Square Root of 2

The square root of 2 is ±1.41421356237, where is 1.41421356237 the positive solution of the equation x² = 2.  

Finding the square root is just the inverse of squaring a number and hence, squaring 1.41421356237 will result in 2.

The square root of 2 is written as √2 in radical form, where the ‘√’  sign is called the “radical”  sign. In exponential form, it is written as (2)^1/2 .

We can find the square root of 2 through various methods. They are:
i) Prime factorization method
ii) Long division method
iii) Approximation/Estimation method
 

The prime factorization of 2 is done by dividing 2 by prime numbers and continuing to divide the quotients until they can’t be separated anymore.

After factorizing 2, make pairs out of the factors to get the square root. If there exist numbers that cannot be made pairs further, we place those numbers with a “radical” sign along with the obtained pairs.

So, Prime factorization of 2 = 2 × 1   
But here in the case of 2, no pairs of factors can be obtained but a single 2 is remaining

So, it can be expressed as  √2 
√2 is the simplest radical form.

This method is used for obtaining the square root for non-perfect squares, mainly. It usually involves the division of the dividend by the divisor, getting a quotient and a remainder too sometimes.

Follow the steps to calculate the square root of 2:

 Step 1: Write the number 2, and draw a bar above the pair of digits from right to left.

               
Step 2: Now, find the greatest number whose square is less than or equal to. Here, it is
               1, Because 1²=1 < 2.

Step 3 : Now divide 2 by 1 (the number we got from Step 2) such that we get 1 as quotient
             and we get a remainder. 

Double the divisor 1, we get 2, and then the largest possible number A1=4 is chosen such that when 4 is written beside the new divisor, 2, a 2-digit number is formed →24, and multiplying 4 with 24 gives 96 which is less than 100.

 Repeat the process until you reach the remainder of 0. We are left with the remainder, 3836 (refer to the picture), after some iterations and keeping the division till here, at this point.

             
Step 4 : The quotient obtained is the square root. In this case, it is 1.4142….

Approximation or estimation of the square root is not the exact square root, but it is an estimate.

Here, through this method, an approximate value of square root is found by guessing.

Follow the steps below:

Step 1:  identify the square roots of the perfect squares above and below 2
             Below : 1→ square root of 1 = 1     ……..(i)
             Above : 4 →square root of 4 = 2     ……..(ii)

Step 2: Dividing 2 with one of 1 or 2      
             If we choose 1 
            We get 2 when 2 is divided by 1    …….(iii)

             
Step 3: Find the average of 1 (from (i)) and 2 (from (iii))
            (1+2)/2 = 1.5 

         
 Hence, 1.5 is the approximate square root of 2

1)Exponential form

An algebraic expression that includes an exponent. It is a way of expressing the numbers raised to some power of their factors. It includes continuous multiplication involving base and exponent.
Ex: 2 ⤬ 2 ⤬ 2 ⤬ 2 = 16
Or, 2 4 = 16, where 2 is the base, 4 is the exponent. 

2)Factorization   
Expressing the given expression as a product of its factors
Ex: 48=2 ⤬ 2 ⤬ 2 ⤬ 2 ⤬ 3

3) Prime Numbers 
Numbers which are greater than 1, having only 2 factors as →1 and Itself. Ex: 1,3,5,7,...

4)  Rational numbers and Irrational numbers

The Number which can be expressed as p/q, where p and q are integers and q not equal to 0 are called Rational numbers. Numbers which cannot be expressed as p/q, where p and q are integers and q not equal to 0 are called Irrational numbers. 

5) Perfect and non-perfect square numbers
Perfect square numbers are those numbers whose square roots do not include decimal places. Ex: 4,9,25 Non-perfect square numbers are those numbers whose square roots comprise decimal places. Ex :3, 8, 24.