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Square Root of 2

The square root of 2 is expressed as √2 in the radical form and as (2)^½ or (2)^0.5 in the exponent form. The square root of 2 rounded up to 10 decimal places is 1.4142135624. It is the positive solution of the equation x² = 2.

Square Root of 2: 1.4142135623730951
Square Root of 2 in exponential form: (2)^½ or (2)^0.5
Square Root of 2 in radical form: √2

1.	What Is the Square Root of 2?
2.	Is Square Root of 2 Rational or Irrational?
3.	Important Notes on Square Root of 2
4.	How to Find the Square Root of 2?
5.	Thinking Out of the Box!
6.	FAQs on Square Root of 2

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 non-terminating and non-repeating.
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 y₁=1

Iteration 1: y₁ = (2 + 1)/2 = 1.5
Iteration 2: y₂ = (4/3 + 3/2)/2 = 1.4166
Iteration 3: y₃= (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
Example: If the area of an equilateral triangle is 2√3 in². Find the length of one of the sides of the triangle.

Solution:

Let 'a' be the length of one of the sides of the equilateral triangle.
⇒ Area of the equilateral triangle = (√3/4)a² = 2√3 in²
⇒ a = ±√8 in
Since length can't be negative,
⇒ a = √8 = 2 √2
We know that the square root of 2 is 1.414.
⇒ a = 2.828 in

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FAQs on the Square Root of 2

What is the Value of the Square Root of 2?

The square root of 2 is 1.41421.

Why is the Square Root of 2 an Irrational Number?

The number 2 is prime. This implies that the number 2 is pairless and is not in the power of 2. Therefore, the square root of 2 is irrational.

If the Square Root of 2 is 1.414. Find the Value of the Square Root of 0.02.

Let us represent √0.02 in p/q form i.e. √(2/100) = 0.02/10 = 0.141. Hence, the value of √0.02 = 0.141

Is the number 2 a Perfect Square?

The number 2 is prime. This implies that the square root of 2 cannot be expressed as a product of two equal integers. Therefore, the number 2 is not a perfect square.

What is the Square Root of -2?

The square root of -2 is an imaginary number. It can be written as √-2 = √-1 × √2 = i √2 = 1.414i
where i = √-1 and it is called the imaginary unit.

What is the Value of 17 square root 2?

The square root of 2 is 1.414. Therefore, 17 √2 = 17 × 1.414 = 24.042.

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