The history of squares and square roots of numbers in Mathematics dates back to the 1900s when Babylonians invented various methods to determine the square root of numbers.

However, Indian Mathematicians have recorded several methods to find the square root of numbers in their works such as ‘Sulabha Sutra’ by Baudhayana and ‘Aryabhatiya’ by Aryabhatta. The number below gives ‘What is the value of root 3’ determined upto 4 decimal places.

Only perfect square numbers have perfect square roots.

An even perfect square always has an even root.

An odd perfect square always has an odd root.

All perfect squares are positive integers and hence the square root of negative numbers cannot be explained with the concept of real numbers.

Square roots of 2 numbers can be multiplied and the product is the square root of the product of two numbers. Example: The product of square root 3 and square root 2 is square root 6. i.e. √3 x √2 = √6

Square and square root of any number are inverse operations. Therefore, the square root of the square of a number is the number itself.

The square root of any positive integer can be found using various methods. A few notable ones are :

Repeated subtraction method

Prime Division or Prime Factorization

Average method

Number Line method

Long division method

First two methods can be used to find the square roots of perfect square numbers only. However, the other three methods can be employed to find the square root of any positive integer.

Example:

Find the square root of 100 using repeated subtraction method.

Solution:

100 - 1 = 99

99 - 3 = 96

96 - 5 = 91

91 - 7 = 84

84 - 9 = 75

75 - 11 = 64

64 - 13 = 51

51 - 15 = 36

36 - 17 = 19

19 - 19 = 0

Find the square root of 36 by prime factorization method.

Solution:

The basic principle of finding the square root of a number is the idea of finding the average of two numbers.

Step 1:

Determine the two perfect square numbers which are very close to the given number on either side.

For the number ‘3’, the immediate perfect square preceding ‘3’ is ‘1’ and the immediate perfect square succeeding 3 is ‘4’.

Step 2:

Note down the square roots of the perfect squares identified in Step 1.

Square root of ‘1’ is ‘1’ and the square root of ‘4’ is ‘2’.

Step 3:

It can be inferred that the square root of a given number lies between the square roots of numbers determined in step 2.

Square root 3 is any number between 1 and 2.

Step 4: Divide the number whose square root is determined by any of the numbers obtained in Step 2.

‘3’ can be divided either by ‘1’ or ‘2’.

Let us divide ‘3’ by ‘2’

Step 5: Find the average of the quotient and divisor in Step 4.

The average of 2 and 1.5 is

Step 6:To get an accurate value of the square root, an average of the answer in step 5 and divisor of step 4 can be found. Finding of average can be continued as many times as required to get a precise value.

However the answer in step 5 is approximately equal to the value of root 3 calculated using the calculator (1.7321).

To find the square root of any number using a number line, a clear understanding of the basic concept of “Pythagorean Theorem” is required.

Pythagorean theorem states that “In a right triangle, the square on the longest side (hypotenuse) is equal to the sum of the squares on the other two sides (base and perpendicular)”.

A series of steps to be followed to determine the square root of any number using a number line is explained below.

Step 1:

Construct a number line with a minimum two units on either side.

Step 2:

Label the unit 1 as A. From the point ‘A’ draw a perpendicular AB of unit length.

Step 3:

Join OB. According to Pythagorean theorem, the length of OB is calculated as

Step 4:

From the point ‘B’ draw a perpendicular BC of 1 unit to the line OB. Join OC. The length of OC is calculated using pythagoras theorem as

Length of the line OC gives the measure of square root of 3.

Long division method is the most convenient and easiest method to find the square root of any number. It gives an accurate value of any number.

Step 1:

‘5’ is represented as 3.00000000 to determine the square root of 3 upto 4 decimal places. Group the digits after the decimal point in pairs as 3. 00 00 00 00

Step 2:

Choose a perfect square whose value is below ‘3’. The perfect square below 3 is ‘1’ and its square root is ‘1’.

Step 3:

Represent that ‘1 times 1 gives 1’ by writing 1 in the place of dividend and quotient. The number ‘1’ is written below ‘3’. ‘1’ is subtracted from ‘3’ and the difference is 2.

Step 4:

The first pair of zeroes in the dividend is carried down and a decimal point is placed in the quotient.

Step 5:

Add 1 to the divisor. The sum will be 2. Now take a number succeeding 2 to get a 2 digit number such that when the two digit number is multiplied by the number taken, the product is less than 200. If we take the digit as ‘7’, 27 x 7 = 189 which is less than 200. So now the remainder is ‘11’ and the next divisor is 34_. And the dividend is 1100 if the next pair of zeroes are taken down.

Step 6:

If the above steps are continued, the final answer obtained in the place of quotient is 1.7320.

What would be the length of the diagonal of one face of a cuboid whose length is 2 cm and width is 1cm?

Square root of 2

Square root of 5

Square root of 3

What is the value of root 3 raised to the power two?

Nine

3

Square root of 3

FAQ (Frequently Asked Questions)

1. What is a Square Root of A Number?.

The product obtained by multiplying a number by itself is called a square number. Square root of a number is that number which when multiplied by itself gives the answer as the number whose square root is to be determined.

Perfect square numbers are those numbers which are obtained by squaring integers.

Computing the square and square roots are inverse operations . Therefore, the square root of the square of a number is the number itself.

Square root of a perfect square is always a positive or negative integer. Square root of an even perfect square is even and that of an odd square number is odd.

2. Does the Square Root of Any Number Have Two Answers?

Square root of any number is that number which when multiplied by itself gives the product as in the question. For example, when the positive integer 3 is multiplied by itself, the product is 9. At the same time, the product still remains 9 when the negative integer ‘-3’ is multiplied by itself. This is due to a known fact that two negative numbers are multiplied the product remains positive. So it can be inferred that the square root of any number has two values either positive or negative. If ‘y’ is the square root of a number ‘x’, it can be represented as:

x = ± √y |