Squares and Square Roots-Notes
Notes For Squares and Square Roots
4.1 Square of a number.
If a natural number m can be expressed as n2 (where n is a natural number), then m is the square root or perfect square.
i.e. if m = n2 (m, n – natural numbers)
E.g. 81 = 3 × 3 × 3 × 3
= 32 × 32 = (3 × 3)2 = 92
Hence, 9 is the square root of 81.
4.2 Properties of Square Root.
Below is the table that has squares of numbers from 1 to 10.
If we see the above results carefully, we can conclude that numbers ending with 0, 1, 4, 5, 6, or 9 at units place are perfect squares None of these end with 2, 3, 7 or 8, So numbers that end with 2, 3, 7, 8 are not perfect squares.
Thus, numbers like 122, 457, 183, 928 are not perfect squares.
4.3 One’s digit in square of a number.
(1). The ones digit in the square of number can be determined if the ones digit of the number is known.
(2). The number of zeros at the end of a perfect square is always even and double the number of zeros at the end of the number
E.g.
Double zero {7002 = 490000} four zero (even)
(3). The square of an even number is always an even number and square of an odd number is always an odd number.
E.g.
4.4 Interesting patterns of Square Root.
Number between square numbers
There are ‘2a’ non perfect square numbers between the square of two Consecutive natural numbers n + (n + 1)
Between 22 = 4 & 32 = 9 → 5, 6, 7, 8
2 × 2 = 4 non square numbers
Between
32 = 9 & 42 = 16 → 10, 11, 12, 13, 14, 15
2 × 3 = 6 non square numbers.
Adding Consecutive odd numbers.
So, we can conclude that the sum of first in odd natural numbers in n2 or, we can say if the number is a square number, it has to be the sum of successive odd numbers.
E.g. 36
Successively substract 1, 3, 5, 7 ... from 36
36 – 1 = 35
35 – 3 = 32
32 – 5 = 27
27 – 7 = 20
20 – 9 = 11
11 – 11 = 0
∴ 36 is a perfect square
4.5 Short Cut Method of Squaring a number:
We can use the below to methods for Calculating square of any natural number
(a + b)2 = a2 + b2 + 2ab
(a � b)2 = a2 + b2 � 2ab
The logic behind using these formulae is to convert square of unknow number into square of a known number.
E.g.
(53)2 = (50 + 3)2
= 502 + 32 + 2.50.3
= 250 + 9 + 300
= 2809
992 = (100 – 1)2
= 1002 + 12 – 2.100.1
= 10000 + 1 – 200
= 9801
4.6 Square Roots
Finding the Square root is inverse operation of squaring.
If 9 × 9 = 81 i.e, 92 = 81
Then the square root of 81 = 9
In other words, if p = q2, the q is called the square root of p
The Square root of a number is the number which when multiplied with itself gives the number as the product. The square root of a number is denoted by the symbol 
Some Properties of square root
1. The Square root of an even perfect square is even and that of an odd Perfect square is odd.
2. Since there is no number whose square is negative the square root of a negative number is not defined.
3. If a number ends with an odd number of zeroes, then it cannot have a square root which is a natural number.
4. If the units digit of a number is 2, 3, 7 or 8 then square root of that number (in natural numbers) is not possible.
5. If m is not a perfect square, then there is no integer n such that square root of m is n.
Finding square root through repeated subtraction Recall the pattern formed while adding Consecutive odd numbers.
1 + 3 + 5 + 7 + 9 = 52 = 25
1 + 3 + 5 + 7 + 9 + 11 = 62 = 36
Sum of first n odd numbers = n2
The above Pattern can be used to find the square root of the given number.
1. Obtain the given perfect square whose square root is to the calculated let the number be a.
2. Subtract from it successively 1, 3, 5, 7, 9 till you get zero.
3. Count the number of times the subtraction is performed to arrive at zero let the number be n.
4. Write 
Finding square root through Prime factorization. In order to find the square root of a perfect square by prime factorization.
We follow the following steps.
1. Obtain the given number
2. Resolve the given number into prime factors by successive division.
3. Make pairs of prime factors such that both the factors in each pair are equal.
4. Take one factor from each pair and find their product.
5. The product obtained is the required square root.
E.g.
Find the square root of 400
Finding the square root by division Method when the numbers are large even the method of finding square root by prime factorisation becomes slightly difficult. To over come this problem we use long division method.
(1). Consider the following steps to find the square root of 784
Step 1: Place a bar over every pair of digits starting from the digit at one�s place. If the number of digits is odd, then the left most single digit too will have a bar. Thus we have 
Step 2: Find the largest number whose square is less than or equal to the number under the extreme left bar (22 < 7 < 32). Take this number as the divisor and the quotient with the number under the extreme left bar as the dividend (here 7) Divide and get the remainder (5 in this case)
Step 3: Bring down the number under the next bar (i.e, 84 in this case) to the right of the remainder. So the new divided is 384.
Step 4: Double the divisor and enter it with a blank on its right
Step 5: Gives a largest possible digit to fill the blank which will also become the new digit in the quotient, such that when the new divisor is multiplied to the new quotient the product is less than or equal to the dividend.
In this case 47 × 7 = 329
As 48 × 8 = 384 so we choose
the new digit as 8. Get the remainder
Step 6: Since the remainder is 0 and no digits are left in the given number, therefore 