sqrt{n}} = a + \frac{b}{a + \textcolor{b l u e}{a + \frac{b}{2 a + \frac{b}{2 a + \frac{b}{2 a + \frac{b}{2 a + \ldots}}}}}} = \sqrt{n}$

Multiplying both ends by $\left(a + \sqrt{n}\right)$ we get:

${a}^{2} + \textcolor{red}{\cancel{\textcolor{b l a c k}{a \sqrt{n}}}} + b = \textcolor{red}{\cancel{\textcolor{b l a c k}{a \sqrt{n}}}} + n$

Hence:

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

In our example, first note that $4783 < 4900 = {70}^{2}$, so let's see what we get when we square $69$:

${69}^{2} = {\left(70 - 1\right)}^{2} = {70}^{2} - 2 \cdot 70 + 1 = 4900 - 140 + 1 = 4761$

So putting $n = 4783$ and $a = 69$ we get:

$b = n - {a}^{2} = 4783 - 4761 = 22$

So:

$\sqrt{4783} = 69 + \frac{22}{138 + \frac{22}{138 + \frac{22}{138 + \ldots}}}$

We can truncate after any number of terms to get a rational approximation for $\sqrt{4783}$:

$\sqrt{4783} \approx 69$

$\sqrt{4783} \approx 69 + \frac{22}{138} = \frac{4772}{69} \approx 69.1594$

$\sqrt{4783} \approx 69 + \frac{22}{138 + \frac{22}{138}} = 69 + \frac{22}{\frac{9533}{69}} = \frac{659295}{9533} \approx 69.1592363$

A calculator tells me:

$\sqrt{4873} \approx 69.1592365487$