Calculating Square Root By Division

I. Introduction

We learnt , how to calculate the root using the factorization method. But the factorization method works only when the given number is a perfect square and the number can be easily factorised.

This does not work when the given number is not a perfect square, and hence does not have an integer root, also sometimes factorising large numbers can be difficult.

Today we will learn a new method, which is similar to long division, to calculate the square root of any given number.

Starting with the decimal point, group the digits of the given number into groups of two.
Find the largest single digit number whose square is less than or equal to the leftmost group. This is the first digit of your square root. Write this digit in place of the quotient, and the square below the number of the leftmost group.
Calculate the remainder, and bring down the next group.
Bring down twice the current square root value in place of the new quotient.
Find the highest digit which when placed after the new quotient and then the whole number multiplied by the same digit will be less than the remainder + the new group.
Stop when you have calculated the whole answer.

If the above steps may seem a little tricky, don't worry, this you will understand this much better once you have gone through all the examples below.

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II. Examples

$\underline{Example\ 1}:$

Let us find the value of $\sqrt{67081}$.

When we divide $67081$ into groups of two digits starting at the decimal point, we get $3$ groups like, $\overline{06}\ \overline{70}\ \overline{81}$

Now we can start our division process.

$\begin{array}{rl} 2\ \ 5\ \ 9 \phantom{0000} & \Leftarrow & \hbox{result}\\ \phantom{000000}\enclose{left top}{\overline{06}\ \overline{70}\ \overline{81} \phantom{00}} \phantom{0} \\[-3pt] 4 \phantom{00000000.} & \Leftarrow & \hbox{the largest perfect square less than 6 is 4, so we write 2 in the result}\\[-3pt] 4\ 5\ \enclose{left top}{2\ 70} \phantom{00000.} & \Leftarrow & \hbox{remainder is 2, bring down the next group 70, and 2} \times \hbox{2 = 4 as divisor}\\ 2\ 25 \phantom{000000} & \Leftarrow & \hbox{we can find that 46}\times\hbox{6=276 and 45}\times \hbox{5=225, so our next digit in the result is 5}\\50\ 9\ \enclose{left top}{45\ 81} \phantom{000} & \Leftarrow & \hbox{remainder is 45, bring down the next group 81, and 25} \times \hbox{2 = 50 as divisor}\\ \underline{45\ 81} \phantom{000.} & \Leftarrow & \hbox{we can find that 509 }\times\hbox{9=4581, so our next digit of the result is 9} \\0 \phantom{0000} & \Leftarrow & \hbox{the remainder is 0, so, we stop here}\\ \end{array}$

Therefore, $\sqrt{67081} = 259$

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$\underline{Example\ 2}:$

We will take one more elaborate example. We will find the value of $\sqrt{16.5}$. Since $16.5$ is not a perfect square, its square root is an $irrational\ number$, which means a never ending series of digits after decimal point. We will learn more about $irrational\ numbers$ later. For now we will calculate the square root of this till $3$ places of decimal, so we will need $3$ groups after the decimal point. This is how we will group the digits of $16.5$

$16.5 = \overline{16}\ .\overline{50}\ \overline{00}\ \overline{00}$

Now, we will get onto the steps of calculating the square root.

Therefore, $\sqrt{16.5}$, calculated up to $3$ places of decimal, is $4.062$.

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$\underline{Example\ 3}:$

Now, let us find the square root of an integer that is not a perfect square using this method. We know that $5$ is not a perfect square. So, we will find the value of $\sqrt{5}$ using the above method.

Like the example before, $\sqrt{5}$ is also an $irrational\ number$ and has an indefinite number of digits after the decimal point, so we will calculate the value only till $3$ places of decimal. To calculate $3$ places of decimal, we will need to have $3$ groups after the decimal point.

We make the following groups from $5$.

$\overline{05}\ .\overline{00}\ \overline{00}\ \overline{00}$

$\begin{array}{rl} 2\ .\ 2\ 3\ 6 \phantom{0000} & \Leftarrow & \hbox{result} \\ \phantom{000000}\enclose{left top}{\overline{05}\ .\overline{00}\ \overline{00}\ \overline{00}} \phantom{0} \\[-3pt] 4 \phantom{0000000000} & \Leftarrow & \hbox{the largest perfect square less than 5 is 4, so we write 2 in the result}\\[-3pt] 4\ 2\ \enclose{left top}{\ \ 1\ 00} \phantom{0000000} & \Leftarrow & \hbox{remainder is 1, add decimal point to the result, bring down 00, and 2 } \times \hbox{2 = 4 as divisor}\\ 84 \phantom{0000000.} & \Leftarrow & \hbox{we can find that 43 }\times\hbox{ 3 = 129 and 42 }\times \hbox{ 2 = 84, so our next digit in the result is 2}\\44\ 3\ \enclose{left top}{\ \ \ 16\ 00} \phantom{0000.} & \Leftarrow & \hbox{remainder is 16, bring down the next group 00, and 22} \times \hbox{2 = 44 as divisor}\\ 13\ 29 \phantom{00000} & \Leftarrow & \hbox{we can find that 443 }\times\hbox{3 = 1329, so our next digit of the result is 3} \\446\ 6\ \enclose{left top}{\ \ \ 2\ 71\ 00} \phantom{00} & \Leftarrow & \hbox{remainder is 271, bring down 00, and 223 } \times \hbox{ 2 = 446 as divisor}\\ \underline{2\ 67\ 96} \phantom{...} & \Leftarrow & \hbox{we can find that 4466 }\times\hbox{ 6 = 26796, so our next digit of the result is 6} \\3\ 04 \phantom{...} & \Leftarrow & \hbox{the remainder is 304, but we reached 3 digits of decimal, so we stop here}\\ \end{array}$

Therefore, we get the value of $\sqrt{5}$, till $3$ places of decimal as $2.236$.

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$\underline{Example\ 4}:$

Let us take one more example of finding square root of a fraction. We will find the value of $\sqrt{12.7}$ upto $3$ places of decimal.

Like before, we will group the digits into groups of $2$, and will have $3$ groups after the decimal point. So, we get:

$\overline{12}\ . \overline{70}\ \overline{00}\ \overline{00}$

We will follow the same steps described previously.

$\begin{array}{rl} 3\ .\ 5\ 6\ 3 \phantom{0000} & \Leftarrow & \hbox{result} \\ \phantom{000000}\enclose{left top}{\overline{12}\ . \overline{70}\ \overline{00}\ \overline{00}} \phantom{0} \\[-3pt] 9 \phantom{0000000000} & \Leftarrow & \hbox{the largest perfect square less than 12 is 9, so we write 3 in the result}\\[-3pt] 6\ 5\ \enclose{left top}{\ \ 3\ 70} \phantom{0000000} & \Leftarrow & \hbox{remainder is 3, add decimal point to the result, bring down 70, and 3 } \times \hbox{2 = 6 as divisor}\\ 3\ 25 \phantom{0000000.} & \Leftarrow & \hbox{we can find that 66 }\times\hbox{ 6 = 396 and 65 }\times \hbox{ 5 = 325, so our next digit in the result is 5}\\70\ 6\ \enclose{left top}{\ \ \ 45\ 00} \phantom{0000.} & \Leftarrow & \hbox{remainder is 45, bring down the next group 00, and 35} \times \hbox{2 = 70 as divisor}\\ 42\ 36 \phantom{00000} & \Leftarrow & \hbox{we can find that 706 }\times\hbox{6 = 4236, so our next digit of the result is 6} \\712\ 3\ \enclose{left top}{\ \ \ 2\ 64\ 00} \phantom{00} & \Leftarrow & \hbox{remainder is 264, bring down 00, and 356 } \times \hbox{ 2 = 712 as divisor}\\ \underline{2\ 13\ 69} \phantom{0..} & \Leftarrow & \hbox{we can find that 7123 }\times\hbox{ 3 = 21369, so our next digit of the result is 6} \\57\ 31 \phantom{0..} & \Leftarrow & \hbox{the remainder is 5731, but we reached 3 digits of decimal, so we stop here}\\ \end{array}$

Therefore, the value of $\sqrt{12.7}$, calculated upto $3$ places of decimal, is $3.563$

This method may appear a little calculation intensive, but this is the only method that can be used manually to calculate square roots of numbers that are not perfect squares.

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You can get more familiar with this method by practicing this method a few times.

Following are some values with their square roots given. You can try to compute the square roots up to $2$ or $3$ places of decimal and verify you answer with what is given here.

$\sqrt{126.8} = 11.26055061$

$\sqrt{25.7} = 5.0695167$

$\sqrt{37.2} = 6.09918027$