What is a Square Root?

In mathematics, the square root of a number a is another number b in such a way that b^2 = a. We can say that when we will take the square of the square root of a number, then we will get the number whose square root was originally computed. The square root of a number can be positive or negative, For instance, the square root of 25 is equal to 5, and -5 because 5 multiplied by 5 is 25 and -5 multiplied by -5 is also 25. Hence, can say that the square root is an inverse operation of taking the square of a number.

Many people today think that they cannot find the square roots of the bigger numbers without calculators. While calculators are the easiest and the quickest way to compute the square root of a number, however, by doing so you cannot understand the logic behind the square root of a number. In this article, we will discuss how to compute the square root of the number using paper and pencil with examples. So, let us get started.

 

The best Maths tutors available
1st lesson free!
Intasar
4.9
4.9 (23 reviews)
Intasar
£42
/h
1st lesson free!
Matthew
5
5 (17 reviews)
Matthew
£25
/h
1st lesson free!
Dr. Kritaphat
4.9
4.9 (6 reviews)
Dr. Kritaphat
£49
/h
1st lesson free!
Paolo
4.9
4.9 (11 reviews)
Paolo
£25
/h
1st lesson free!
Petar
4.9
4.9 (9 reviews)
Petar
£27
/h
1st lesson free!
Rajan
4.9
4.9 (11 reviews)
Rajan
£15
/h
1st lesson free!
Farooq
5
5 (13 reviews)
Farooq
£35
/h
1st lesson free!
Myriam
5
5 (15 reviews)
Myriam
£20
/h
1st lesson free!
Intasar
4.9
4.9 (23 reviews)
Intasar
£42
/h
1st lesson free!
Matthew
5
5 (17 reviews)
Matthew
£25
/h
1st lesson free!
Dr. Kritaphat
4.9
4.9 (6 reviews)
Dr. Kritaphat
£49
/h
1st lesson free!
Paolo
4.9
4.9 (11 reviews)
Paolo
£25
/h
1st lesson free!
Petar
4.9
4.9 (9 reviews)
Petar
£27
/h
1st lesson free!
Rajan
4.9
4.9 (11 reviews)
Rajan
£15
/h
1st lesson free!
Farooq
5
5 (13 reviews)
Farooq
£35
/h
1st lesson free!
Myriam
5
5 (15 reviews)
Myriam
£20
/h
First Lesson Free>

Procedure of Finding the Square Root of a Number without Calculator

In this section, we will discuss the steps involved in finding the square root of a number without a calculator. We will explain these steps through examples.

Example 1

Calculate the square root of the number 34466.

Solution

It is a five-digit number. If we want to calculate its square root without a calculator, then we have to follow the following steps.

Step 1

Group the digits in pairs of twos from the left. In case you are calculating the square root of a number having an odd number of digits, then the left-most digit will be left alone, i.e. it will not be paired.

Step 2

Now, in this step, you need to start with the first group (the left-most group). If you have an odd number of digits in a number, then the group may have a single-digit only. Find the greatest square that is less than or equal to that group and find its square root. Write the square below that group and the square root of the number above. In this example, the greatest square that is less than or equal to 3 is 1. We will write 1 above and below like this:

\begin{array}{ccccccccccc} & & &  1&\\ \cline{3-10} \multicolumn{2}{r}{ \surd} & 3&44&66 && \\ & &1& & & & \\ \cline{3-7} \cline{4-8} & & &244&&&&&&\\ \end{array}

 

Step 3

Now, multiply the number written above the square root symbol by 2 and write the number in paranthesis on the left side with a blank. In this example, the number above the square root symbol is 1. When we will multiply it by 2, we will get 2. Hence, we will write (2 _) on the left side like this:

\begin{array}{ccccccccccc} & & &  1&\\ \cline{3-10} \multicolumn{2}{r}{ \surd} & 3&44&66 && \\ & &1& & & & \\ \cline{3-7} \cline{4-8} & &(2 \textendash) &244&&&&&&\\ \end{array}

Step 4

In this step, we will think which single-digit number could come in the blank, so that twenty something multiplied by  something is equal to or less than 244. 28 multiplied by 8 is equal to 224, so we will place 8 in the blank.

\begin{array}{ccccccccccc} & & &  1&\\ \cline{3-10} \multicolumn{2}{r}{ \surd} & 3&44&66 && \\ & &1& & & & \\ \cline{3-7} \cline{4-8} (28)& &&244&&&&&&\\ \end{array}

 

Step 5

Multiply 28 by 8 to get 224. Place 8 above the square root symbol and 224 below 244 and subtract the two numbers like this:

 

\begin{array}{ccccccccccc} & & &  18&\\ \cline{3-10} \multicolumn{2}{r}{ \surd} & 3&44&66 && \\ & &1& & & & \\ \cline{3-7} \cline{4-8} (28)& &&244&&&&&&\\ & &&224&& & &&& \\ \cline{4-8} & &&20& && &&& \\ \end{array}

 

Step 6

Double the digit above the square root symbol and write the number again in parenthesis on the left side. Put blank on the right hand side of the digit because the third digit is unknown.  In this example, the number above the square root is 18, so by doubling it, we will get 36. We will write (36 _) on the left side like this:

\begin{array}{ccccccccccc} & & &  18&\\ \cline{3-10} \multicolumn{2}{r}{ \surd} & 3&44&66 && \\ & &1& & & & \\ \cline{3-7} \cline{4-8} (29)& &&244&&&&&&\\ & &&224&& & &&& \\ \cline{4-8} (36 \textendash)& &&2066&&& && &&& \\ \end{array}

 

Step 7

Think of the digit that could go in the blank in such a way that three hundred and sixty something multiplied by something is equal to 2066.

\begin{array}{ccccccccccc} & & &  185&\\ \cline{3-10} \multicolumn{2}{r}{ \surd} & 3&44&66 && \\ & &1& & & & \\ \cline{3-7} \cline{4-8} (29)& &&244&&&&&&\\ & &&224&& & &&& \\ \cline{4-8} (368)& &&2066&&& && &&& \\ & &&1840&&& && &&& \\ \cline{4-8} & &&226&&& && &&& \\ \end{array}

 

Step 8

Now, it's time to get the values of the square root after the decimal point. Put 00 at the end of the dividend and also after the number above the square root symbol. Double the number 185 and place blank at the end and write the number in parenthesis like this:

\begin{array}{ccccccccccc} & & &  185&\\ \cline{3-10} \multicolumn{2}{r}{ \surd} & 3&44&66 && \\ & &1& & & & \\ \cline{4-8} (29)& &&244&&&&&&\\ & &&224&& & &&& \\ \cline{4-8} (368)& &&2066&&& && &&& \\ & &&1840&&& && &&& \\ \cline{4-8} (370 \textendash)& &&226 00&&& && &&& \\ \end{array}

Step 9

Think of the fourth digit that will come after 370, so that three thousand seven hundred something is multiplied by something to get a number smaller than or equal to 221000. 3709 multiplied by 6 is equal to 22,254. Hence, place 6 after the decimal point in the quotient's place and write 22254 below the number 22600 and subtract them:

\begin{array}{ccccccccccc} & & &  185.6&\\ \cline{3-10} \multicolumn{2}{r}{ \surd} & 3&44&66 && \\ & &1& & & & \\ \cline{3-7} \cline{4-8} (29)& &&244&&&&&&\\ & &&224&& & &&& \\ \cline{4-8} (369)& &&2066&&& && &&& \\ & &&1840&&& && &&& \\ \cline{4-8} (3709)& &&226 00&&& && &&& \\ & &&22254&&& && &&& \\ \cline{4-8} & &&346&&& && &&& \\ \end{array}

Hence, the square root of the number 34466 up to one decimal place is 185.6.

 

Example 2

Calculate the square root of the number 54.

Solution

Follow these steps to calculate the square root of the above number without a calculator.

Step 1

Group the numbers. In this example, we have a 2 digit number, so we will have only one group 54.

Step 2

Find the perfect square that is less than or equal to 54. In this example, that perfect square is 49. Take the square root of 49 and place the number above the square root symbol. Write 49 below 54 and subtract the two numbers.

\begin{array}{ccccccccccc} && 7&&&\\ \cline{3-10} \multicolumn{2}{r}{ \surd} & 54&& && \\ & &49& & & & \\ \cline{3-7} \cline{4-8} && 5&&&&&&&&\\ \end{array}

 

 

Step 3

Double the number above the square root symbol and write it on the left hand side with a dash at the end because we still have to figure out the third digit. In this example, the 2 multiplied by 7 is 14, so we will write (14_) on the left hand side like this:

 

\begin{array}{ccccccccccc} && 7&&&\\ \cline{3-10} \multicolumn{2}{r}{ \surd} & 54&& && \\ & &49&&&& \\ \cline{3-7} (14 \textendash) &&500&&&&&&&&\\ \end{array}

 

Step 4

In this step, we will think which single-digit number could come in the blank, so that one forty something multiplied by  something is equal to or less than 500.  149 into 3 is equal to 447, so 7 in the blank will work out. We will also put 3 after the decimal point above the square root symbol as shown below:

 

\begin{array}{ccccccccccc} && 7.3&&&\\ \cline{3-10} \multicolumn{2}{r}{ \surd} & 54.00&& && \\ & &49& & & & \\ \cline{3-7} (149) &&500&&&&&&&&&\\   &&447&&&&&&&&&\\ \cline{3-7}   &&53&&&&&&&&\\ \end{array}

We will stop our calculation here because we were asked to compute the square root of the number 54 up to 1 decimal point. Hence, the square root of 54 up to a single decimal place is 7.3.

 
Need a Maths teacher?

Did you like the article?

1 Star2 Stars3 Stars4 Stars5 Stars 5.00/5 - 1 vote(s)
Loading...

Emma

I am passionate about travelling and currently live and work in Paris. I like to spend my time reading, gardening, running, learning languages and exploring new places.