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To find square root through repetitive subtraction, consecutive odd numbers are subtracted from a number till the output is 0. Square Root = Total Numbers Subtracted. A number is said to be squared when it is raised to the power of 2. In other words, when a number is multiplied by itself, the product is called its square. Most used squares of the first few natural numbers can be remembered easily. Series of numbers including 1,4,9,16,25 . . . is called a series of perfect squares. However, every square number can break down into the sum of successive odd natural numbers starting from 1.
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Perfect Square
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A natural number that can essentially be expressed as the product of an integer by itself or the integer raised to its second power is called a perfect square. Perfect squares are also known as square numbers. 16 is a perfect square which is the same as 4 multiplied by 4 (4x4) or 4 to the power of 2 (42). Given below is a list of perfect squares corresponding to their square roots.
| No | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Square | 1 | 4 | 9 | 16 | 25 | 36 | 49 | 64 | 81 | 100 | 121 | 144 | 169 | 196 | 225 |
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Square Roots
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The square root of a number is another numerical value that when multiplied by itself will give the initial number. Consider a number 25. The square root of 25 is 5, because when 5 is multiplied by 5, we get 25. However, -5 is also the square root of 25 because -5 multiplied by itself gives 25.
Finding Square Root Through Repetitive Subtraction
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We can easily approach the square root of a perfect square by using the Repetitive Subtraction method. Let us consider a square number P which we require to find the square root. It is safe to say that the sum of the first P odd natural numbers will give us P2. Hence following this concept by subtraction of odd numbers starting from 1, and continuing till P becomes 0 we can reach a solution where the square root is equal to the number of odd numbers.
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Step Wise Repetitive Subtraction Method
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In the previous section, we learned about the concept used in the repetitive subtraction method. Let us not consider the number P = 81. According to the rules we require to start subtracting 1 from P.
Hence the first step is given below.
Step 1: 81-1=80
Now, from the result of the first step, we subtract the next odd number succeeding 1 i.e. 3.
Step 2: 80-3=77
We keep subtracting the successive odd number in the succeeding steps until we get the result 0.
Step 3: 77-5=72
Step 4: 72-7=65
Step 5: 65-9=56
Step 6: 56-11=45
Step 7: 45-13=32
Step 8: 32-15=17
Step 9: 17-17=0
This was the last step because we finally got the difference equal to 0. Now, it is time we count how many odd numbers had to be subtracted to finally reach at 0.
Since, 9 odd numbers were used, hence the square root of 81 is 9.
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Things to Remember
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- A natural number that can essentially be expressed as the product of an integer by itself or the integer raised to its second power is called a perfect square or a square number.
- Square root of a number is another numerical value that when multiplied by itself will give the initial number.
- The sum of the first P odd natural numbers is equal to P squared.
- Repetitive Subtraction of successive odd natural numbers from a given perfect square is performed starting from 1 till the final result is 0. The total number of odd numbers used in this method is equal to the square root of the number.
- Repetitive Subtraction Method is only valid for the perfect squares.
Check Also: Number System
Sample Questions
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Ques. What is a perfect square? Give one example. (1 Mark)
Ans. A perfect square of an integer is the value obtained by multiplying an integer by itself or raising it to the power of 2. For example, 25 is a perfect square which is 5 raised to the power of 2.
Ques. What do you mean by square root? Give one example. (1 Mark)
Ans. The numerical value which when multiplied by itself gives a number is called the square root of the number. For example, 4 and -4 are the square roots of the number 16.
Ques. If a number X is equal to the sum of the first 8 odd natural numbers, what is the value of X? (2 Marks)
Ans. As we have already studied in the earlier sections the sum of the first P odd natural numbers is equal to the square of P.
Here P = 8.
Hence, X = Sum of the first 8 odd natural numbers = Square of 8 = 8 x 8 = 64 (Answer).
Ques. Use the Repetitive Subtraction Method to find the square root of 100. (3 Marks)
Ans. To solve the problem let's start by subtracting 1 from the given number 100 and then successively subtracting 3,5,7. . . and so on until the final value is equal to 0.
Step 1: 100 - 1 = 99
Step 2: 99 - 3 = 96
Step 3: 96 - 5 = 91
Step 4: 91 - 7 = 84
Step 5: 84 - 9 = 75
Step 6: 75 - 11 = 64
Step 7: 64 - 13 = 51
Step 8: 51 - 15 = 36
Step 9: 36 - 17 = 19
Step 10: 19 - 19 = 0
Hence we needed 10 odd natural numbers in the process which means the square root of 100 is 10.
Ques. Find the square root of 121. (3 Marks)
Ans. To solve the problem let's start by subtracting 1 from the given number 121 and then successively subtracting 3,5,7. . . and so on until the final value is equal to 0.
Step1 : 121 – 1 = 120
Step 2: 120 – 3 = 117
Step 3: 117 – 5 = 112
Step 4: 112 – 7 = 105
Step 5: 105 – 9 = 96
Step 6: 96 – 11 = 85
Step 7: 85 – 13 = 72
Step 8: 72 – 15 = 57
Step 9: 57 – 17 = 40
Step 10: 40 – 19 = 21
Step 11: 21 – 21 = 0
Because 11 odd natural numbers were required to reach the final difference of 0, the square root of 121 is 11.
Ques. Find the square root of 36 using repetitive subtraction. (3 Marks)
Ans. Let's start by subtracting 1 from the given number 36 and then successively subtracting 3,5,7. . . and so on until the final value is equal to 0.
Step1 : 36 - 1 =35
Step 2: 35 - 3 = 32
Step 3: 32 - 5 =27
Step 4: 27 - 7 = 20
Step 5: 20 - 9 = 11
Step 6: 11 – 11 = 0
6 odd natural numbers were required to end the repetitive subtraction.
Hence, the square root of 36 is 6.
Ques. What is the sum of the first 9 odd natural numbers? (1 Mark)
Ans. The sum of the first P odd natural numbers is equal to P squared. Here P = 9.
Hence, the required sum is = 9 x 9 = 81.
Ques. Find the square root of 169. (3 Marks)
Ans. Starting the repetitive subtraction and following till we receive 0 as the final result here are the steps.
Step 1: 169 - 1 = 168
Step 2: 168 - 3 = 165
Step 3: 165 - 5 = 160
Step 4: 160 - 7 = 153
Step 5: 153 - 9 = 144
Step 6: 144 - 11 = 133
Step 7: 133 - 13 = 120
Step 8: 120 - 15 = 105
Step 9: 105 - 17 = 88
Step 10: 88 - 19 = 69
Step 11: 69 - 21 = 48
Step 12: 48 - 23 = 25
Step 13: 25 - 25 = 0
13 odd natural numbers were required. Hence, the square root of 169 is 13.
Ques. Can we find the square root of 21 through the repetitive subtraction method? (1 Mark)
Ans. 21 is not a perfect square. Hence, the repetitive subtraction method is not feasible to find the square root of 21.
Ques. Find the square root of 225. (3 Marks)
Ans. We can follow the steps of the repetitive subtraction method to find the square root of 225.
Step 1: 225 - 1 = 224
Step 2: 224 - 3 = 221
Step 3: 221 - 5 = 216
Step 4: 216 - 7 = 209
Step 5: 209 - 9 = 200
Step 6: 200 - 11 = 189
Step 7: 189 - 13 = 176
Step 8: 176 - 15 = 161
Step 9: 161 - 17 = 144
Step 10: 144 - 19 = 125
Step 11: 125 - 21 = 104
Step 12: 104 - 23 = 81
Step 13: 81 - 25 = 56
Step 14: 56 - 27= 29
Step 15: 29 - 29 = 0
Hence, 15 is the square root of 225 as 15 odd natural numbers were required to obtain the final difference of 0.
Ques. If N squared is equal to N. Find N. (1 Mark)
Ans. Given that the N2= N. We know that N2 = Sum of first N natural odd numbers. Which implies N= Sum of first N natural odd numbers. It can only be possible if N = 1. Hence, the value of N = 1.
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