What is Root of 2

When you search “What is the root of 2?” on Google, the answer is:

👉 √2 = 1.414213562… (approximately)

This value is irrational, meaning it cannot be written exactly as a fraction, and its decimal expansion never ends or repeats.

But what does “root” mean?
Let’s understand everything about square roots and how to find roots of any number step by step.

🔢 What is a Root? (Definition)

The root of a number is a value that, when multiplied by itself a certain number of times, gives the original number.

📘 For Example:

The square root of 9 is 3, because 3 × 3 = 9.
The cube root of 8 is 2, because 2 × 2 × 2 = 8.

🧠 What is the Square Root of 2?

The square root of 2 (written as √2) is the number which, when multiplied by itself, equals 2.

[
√2 × √2 = 2
]

✅ Value:

√2 = 1.4142135623730951…
This value continues infinitely without repeating — that’s why √2 is an irrational number.

📏 Square Root Symbol and Meaning

Symbol	Meaning	Example
√	Square root	√4 = 2
³√	Cube root	³√8 = 2
⁴√	Fourth root	⁴√16 = 2

⚙️ Methods to Find the Square Root of Any Number

There are several methods to find the square root of a number — from basic manual techniques to calculator or programming methods.

Let’s understand them one by one 👇

🔹 1. Prime Factorization Method

Used for perfect squares.

Example: Find √144

👉 Step 1: Factorize 144
144 = 2 × 2 × 2 × 2 × 3 × 3
= (2² × 2² × 3²)

👉 Step 2: Take one factor from each pair
√144 = 2 × 2 × 3 = 12

✅ Answer: √144 = 12

📘 Note: This method works best when the number is a perfect square (like 4, 9, 16, 25, 36…).

🔹 2. Long Division Method (Manual Square Root)

Used for numbers that are not perfect squares, like √2, √3, √5, etc.

Let’s find √2 step by step:

Step 1: Group digits in pairs from the right

For 2 → write as 2.000000…

Step 2: Find the largest square less than 2

1² = 1 → write 1 as divisor and quotient

Step 3: Subtract and bring down next pair of zeros

(2 − 1) = 1 → bring down “00” → 100

Now, double the quotient (1 → 2)
Find a digit x such that (20 + x) × x ≤ 100
Here, x = 4 → (24 × 4 = 96)

Write x = 4 beside the quotient → 1.4
Subtract → 100 − 96 = 4
Bring down next “00” → 400

Repeat the same steps…
You’ll get √2 = 1.414213…

✅ Result: √2 ≈ 1.414213562

🔹 3. Approximation or Estimation Method

If you need an approximate value quickly:

Example: √50

We know:
√49 = 7
√64 = 8

So √50 lies between 7 and 8, and closer to 7.

Using estimation, √50 ≈ 7.07

✅ Fast method for rough calculations.

🔹 4. Using the Exponent Formula

Every root can be expressed as a fractional power.

Example:
√2 = 2¹ᐟ² = 1.414…
³√8 = 8¹ᐟ³ = 2

This is useful in algebra and calculators.

🔹 5. Using Calculator or Computer

Simply press the √ button on any scientific calculator or use programming tools.

Example in Python:

import math
print(math.sqrt(2))

Output: 1.4142135623730951

📘 Common Square Roots (1 to 25)

Number	Square Root	Perfect Square?
1	1	✅
2	1.414	❌
3	1.732	❌
4	2	✅
5	2.236	❌
6	2.449	❌
7	2.645	❌
8	2.828	❌
9	3	✅
10	3.162	❌
16	4	✅
25	5	✅

💡 Fun Facts About √2

√2 is the first known irrational number.
It was discovered by the ancient Greek mathematician Pythagoras’s followers.
√2 is used in geometry — especially in the diagonal of a square.
If each side of a square is 1 unit, the diagonal is √2 units.
√2 never ends or repeats — its decimal expansion goes on forever.

🧮 Root of Any Number – General Formula

Type of Root	Formula	Example
Square root	√x or x¹ᐟ²	√9 = 3
Cube root	³√x or x¹ᐟ³	³√27 = 3
Fourth root	⁴√x or x¹ᐟ⁴	⁴√16 = 2

🔍 Summary

Concept	Description
√2 Value	1.414213562…
Nature	Irrational number
Symbol	√ (square root)
Found by	Long division or calculator
Used in	Geometry, algebra, trigonometry, computing

🎯 Real-Life Uses of Square Roots

Finding diagonal of square or rectangle
Trigonometric calculations
Physics and Engineering formulas
Finance – to find standard deviation or growth rates

🏁 Conclusion

The square root of 2 (√2) is one of the most famous and important numbers in mathematics.
It represents the diagonal of a unit square, has an infinite non-repeating decimal, and forms the foundation of irrational number theory.

Whether you find it by long division, factorization, or calculator, learning how to find roots helps you understand the beauty and logic of mathematics.