The square root of 2 is one of the most famous numbers in mathematics. It is an irrational number, meaning it cannot be expressed as a simple fraction and its decimal goes on forever without repeating.
In this guide, you’ll learn the exact value of √2, how to calculate it, and where it appears in real life. You can also use our free square root calculator to verify instantly.
What is the Square Root of 2?
√2 is the number that, when multiplied by itself, gives 2.
Also commonly searched as squareroot 2, sq root of 2, or even sqaure root 2 — the value remains the same:
- √2 = 1.41421356…
- It is an irrational number
- It cannot be expressed as a fraction
- Its decimal goes on forever
| Form | Value |
|---|---|
| Decimal Form | 1.41421356… |
| Simplified Radical | √2 (already simplified) |
| Polar Form | 1.4142 ∠ 0° |
| Perfect Square? | ❌ No |
Is Square Root of 2 Rational or Irrational?
√2 is irrational. This was first proved by ancient Greek mathematicians. An irrational number:
- Cannot be written as a fraction p/q
- Has a non-terminating, non-repeating decimal
- Goes on forever: 1.41421356237309504880168…
How to Calculate Square Root of 2?
Method 1 — Using a Calculator
The easiest way is to use our free square root calculator. Simply enter 2 and get instant results with step-by-step working.
Method 2 — Long Division Method
You can calculate √2 manually using the long division method:
- Write 2 as 2.000000
- Find largest number whose square ≤ 2 → 1 (1×1=1)
- Remainder = 2-1 = 1. Bring down 00 → 100
- Double quotient (1×2=2). Find X: 2X × X ≤ 100 → X=4 (24×4=96)
- Remainder = 100-96 = 4. Bring down 00 → 400
- Continue → √2 ≈ 1.414213…
Method 3 — Estimation Method
Quick estimation of √2:
- 1² = 1 and 2² = 4
- So √2 is between 1 and 2
- 1.4² = 1.96 (close to 2)
- 1.41² = 1.9881 (very close!)
- 1.414² = 1.999396 ≈ 2 ✅
Square Root of 2 — Step by Step
- Input number: 2
- Apply square root: √2
- Decimal result: 1.414214
- Simplified radical: √2 (cannot be simplified further)
- Polar form: 1.4142 ∠ 0°
- Perfect square check: No
Why is √2 Important?
The square root of 2 appears in many real-world applications:
- Geometry — Diagonal of a unit square = √2
- Trigonometry — sin(45°) = cos(45°) = 1/√2
- Paper sizes — A4, A3 paper ratio is based on √2
- Engineering — Used in electrical circuits
- Architecture — Used in building proportions
Square Root of 2 vs Other Square Roots
| Number | Square Root | Type |
|---|---|---|
| 1 | 1.000000 | ✅ Rational |
| 2 | 1.414214 | ❌ Irrational |
| 3 | 1.732051 | ❌ Irrational |
| 4 | 2.000000 | ✅ Rational |
| 5 | 2.236068 | ❌ Irrational |
Frequently Asked Questions (FAQ)
What is the square root of 2?
The square root of 2 is 1.41421356… It is an irrational number that cannot be expressed as a simple fraction.
Is square root of 2 a rational number?
No, the square root of 2 is an irrational number. It was one of the first numbers proved to be irrational by ancient Greek mathematicians.
What is √2 in fraction form?
√2 cannot be expressed as an exact fraction. A common approximation is 99/70 ≈ 1.4142857, which is very close but not exact.
Is 2 a perfect square?
No, 2 is not a perfect square. Perfect squares are numbers like 1, 4, 9, 16, 25 whose square roots are whole numbers.
What is the square root of 2 to 10 decimal places?
√2 = 1.4142135624 (to 10 decimal places)
How do you simplify √2?
√2 is already in its simplest form. It cannot be simplified further because 2 has no perfect square factors other than 1.
Q: What is the square root of 2 in fraction form?
The square root of 2 in fraction form simply does not exist — and here is why. √2 is an irrational number, which means no matter how hard you try, you cannot write it as a clean fraction like 1/2 or 3/4. The decimal value of squareroot 2 is 1.41421356… and it goes on forever without repeating — making an exact fraction impossible. If someone needs a quick approximation, 99/70 comes closest (≈ 1.41428), but even that is not perfectly equal to sq root of 2.
Q: Can sqaure root 2 be simplified into a fraction?
No — sqaure root 2 (commonly misspelled but meaning √2) cannot be simplified into any fraction. Mathematicians proved this over 2,000 years ago: if you assume √2 = p/q in its simplest form, the math breaks down into a contradiction — both p and q end up being even numbers, which is impossible in a truly simplified fraction. So what is a square root of 2 as a fraction? The honest answer is: it cannot be one.
Conclusion
√2 is approximately 1.414214. It is an irrational number, meaning its decimal never ends or repeats. It appears in geometry, trigonometry, engineering, and everyday life.
According to Wikipedia, the square root of 2 is also known as Pythagoras’s constant.
Use our free square root calculator for instant results with full step-by-step working!
√2 as a Fraction — Can squareroot 2 be Written as a Fraction?
Many users ask: what is a square root of 2 in fraction form?
The short answer is — No. √2 cannot be written as any fraction.
√2 = 1.41421356… is a non-terminating, non-repeating decimal.This is exactly what makes sq root of 2 an irrational number —it has no exact fraction equivalent.
The closest rational approximation of squareroot 2 as a fraction
is 99/70 ≈ 1.41428, but even this is not perfectly equal to √2.
So if anyone searches for sqaure root 2 as a fraction —
the mathematically correct answer will always be: it is impossible.
