A number is a mathematical value used for counting and measuring objects, and for performing arithmetic calculations. It is a system of writing for expressing numbers. It gives a different representation to every number and represents the arithmetic and algebraic structure of the integer. In the number system, operating arithmetic operations like addition, subtraction, multiplication, and division also take place.
In everyday situations, people are facing problems in calculating the square root of a number. What if one doesn’t have a calculator or mobile phone? It can be done by using paper and pencil in a long division style. Yes, there are a variety of ways to do so. Let’s first discuss what is square root is and its properties.
Square Root
The square root is a value, which on multiplication by itself gives the original number. For example, the square of 5 is 25, 5² = 25, and the square root of 25, √25 = 5. The original number is obtained from the square root of the square of a positive number.
How to represent the square root, Suppose, c is the square root of d, then it is represented as,
c = √d
c² = d
Let the square of 4 is 16 so the square root of 16 will be 4 i.e.
√16 = 4
The following are the square roots of the first 10 digits,
| Square root | Value |
|
√1 |
1 |
|
√2 |
1.4142 |
|
√3 |
1.7320 |
|
√4 |
2 |
|
√5 |
2.2361 |
|
√6 |
2.4495 |
|
√7 |
2.6458 |
|
√8 |
2.8284 |
|
√9 |
3 |
|
√10 |
3.1622 |
Hence, the square root of the square of a positive number gives the original number. However, the square root of a negative number gives a complex number.
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 root in N(the set of natural numbers). Example: 122, 253, 788 does not have perfect square roots as unit digit are 2, 3, and 8 respectively.
- Property 2: At the end of a number if there is an odd number of zeroes, then it does not have a square root. If a square number is followed by an even number of zeroes, it has a square root in which the number of zeroes, in the end, is half the number of zeroes in the number. Example: 4000 does not have a perfect square root as the number of zeroes is 3 (odd). 400 have a perfect square root as the number of zeroes is 2 (even). So the square root of 400 will contain only 1 zero. (half of two zeroes). √400 = 20.
- Property 3: Even square root is obtained by an even square number and we get odd square root by an odd square number. Example: √4 = 2 (both are even numbers) and √9 = 3 (both are odd numbers).
- Property 4: If a number has N in a square root, then its unit digit must be 0, 1, 4, 5, or 9. Example: Unit digit of √1024 is 2 as unit digit of 1024 is 4 and its square root is 2.
- Property 5: In the system of rational numbers negative numbers have no square root. Example: √(-9) is not a rational number. It will be a complex number.
- Property 6: The sum of the first n odd number is n². Example: 1 + 3 + 5 = 9 = 3²
Methods to find the Square Root of a number
To know if a given number is a perfect square or an imperfect square, we must first check out if it is a perfect square or an imperfect square. If it is a perfect square, such as 4, 9, 16, etc., Use the prime factorization process to factorize it, if it is an incomplete square use the long division method to find the root, such as 2, 3, 5, and so on.
- Repeated Subtraction Method
- Prime Factorization Method
- Division Method
Repeated Subtraction Method
The sum of the first n odd natural numbers is known to be n2. Do this to calculate the square root of a number by subtracting it several times. Let’s check out an example and see how this works. Let’s find the square root of 25, which is √25. Let’s consider the following examples to understand the repeated subtraction method to determine the square roots,
Example: Determine the square root of 16 using the repeated subtraction method.
Solution:
Find the square root of 16 as 16 is an even number. Therefore, the steps to find the square root of 16,
16 – 2 = 14
14 – 4 = 10
10 – 6 = 4
4 – 4 = 0
Here it takes four steps to get the 0.
Therefore, the square root of 16 is 4.
Prime Factorization Method
Prime factorization means expressing numbers as a function of their prime factors. Prime factorization is defined as a way of finding the prime factors of a number, such that the original number is evenly divisible by these factors
Example: What is the prime factor of 420?
Solution:
The prime factor of 420 will be 2, 3, 5 and 7 as
2 × 2 × 3 × 5 × 7 = 420 and 2, 3, 5 and 7 are prime n numbers.
Division Method
When the numbers are large, use the long division method to obtain the square root of a perfect square, because calculating square roots through factorization becomes difficult and complicated. To overcome this problem, a new method is developed for finding the square root. In this method divisor uses the division operation whose square is either less than or equal to the dividend.
Following are the steps to for division method
- Step 1: Take a number to find the square root. Place a bar covering every pair of the digit of the number starting from the rightmost side.
- Step 2: Now divide the leftmost number by the largest number whose square is equal to the number or it is less than the number under the leftmost bar. Now take this number as the divisor and the quotient. The dividend is the number under the leftmost bar.
- Step 3: Divide and get the number. Now bring down the next dividend under the next bar to the right of the remainder to complete the method.
- Step 4: Now add the divisor to itself (or double the divisor). Form a new divisor by finding a suitable number to the right of this divisor which together forms a new divisor for the new dividend. The new number which is in the quotient has the same number as selected in the divisor. The state is the same as being either less or equal to that of the dividend.
-
Step 5: Until the remainder as 0 is obtained, continue this process. The square root of the number is the quotient obtained.
What are the 2 square roots of 400?
Solution:
Square roots of a number are the numbers that when multiplied by itself gives the initial number.
Example: b is said to be the square root of a number c if
b × b = c
Square roots of 400 are 20 and -20
Since 20 × 20 = 400
And (-20) × (-20) = 400
Sum: 20 + (-20) = 0
Product: 20 × (-20) = -400
Similar Problems
Question 1: Find the square roots of 400?
Solution:
Two square roots of 400 are 20 and -20
Since 20 × 20 = 400
And (-20) × (-20) = 400
Sum: 20 + (-20) = 0
Product: 20 × (-20) = -400
Question 2: Find the square roots for 900?
Solution:
Two square root of 900 are 30 and -30
Since 30 × 30 = 900
And (-30) × (-30) = 900
Sum: 30 + (-30) = 0
Product: 30 × (-30) = -900