Value of Root 2 – How to calculate it

In this article, we will talk about the concept of root and square root. 

To understand the concept of root properly, we must be aware of the concept of power. 

Consider any number ‘p’ and any natural number ‘n’, then

pⁿ = p x p x p …… (multiplied ‘n’ number of times)

Here, ‘p’ is the base, and ‘n’ is the exponent, generally called ‘p raised to the power n.’

for instance, 2² = 2 x 2 = 4

Therefore, 2 raised to the power 2 is equal to 4.

What do you mean by the word ‘root’?

The n^th root of a numeric value ‘x’ is equal to another numeric value that, when multiplied by itself ‘n’ number of times, equals the numeric value ‘x’ for which we were finding the root. 

Furthermore, 

• The second root is also called the square root of the number. 
• The third root is called the cubic root of the number.

What is the value of root 2?

Owing to the nature of the number, root 2 is an irrational number, and we cannot find the value of root 2 in fraction form. Therefore, we find an approximate value of \sqrt2 in decimals, i.e., \sqrt2 =1.414. At present, mathematicians have derived the value of \sqrt2 up to 1 trillion decimal places and the digits keep increasing. There is no defined value of \sqrt2 thereby making it an irrational number. The value of root 2 can be positive or negative. Here, we are considering its positive value in the article.

But, how do we conclude that \sqrt2 =1.414?

How to find the value of root 2?

To determine a number’s square root value, we generally verify whether it is a perfect square or not. The value of perfect squares is easy to locate, but we have to apply the long division method to get a root value for non-perfect squares. 

For example, a number like 2, 3, 5, 20, etc. is not a perfect square; however, 4, 9, 25 etc., are perfect squares, giving a whole number when we calculate their root.

Finding root 2 value by long division method

Step 1: Put a bar on every pair of numerals starting from right to left. When we have an odd number of digits, the extreme left single digit has a bar.

Step 2: Find out the largest number whose square is less than or equal to the extreme left digit.

Step 3: Take the number as the new divisor and quotient, 2 as the dividend. Then divide with the next remaining number below the extreme left. Next, we’ll place a decimal point and a pair of zeros next to it and continue our division.

Step 4: Put the number under the next bar to the right of the remainder.

Step 5: Double the quotient and place it with a blank on its right-hand side.

Step 6: Keep repeating steps 2, 3, 4 and 5 until the remainder is 0 or repetitive.

Step 7: The quotient you thus obtain will be the value of the square root by the long division method.

We can also calculate the value of root 2 up to 100 decimal places by this method, i.e., 1.4142135623730950488016887242096980785696718753769480731766797379907324784621070388503875343276415727

But for general calculations, we consider 1.414 as the value of root 2.

Value of root 2 in fraction form

We can find the rational approximation of an irrational number by the expansion of its continued fraction. Therefore, the continued fraction for \sqrt2 is given by:

Examples

Example 1: Which types of numbers have perfect square roots?

Solution: A perfect square such as 4, 9, 16, 25, etc., has exact square roots and gives a whole number when calculating the value of their root.

Example 2: Calculate the length of the diagonal of a square sheet if its sides are 4 cm each.

Solution: Side of the square sheet ‘ a’ = 4 cm.

By applying Pythagoras theorem, the diagonal of the square

=√2 x a

=√2 x 4

= 5.656 cm

Therefore, the diagonal of the square sheet is 5.656 cm.