Square Root of 2: Value, Formula, Methods and Examples

The square root of 2 (√2) is the number that, when multiplied by itself, gives 2. It is an irrational number, which means its decimal value never ends and never repeats. The square root of 2 is widely used in geometry, algebra, and many real-life calculations.

In this article, you'll learn the value of √2, different methods to calculate it, its properties, solved examples, and practical applications.

Table of Contents

• What is the Square Root of 2?
• Why is the Square Root of 2 Important?
• Approximate Value of Square Root of 2
• How to Calculate the Square Root of 2
• Square Root of 2 Formula
• Simplify the Square Root of 2
• Common Misconceptions about √2
• Fun Facts: Square Root of 2
• Solved Examples of the Square Root of 2
• Conclusion
• Frequently Asked Questions on the Square Root of 2

What is the Square Root of 2

The square root of 2 (written as √2) is a number that, when multiplied by itself, gives 2.  

• “It is an irrational number, which means we can’t write it exactly as a fraction like ½ or ¾.

• In decimal form, √2 ≈ 1.414...  

• It is the length of the diagonal of a square with a side length of 1.
  
  

Read more:

• Square Root 1 To 30
• Value Of Root 3
• Surds
  
  

Why is the Square Root of 2 Important

• It is the first number proven to be irrational in mathematics.  

• It is widely used in geometry, especially in right triangles.  

• It commonly appears in physics, computer graphics, and engineering.  

• It is essential for understanding roots in algebra.  

Approximate Value of Square Root of 2

Approximation	Value of   $\sqrt{2}$
Up to 2 decimal places	1.41
Up to 4 decimal places	1.4142
Up to 10 decimal places	1.4142135624
Up to 50 decimal places	1.41421356237309504880168872420969807856967187537695

How to Calculate the Square Root of 2

Long Division Method  

This is a step-by-step method to manually calculate the square root of 2.  

Write the number 2 as 2.0000… (adding zeros) and group digits in pairs.

• Start the division like regular long division.  

• Estimate the digits one at a time.  

• Continue the steps to get more precise digits of √2.  

This way, you can find √2 step-by-step, correct up to many decimal places.

Approximation Method  

A calculator gives √2 ≈ 1.414.  

We can also use a guessing method called the Babylonian method.

Step-by-step process:  

• Let x = 2, guess = 1.5.  

• New guess = (1.5 + 2/1.5)/2 = 1.4167.  

• Repeat until you reach the desired accuracy.  

This is a very effective technique if you're learning how to find the root of 2 manually.

Square Root of 2 Formula

There is no exact formula for √2, but we can use it in equations and geometry problems.

The square root of 2 formula helps in solving quadratic equations or determining diagonal lengths.  

Simplify the Square Root of 2

The number √2 is already in its simplest form.  

However, many roots can be simplified into forms involving √2:  

Examples:  

√8 = √(4 × 2) = 2√2  

√50 = √(25 × 2) = 5√2  

Learning how to simplify square roots like √8 = 2√2 is an important maths skill.

Common Misconceptions about √2

• Misconception: √2 is a rational number.  

Reality: It is irrational and cannot be expressed as a fraction.  

• Misconception: √2 = 1.41 exactly.  

Reality: 1.41 is close, but the decimal never ends.  

• Misconception: √(a + b) = √a + √b.  

Reality: This is incorrect. The square root does not distribute over addition.   

• Misconception: You can convert √2 into a perfect decimal.  

Reality: It has no exact decimal representation.  

• Misconception: √2 can be simplified further.  

Reality: It is already in its simplest form.  

 

Fun Facts: Square Root of 2

• Geometry: Diagonal of a Square  

In a square with a side of 1 unit, the diagonal is √2 units.  

• A4 Paper Size  

The ratio of height to width of A4 paper is √2, allowing consistent scaling.  

• Historical Significance  

The square root of 2 was known and approximated by ancient Babylonians and Indians.  

• Used in Trigonometry and Algebra  

√2 appears in 45-degree right triangles where the legs are equal.  

• Engineering and Architecture  

It is used to calculate slants, diagonal distances, and dimensions.  

Solved Examples of the Square Root of 2

Example 1: Approximate √2 using long division  

Solution:  

Start with 2.  

Use the long division method step-by-step.  

The result converges to 1.4142.  

Answer: √2 ≈ 1.4142  

Example 2: Simplify √8  

Solution:  

√8 = √(4 × 2)  

= √4 × √2  

= 2√2  

Answer: √8 = 2√2  

Example 3: Find the diagonal of a square with a side of 5 cm  

Solution:  

Diagonal = side × √2  

= 5 × √2  

= 5 × 1.414  

= 7.07 cm  

Answer: Diagonal = 7.07 cm  

Example 4: Evaluate 3√2 + 2√2  

Solution:  

Combine like terms: 3√2 + 2√2 = (3 + 2)√2  

= 5√2  

Answer: 3√2 + 2√2 = 5√2  

Example 5: Find x if x² = 2  

Solution:  

x² = 2  

Taking the square root on both sides: x = ±√2  

x ≈ ±1.414  

Answer: x = ±√2 or approximately ±1.414  

Conclusion

The square root of 2 is a fundamental concept in both pure and applied mathematics. From learning how to calculate the square root of 2 using the long division method to using the square root of 2 formula in geometry and simplifying expressions like √8 or √50, this number holds great significance.  

Understanding the root maths symbol, identifying what 2 square root, and knowing how to find the root of 2 enables students to solve complex problems confidently. Whether in class problems or real life, √2 is very useful in maths.

Discover the Square Root of 2 and its applications with Orchids The International School.