Finding squAre roots by Vedic Methods
Square root of any number means to get a number which is multiplied by itself gives the
given number. In the conventional method of finding the square root, the divisor goes on
becoming larger at each step. This increases the calculation time as well as the complexity of
the problem. Here, we shall try to learn some speedy Vedic Methods of finding the square
roots of perfect square numbers. Before proceeding for finding square roots, let us have a
look into the known facts of squares and square roots.
The basic rules for extracting square roots are :
(i) The given number is arranged in two-digit groups from right to left; and a single
digit (if any) left over at the left and is counted as a group by itself.
(ii) The number of digits in the square root will be the same as the number of two-
digit groups in the given number including a single digit group (if any). Thus,
36 will count as one group, 169 as two groups and 1225 as two groups.
(iii) If the number contains n digits then the square root will contain n
2 (when n is
even) and n + 1
2 (when n is odd) digits. Thus, one or two digit number will have
the square root of one digit, three and four digit number will have the square root
of two digits, 5 and 6 digit number will have the square root of 3 digits and so
on.
(iv) The squares of first nine natural numbers are :
12 = 1, 2 2 = 4, 3 2 = 9, 4 2 = 16, 5 2 = 25, 6 2 = 36, 7 2 = 49, 8 2 = 64, 9 2 = 81.
This means :
(a) unit digit of the perfect square number is 1, 4, 5, 6, 9 or 0.
(b) a perfect square number cannot end in 2, 3, 7 or 8.
(c) the relation between the unit digit of a perfect square number and the unit
digit of its square root is as follows :
Unit digit of the number 1 4 5 6 9 0
Unit digit of square root 1 or 9 2 or 8 5 4 or 6 3 or 7 0
(d) If there are odd number of zeros at the end (on right side) of a number, then it
will not be a perfect square.
VilokAnAM (obserVAtion) Method
Square root of one or two digit number is well known from the table given in the beginning.
Now we shall learn the method of finding square root of 3 or 4 digit perfect square number
by Vilokanam method.
Look at the unit digit of the given number and decide about the unit digit of the square
root from the following data :
Unit digit of the number 0 1 4 5 6 9
Unit digit of square root 0 1 or 9 2 or 8 5 4 or 6 3 or 7
Now ignore the last two digits (unit digit and ten’s digit) of the given number and
find out the greatest number whose square is less than or equal to the remaining part of
the given number. Then adjust the above obtained unit digits on its right side and get two
numbers. Find out the unique number with its unit digit 5 which lies between these two Supplement to ICSE Mastering Mathematics – VIII10
numbers and obtain the square of this unique number. If the given number is less than this
square number then the smaller number among above obtained two numbers is the square
root of the given number; otherwise another one is the required square root of the given
number. Let us learn it with the help of some examples.
IlluStratIvE ExamplES
Example 1. Find the square root of 841.
solution. The given number is 841.
Its unit digit is 1, therefore, the unit digit of the square root will be 1 or 9.
Ignoring the last two digits (unit digit and ten’s digit) we get 8.
The greatest number whose square is less than or equal to 8 is 2.
Adjusting above obtained two unit digits 1 or 9 to the right of 2, we get two numbers
21 and 29.
The unique number with unit digit 5 which lies between 21 and 29 is 25.
(25)2 = 625
(By Ekadhikena sutra : (25) 2 = (2 × 3) 25 = 625).
Since 841 greater than 625, therefore, the required square root is 29.
Hence, 841 = 29.
Example 2. Find the square root of 4356.
solution. The given number is 4356.
Its unit digit is 6, therefore, the unit digit of the square root will be 4 or 6.
Ignoring the last two digits (unit digit and ten’s digit), we get 43.
The greatest number whose square is less than or equal to 43 is 6.
On adjusting above obtained two unit digits 4 or 6 to the right of 6, we get two numbers
64 and 66.
The unique number with unit digit 5 which lies between 64 and 66 is 65.
(65)2 = 4225.
(By Ekadhikena sutra : (65) 2 = (6 × 7) 25 = 4225)
Since 4356 greater 4225, therefore, the required square root is 66.
Hence, 4356 = 66.
Thank You....
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