The square root of any positive integer can be found using various methods. A few notable once are :

Average method

Number Line method

Long division method

A sequence of steps are to be followed to determine the square root of a positive integer using the average method which. To determine ‘What is the value of root 5?’ Using the average method, the steps to be followed are summarized below.

Step 1:

Find Out the Two Perfect Square Numbers Which are Very Close to the Given Number on Either Side.

For the number ‘5’, the immediate perfect square lesser than 5 is ‘4’ and the immediate perfect square greater than 5 is ‘9’.

Step 2:

Note Down the Square Roots of the Perfect Squares Identified in Step 1.

Square root of ‘4’ is ‘2’ and the square root of ‘9’ is ‘3’.

Step 3:

It Can be Inferred that the Square Root of a Given Number lies Between the Square Roots of Numbers Determined in Step 2.

Square root of ‘5’ is any number between 2 and 3.

Step 4: Divide the Number Whose Square Root is Determined by Any of the Numbers Obtained in Step 2.

‘5’ can be divided either by ‘2’ or ‘3’.

Let us divide ‘5’ by ‘2’

\[\frac{5}{2}\] = 2.5

Step 5: Find the Average of the Quotient and Divisor in Step 4.

The average of 2 and 2.5 is to be found

Average = \[\frac{2 + 2.5}{2}\] = \[\frac{4.5}{2}\]

Step 6: To Get an Accurate Value of the Square Root, an Average of the Answer in Step 5 and Divisor of Step 4 Can be Found. Finding of Average Can be Continued as Many Times as Required to Get a Precise Value.

The average of 2 and 2.25 is to be found if the answer in step 5 for the value of under root 5 is not precise.

However the answer in step 5 is almost the same as the value of under root 5 calculated using the calculator (2.2361).

(\[\sqrt{5}\])=2.25 by average method.

To find the square root of any number using a number line, a clear understanding of the basic concept of “Pythagorean Theorem” is required.

Pythagorean theorem states that “In a right triangle, the square on the longest side (hypotenuse) is equal to the sum of the squares on the other two sides (base and perpendicular)”.

A series of steps to be followed to determine the square root of any number using a number line is explained below.

Step 1:

Construct a number line with a minimum number of units equal to the square root of the perfect square which is at the immediate right of the number whose square root is to be determined.

The immediate perfect square greater than 5 is ‘9’ whose square root is 3. So construct a number line with a minimum of 3 units towards the right and left of the reference point (zero) labeled as ‘O’.

Step 2:

Represent the number whose perfect square is to be determined as the sum of two perfect square numbers. Also determine the square roots of the two numbers.

The number ‘5’ can be expressed as the sum of two perfect square numbers 4 and 1.

Square root of ‘4’ is ‘2’ and the square root of ‘1’ is ‘1’.

Step 3:

From the point representing one of the answers in Step 2, draw a perpendicular of length measuring the other answer to the number line.

Label the point representing ‘2’ as ‘A’. From the point ‘A’, draw a perpendicular to the number line (AB) Length of the perpendicular should be 1 unit.

Step 4:

Join the tip of the perpendicular to the reference point (zero). The length of this hypotenuse given the square root of the number.

Join OB. The measure of OB gives the value of root 5.

The length of this hypotenuse measures around 2.3 to 2.4 which is the value of under root 5.

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Long division method is the most convenient and easiest method to find the square root of any number. It gives an accurate value of any number. For a better understanding of ‘How to find square root of 5 by Long Division method?’

Step 1:

‘5’ is represented as 5.00000000 to determine the square root of 5 upto 4 decimal places. Group the digits after the decimal point in pairs as 5. 00 00 00 00

Step 2:

Choose a perfect square whose value is below ‘5’. The perfect square below 5 is ‘4’ and its square root is ‘2’.

Step 3:

Represent that ‘2 times 2 gives 4’ by writing 2 in the place of dividend and quotient. The number ‘4’ is written below ‘5’. ‘4’ is subtracted from ‘5’ and the difference is 1.

Step 4:

The first pair of zeroes in the dividend is carried down and a decimal point is placed in the quotient.

Step 5:

Add 2 to the divisor. The sum will be 4. Now take a number succeeding 4 to get a 2 digit number such that when the two digit number is multiplied by the number taken, the product is less than 100. If we take the digit as ‘2’, 42 x 2 = 84 which is less than 100. So now the remainder is ‘16’ and the next divisor is 44_. And the dividend is 1600 if the next pair of zeroes are taken down.

Step 6:

If the above steps are continued, the final answer obtained in the place of quotient is 2.2360.

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Square roots of negative numbers are not real numbers. They are imaginary numbers. Hence, the knowledge of complex numbers is required for understanding their square roots.

It is believed that the symbol for square root is derived from the first letter of the word ‘radix’ in Greek and Latin which means a ‘root’ or ‘base’.

FAQ (Frequently Asked Questions)

1. Where is the Concept of Square Roots Used in Real Life?

Square roots are used in all fields of mathematics. A few notable ones are listed below.

Square roots are used in finding the sides of a square of a given area.

It is used to calculate the value of diagonals of squares, rectangles and other parallelograms.

It is the most important concept of Pythagoras theorem to find the sides of a right triangle.

In statistical calculations, the understanding of finding the square root of 5 by long division method and other numbers is used to find standard deviation when the variance of a group of data is given.

Square roots are also used to find the solution of quadratic equations.

2. Is ‘5’ a Perfect Square Number?

A perfect square number is that number obtained by multiplying any integer by itself. So, if an integer ‘a’ is multiplied by ‘a’ to get the product as ‘b’, then the number ‘b’ is said to be a perfect square number. ‘5’ is not a perfect square number because any integer in mathematics cannot be multiplied by itself to give the product as ‘5’. The perfect square number preceding ‘5’ is ‘4’. Square root of 4 is 2 (i.e. 2 times 2 is equal to 4) and the perfect square number succeeding ‘5’ is ‘9’. ‘3’ times ‘3’ is ‘9’. So, the square root of 9 is 3. From these approximations, it can be inferred that the value of the square root of 5 lies in between 2 and 3.