History of Square Root
The origin of the square root and square root symbol is largely speculative. The first clue of using square roots was seen in Babylonia. From 1800 BCE-1600 BCE they found the square root of 2 up to 5 decimal places as, \( \sqrt{2}=1.41421 \). Later the ancient Indians, Egyptians, Greeks and Chinese also used square roots.
The origin of the square root symbol is also doubtful. Some theories say that the square root symbol originated from the Arabic letter ‘ج’. Some scholars believe that the square root symbol came from the first letter ‘r’ of the Latin word ‘radix’.
What is Square Root?
Definition: The square root of a number p is a number whose square is equal to p. So, a square root of a number p is a solution of the equation. Every positive number has a positive and negative square root.
We write the equation
Then x is called a root or radical because it is like the hidden base of a.
The square root of a number is that factor of a number which when multiplied by itself gives the original number. The square root of a number is the reverse of the square of a number. The square root symbol is also called the radical symbol. The number inside this symbol is called radicand. A perfect square number is a number whose square root is an integer. For every positive real number there exists two square roots. The square roots are numerically equal but opposite in sign. One of the roots is positive and the other is negative.
For example, when 5 is multiplied with 5 gives the product 25. Therefore, square root of 25 is given as,
If the square root of any number is a whole number then that number is called the perfect square number. 4, 9, 16, 25 etc. are perfect square numbers.
Square Root Formula
Square root formula:
If a number y is square root of x then the square root formula is
Where ‘⟺’ denotes ‘if and only if’.
How to find the Square Root of a Number?
There are several methods to find the square root of a number. The method of prime factorization and method of approximation are commonly used methods for finding square roots.
Find the Square Root of a Number by Approximation of the Prime Number Square Roots
This method is majorly used to find the square root of any number. The steps to find the square root by approximation are as follows-
Step 1: First, factorize the number in such a way that each number should be a perfect square or multiple of the prime number and perfect square.
Step 2: Find the square roots of the perfect square numbers. If there is no non-perfect square number inside the radical symbol, then multiply the obtained numbers. The product is the square root of the original number.
Step 3: If there are non-perfect square numbers or prime numbers then find the approximate calculated square root from the table. Multiply the result to the square root of perfect squares numbers.
Example 1: Find a square root of 45.
\( \sqrt 45 \)
\( = \sqrt 9\times 5 \) [Factorise]
\( = 3\times \sqrt 5 \) [Square root of 9 is 3]
\( = 3\times 2.24 \) [Replace 5 with 2.24]
\( = 6.72 \) [Simplify]
Square root of 45 is 6.72.
Square Root by Trial and Error method
This method is based on approximation and estimation. We find the square root of the number guessing the values. This is a time consuming and long method.
For example, to find the square root of 8 as,
8 is between two perfect squares, 4 and 9.
Since \(2^2=4\) and \(3^2=9\). The square root of 8 lies between 2 and 3.
Then reduce 3 and increase 2 by 0.2 (guess a small number)
So, \(2.2^2=4.84\) and \(2.8^2=7.84\)
The value 7.84 is very near to 8. Therefore, we estimate square root of 8 as,
\(\sqrt 8 \approx 2.8\)
Solved Examples of Square Root
Example 1: Find the square root of 900.
Solution:
\(\sqrt {900}\)
\( =\sqrt {2\times 2\times 3\times 3\times 5\times 5} \)
\(=\sqrt {(2\times2)\times(3\times3)\times(5\times5)}\)
\( = 2\times 3\times 5 \)
\( = 30 \)
The square root of 900 is 30.
Example 2: Find the square root of \( \frac{16}{25} \).
Solution:
\(\sqrt {\frac{16}{25}}\)
\(=\sqrt {\frac{2\times2\times2\times2}{5\times5}}\)
\(=\sqrt{\frac{(2\times 2)\times(2\times 2)}{(5\times 5)}}\)
\(=\frac{2\ \times\ 2}{5}\)
\(=\frac{4}{5}\)
The square root of \(\sqrt {\frac{16}{25}}\) is \(\frac{4}{5}\).
Example 3: Find the square root of \(\frac{100}{169}\).
Solution:
\(\frac{100}{169}\)
\(=\sqrt{\frac{2\times2\times5\times5}{13\times13}}\)
\(=\sqrt{\frac{(2\times2)\times(5\times5)}{(13\times13)}}\)
\(=\frac{2 \times\ 5}{13}\)
\(=\frac{10}{13}\)
The square root of \(\frac{100}{169}\) is \(=\frac{10}{13}\).
Example 4: If there are 144 plants in a square garden. The plants are in such a way that there are the same number of plants in row and column. Find the number of plants in each row.
Solution:
If the plants are arranged in square or rectangular form. Let the number of plants in each row is . The number of rows and number of columns are the same. So, there will be \(n \times n=n^2\) plants in total. But the given number of plants is 144.
Therefore, the equation can be obtained as,
\(n^2=144\)
\(n=\pm\sqrt {144}\) [Take square root each side]
\(n=\pm 12\) [Simplify]
The solution of the equation is +12 and -12. The number of plants cannot be negative. Therefore, the number of plants in each row is 12.
The square root of a negative number is imaginary. So using complex numbers we can find the square root of a negative number.
No, the square root of numbers may and may not be rational. For example, square root of 2 i.e. \(\sqrt 2\) is not rational. However, the square root of perfect square numbers are rational. For example, square root of 9 i.e. \(\sqrt 9=\pm 3\) is a rational number.
Yes, the square root of decimal numbers can be found by converting them to fractions. First convert the decimals to fractions. Then take the square roots of both the numerator and denominator. The square root is then obtained in the form of a fraction that can be converted in decimals later on.