# How do you find a square root of a decimal?

Apr 26, 2018

We can use special long division method (other than using a calculator) described below.

#### Explanation:

Let us consider a number $0.04783$, which is not a perfect square, to find the square root of $0.04783$, we should do a special long division, where we pair, the numbers in two, starting from decimal point in either direction. Here, however, we have chosen a number which has $0$ to the left of decimal point.

To the right of decimal the pairs formed are $0. \textcolor{red}{04} \textcolor{b l u e}{78} \textcolor{red}{30}$. We write the number in long form of division as shown below.

$\textcolor{w h i t e}{\times} \underline{0. \textcolor{w h i t e}{x} 2 \textcolor{w h i t e}{\times} 1 \textcolor{w h i t e}{\times} 8 \textcolor{w h i t e}{\times} 7 \textcolor{w h i t e}{\times} 0}$
$\underline{2} | 0. \textcolor{red}{04} \textcolor{w h i t e}{x} \textcolor{b l u e}{78} \textcolor{w h i t e}{x} \textcolor{red}{30} \textcolor{w h i t e}{x} \overline{00} \textcolor{w h i t e}{.} \overline{00} \textcolor{w h i t e}{.} \overline{00} \textcolor{w h i t e}{.} \overline{00}$
$\textcolor{w h i t e}{\times \times} \underline{04} \textcolor{w h i t e}{X} \downarrow$
$\textcolor{red}{4} 1 | \textcolor{w h i t e}{X} 00 \textcolor{w h i t e}{x} 78$
$\textcolor{w h i t e}{\times \times} \underline{00 \textcolor{w h i t e}{x} 41}$
$\textcolor{w h i t e}{x} \textcolor{red}{42} 8 | \textcolor{w h i t e}{\times} 37 \textcolor{w h i t e}{.} 30$
$\textcolor{w h i t e}{\times \times \times .} \underline{34 \textcolor{w h i t e}{.} 24}$
$\textcolor{w h i t e}{\times} \textcolor{red}{436} 7 | \textcolor{w h i t e}{.} 3 \textcolor{w h i t e}{x} 06 \textcolor{w h i t e}{.} 00$
$\textcolor{w h i t e}{\times \times x X x} \underline{3 \textcolor{w h i t e}{.} 05 \textcolor{w h i t e}{.} 69}$
$\textcolor{w h i t e}{\times} \textcolor{red}{4374} 0 | \textcolor{w h i t e}{x} \textcolor{w h i t e}{.} 00 \textcolor{w h i t e}{.} 31 \textcolor{w h i t e}{.} 00$

Here we have first pair is $04$ and the number whose square is just equal or less than it is $02$, so we get $2$ and write its square $04$ below $04$. The difference is $00$ and now we bring down next two digits $78$.

As a divisor we first write double of $2$ i.e. $4$ and then find a number $x$ so that $4 x$ (here $x$ stands for single digit in units place) multiplied by $x$ is just less than the number, here $0078$. We find for $x = 1$, we have $41 \times 1 = 41$ and get the difference as $37$.

Now as we have a remainder of $37$, we bring $30$ making the number $3730$. Observe that we had brought as divisor $2 \times 2 = 4$, but this time we have $21$, whose double is $42$, and so we make the divisor as $42 x$ and identify an $x$ so that $42 x$ multiplied by $x$ is just less than $3730$. This number is just $8$, as making it $9$ will make the product $429 \times 9 = 3861 > 3760$. With $8$, we get $428 \times 8 = 3424$ and remainder is $306$.

Next, we do not have anything, but for accuracy we can bring $00$. The divisor will now be $218 \times 2 = 436$ and identify an $x$ so that $436 x$ multiplied by $x$ is just less than $30600$. The number we get is $7$.

We continue to do similarly and find remainder is just $31$, while we will have $2187 \times 2 = 4374$ i.e. $4374 x$ and this $x$ ought to be zero.

It is evident that we have reached desired accuracy and $\sqrt{0.04783} \approx 0.2187$

Another example is here.