Square Root :
The square root of a number ‘a’ is that number which when multiplied by itself gives ‘a’ as the product.
Thus, if b is the square root of a number ‘a’, then
b x b = a or \( { b }^{ 2 } \) = a
The square root of a number ‘a’ is denoted by
It follows from this that
b = \(\sqrt { a } \)<=> \( { b }^{ 2 } \) = a.
i.e., b is the square root of ‘a’ if and only if ‘a’ is the square of b.
Illustration :
(i) \( \sqrt { 4 } \) =2, because \( { 2 }^{ 2 } \) = 4
(ii) \( \sqrt { 9 } \)=3, because \( { 3 }^{ 2 } \) =9.
(iii) \( \sqrt { 324 } \) = 18, because \( { 18 }^{ 2 } \) = 324.
Remark: Since 4 = \( { 2 }^{ 2 } \) = \( { (-2) }^{ 2 } \), therefore 2 and —2 can both be the square roots of 4. However, we agree that the square root of a number will be taken to be positive square root only. Thus, we have \( \sqrt { 4 } \) = 2.
Properties Of Square Root :
Property 1:- If the units digit of a number is 2, 3, 7 or 8, then it does not have a square root in N (the set of natural numbers).
Explanation: By property 1 of square numbers, a number having 2, 3, 7 or 8 at unit’s place cannot be a perfect square. Hence, a number having 2, 3, 7 or 8 at units place does not have a square root in N.
Property 2:-If a number ends in an odd number of zeros, then it does not have a square root. If a square number is followed by an even number of zeros, it has a square root in which the number of zeros in the end is half the number of zeros in the number.
Explanation: By property 2 of square numbers, the number of zeros at the end of a perfect square is always even and is twice the number of zeros at the end of the number.
Property 3:- The square root of an even square number is even and that square root of an odd square number is odd.
Explanation: By property 3 of square numbers, the squares of even numbers are even numbers and that of odd numbers are odd numbers.
Property 4:- If a number has a square root in N, then its units digit must be 0, 1, 4, 5 or 9.
Explanation: By property 6, the units digits of the square and square root are related as below
Units digit of square: |
0 |
1 |
4 |
5 |
6 |
9 |
Units digit of square root: |
0 |
1 or 9 |
2 or 8 |
5 |
4 or 6 |
3 or 7 |
Property 5:- Negative numbers have no square root in the system of rational numbers.
Explanation: We have, \( { 2 }^{ 2 } \) =4, \( { 3 }^{ 2 } \) =9, \( { 4 }^{ 2 } \) =16 and so on. Also, \( { (-2) }^{ 2 } \) = (—2) x (—2) =4, \( { (-3) }^{ 2 } \)= (—3)x(—3) = 9, \( { (-4) }^{ 2 } \) = (—4)x(—4) = 16 and so on.
This means that the square of a number whether positive or negative is always positive. Consequently, negative numbers are not perfect squares.
Hence, negative numbers have no square roots.
Property 6:- The sum of first n odd natural numbers is \( { n }^{ 2 } \) i.e.,
1+3+5+7+…+(2n—1)= \( { n }^{ 2 } \).
Some Short-Cuts To Find Squares:
In order to find the square of a number, we multiply the given number by itself. The multiplication is convenient for small numbers only. For large numbers, multiplication may be laborious and time-consuming. In this section, Some short methods for finding the squares of natural numbers without using actual multiplication is shown here.
Column Method :
This method is based upon an old Indian method of multiplying two numbers. It is convenient for finding squares of two digit numbers only. As the number of digits increases, this method becomes difficult and time-consuming.
This method uses the identity \( { (a+b) }^{ 2 }\quad =\quad { a }^{ 2 } \) = 2ab + \(\quad { b }^{ 2 } \) for finding the square of a two digit number ab (where a is the tens digit and b is the units digit). We follow the following steps to find the square of a two digit number ab (where a is the tens digit and b is the units digit).
Step I– Make three columns and write the value of \(\quad { a }^{ 2 } \), 2 x a x b and \(\quad { b }^{ 2 } \) respectively in these columns as follows:
As an illustration let us take ab = 57.
\( \therefore \) a=5 and b=7.
Step II– Underline the units digit of b2 (in column III) and add its tens digit, 2xaxb (in column II).
if any, to 2xaxb (in column II).
Step III- Underline the units digit in column II and add the number formed by tens and other digit, if any, to \(\quad { a }^{ 2 } \) in column I.
Step IV- Under the number in column I.
Step V- Write the underlined digits at the bottom of each column to obtain the square of the given number.
In this case, we have \(\quad { 57 }^{ 2 } \) = 3249.
Illustrative Examples :
Example : Find the squares of the following numbers usii column method:
(i) 25 (ii) 96
Solution : (i) Here, a= 2 and b = 5.
We have,
\( \therefore \) \(\quad { 25 }^{ 2 } \) = 625.
(ii) Here, a=9 and b=6.
We have,
\( \therefore \) \(\quad { 96 }^{ 2 } \) = 9216.
Example 2: Find the squares of the following numbers using column method:
(i) 99 (ii) 89
Solution: (i) Here, a=9 and b=9.
We have,
\( \therefore \) \(\quad { 99 }^{ 2 } \) = 9801.
(ii) Here, a=8 and b=9
We have,
\( \therefore \) \(\quad { 89 }^{ 2 } \) = 7921.
Visual Method :
In the column method,the algebraic identity \( { (a+b) }^{ 2 } \) = \( { a }^{ 2 } \)+ 2ab + \( { b }^{ 2 } \) is used to compute the square of a two digit number. The square of a positive integer can also be computed by closely following the visual representation of \( { (a+b) }^{ 2 } \). In order to represent \( { (a+b) }^{ 2 } \), we draw a square of side a + b and divide it into two rectangles of size a x(a + b) and b x (a + b) by drawing a vertical line as shown in Fig. 1 We also draw a horizontal line to divide the square into two rectangles of size (a + b) x b and (a + b) x a as shown in Fig. 3.1. These two lines divide the square into four parts, namely, two squares of size ax a and b x b and two rectangles of size ax b and b x a. The sum of the areas of these four parts is
a x a + a x b +b x a + b x b = \( { a }^{ 2 } \) + 2ab + \( { b }^{ 2 } \)
We use this visual representation of \( { (a+b) }^{ 2 } \) to find the square of a number.
Suppose we wish to find the square of 105.
We have, 105 = 100 + 5
So, we draw a square of side 105 units and divide it into four parts as shown in Fig. The sum of the areas of these four parts is the square of 105.
\( \therefore \) \(\quad { 89 }^{ 2 } \) = 10000+500+500+25 = 11025
Following examples will illustrate the above method.
Illustrative Examples :
Example 1: Find the square of the following numbers by Visual method:
(i) 54 (ii) 97
Solution: (i) We have, 54=50+4
So, we draw a square of side 54 units and divide it into parts as shown in Fig.
The sum of the areas of these four parts is the square of 54.
\( \therefore \) \(\quad { 54}^{ 2 } \) = 2500+200+200+16 = 2916
(ii) We have, 97 =90+7
So, we draw a square of side 97 units and divide it into parts as shown in Fig.
The sum of the areas of these four parts is the square of 97.
\( \therefore \) \(\quad { 97 }^{ 2 } \) =8100+630+630+49=9409
Example 2: Find the square of the following numbers by Visual method:
(i) 205 (ii) 315
Solution: (i) We have, 205 = 200 + 5
So, let us draw a square of side 205 units and divide it into 4 parts as shown in Fig.
\( \therefore \) \(\quad { 205 }^{ 2 } \) = 40000+1000+1000+25 = 41025
(ii) We have, 315 =300+15
So, let us draw a square of side 315 units and divide it into parts as shown in Fig.
The sum of the areas of these four parts is the square of 315.
\( \therefore \) \(\quad { 315 }^{ 2 } \) = 90000+4500+4500+225=99225.
Diagonal Method :
This method is applicable to find the square of any number irrespective of the number of digits in the number. The following steps are to be followed to find the square of a number by this method.
Step I– Obtain the number and count the number of digits in it. Let there be n digits in the number to be squared.
Step II– Draw square and divide it into \(\quad { n }^{ 2 } \) sub-squares of the same size by drawing (n —1) horizontal and (n —1) vertical lines.
Step III– Draw the diagonals of each sub-square. As an illustration, let the number to be squared be 479.
Step IV– Write the digits of the number to be squared along left vertical side and top horizontal side of the squares as shown below.
Step V– Multiply each digit on the left of the square with each digit on top of the column one-by-one. Write the units digit of the product below the diagonal and tens digit above the diagonal of the corresponding sub-square a shown below.
Step VI- Starting below the lowest diagonal, sum the digits along the diagonals so obtained. Write the units digit of the sum and take carry, the tens digit (if any) to the diagonal above as shown below.
Step VII- Obtain the required square by writing the digits from the left-most side.
\( \therefore \) \(\quad { 479 }^{ 2 } \) = 229441
Illustrative Examples :
Example 1: Find the squares of the following using diagonal method:
(i) 89 (ii) 68
Solution (i) Using diagonal method, we have (ii) Using diagonal method, we have
Example 2: Find, the square of the following numbers by diagonal method:
(i) 349 (ii) 293
Solution: (i) Using diagonal method, we have
\( \therefore \) \(\quad { 349 }^{ 2 } \) = 121801
(ii) Using diagonal method, we have
\( \therefore \) \(\quad { 293 }^{ 2 } \) = 85849