Square Root of 2
The square root of 2 is written as √2.The value of √2 has an infinite number of decimals and it cannot be written as a fraction. In this chapter, we will calculate the square root of 2 by long division method along with solved examples. Let us see what the square root of 2 is.
 Square Root of 2: √2 = 1.4142
 Square of 2: 2^{2} = 4
What Is the Square Root of 2?
Square root is just an inverse operation of square. The square root of 2 is represented as √2. This is a number which when multiplied to itself gives us the result as 2. During ancient times, Greeks found a number that can never ever be written in the form of p/q, where p,q are integers and q is not equal to 0. This means √2 is not rational. √2 happens to be very useful in geometry. Let's say we have a square with side length 1 and you want to find the length of the diagonal.
To find the third side we use Pythagoras Theorem. The third side will be √2. Let's locate the number √2 on a number line. We will use transfer the square we used to discover √2. Let's name the vertices of the square as shown. Keep the vertex O at 0. We already discovered that OB=√2
Using a compass with centre O and radius OB, draw an arc intersecting the number line at the point P.
The point P corresponds to √2 on the number line.
Is the Square Root of 2 Rational or Irrational?
The actual value of √2. is undetermined. The value of √2 upto 25 decimal places is 1.4142135623730950488016887..
At present, the value of √2 is known up to 1 trillion decimal place.
Hence, √2 is irrational.
Important Notes:
 √2 is also called Pythagoras' constant.
 √2 represents the diagonal of a unit square.
 √2 was the first number to be discovered as an irrational number.
 Its decimal representation is nonterminating and nonrepeating.
 The ratio of longer edge and shorter edge of an A4 size sheet paper is equal to √2.
How to Find the Square Root of 2?
We can find the square root of 2 by the following two methods:
 Long Division Method
 Estimation and Approximation Method
Square Root of 2 by Long Division Method
The value of square root of 2 by long division method consists of following steps:
 Step 1: Find the largest number whose square is less than or equal to the number 2. Take this number as the divisor and the quotient, (1 in this case). Divide and write the remainder.
 Step 2: In the quotient, put a decimal point after 1. Bring down two zeroes to the right of the remainder. So, the new dividend is 100
 Step 3: Double the divisor and enter it with a blank on its right. Guess a largest possible digit to fill the blank which will also become the new digit in the quotient, such that when the new divisor is multiplied to the new quotient the product is less than or equal to the dividend. Divide and write the remainder. Repeat this process to get the decimal places you want.
Square Root of 2 by Estimation and Approximation Method
We can use the formula y=√x to find the value of √2.
This formula can be written as
((x/y)+y)/2
The iteration formula is
y_{n+1} = ((x/y_{n}) + y_{n})/2
The first three iterations give the result as shown below.First set y_{1}=1
 Iteration 1: y_{1} = (2 + 1)/2 = 1.5
 Iteration 2: y_{2} = (4/3 + 3/2)/2 = 1.4166
 Iteration 3: y_{3 }= (24/17 + 17/12)/2 = 1.414215...
Have you noticed that it has somewhat starting to shape into √2 =1.41421356237309 ?
Explore Square roots using illustrations and interactive examples
Think Tank:
 Can you think of any quadratic equation which has a root as √2?
 Since (√2)^{2}=2, can we say that √2 is also a square root of 2?
Square Root of 2 Solved Examples

Example 1: Find the length of diagonal of a square made up of 4 unit squares.
Solution
We know that the length of diagonal of 1 unit square is √2 units. To find the diagonal, we need to consider the length of diagonal of 2 unit squares.
Diagonal of 1 unit square = √2 units
Sum of Diagonal of 2 unit squares = 2√2 units
Hence, length of diagonal is 2√2 units. 
Example 2: What would be the length of the diagonal of a square shape cake, if each side is of 2 units? (Write your answer in decimal form upto 3 decimal places)
Solution
Given, Side of the Square Cake = 2 units
Using Pythagoras Theorem,
Diagonal of a square =√2a
Diagonal = √2 × 2 = 2.828 units
FAQs On Square Root of 2
Is the square root of 2 infinite?
The decimal expansion of √2 is infinite because it is nonterminating and nonrepeating.
How do you solve an equation with two square roots?
Use the following steps to solve an equation with two square roots.
Step 1: Isolate one of the square roots in the equation and square both sides.
Step 2: Simplify the equation obtained in Step 1.
Step 3: If the square root still exists, repeat the above steps.
Why is √2 an irrational number?
A number with decimal expansion as nonterminating and nonrepeating is always an irrational number. So, √2 is an irrational number.
Can the square root of 2 be simplified?
The rational number 99/70 is used as a good approximation for √2.
What is the square root of 2 divided by 2?
The value of square root 2 divided by 2 is 0.70710...
Is square root of 2 a real number?
Yes, the square root of 2 is a real number.