Square Root Calculator
By Talcart · Last updated July 10, 2026
Square Root Calculator
A square root calculator returns the principal (non-negative) square root of any number you enter: the square root of 144 is exactly 12, while the square root of 2 is the irrational value 1.41421356... For perfect squares the answer is a whole number; for everything else the calculator reports a decimal accurate to about 15 significant digits.
Key facts
- The square root of 2 is approximately 1.41421356237 and is irrational - a fact the Pythagoreans proved around the 5th century BC, reputedly to their own dismay.
- The radical symbol first appeared in print in 1525, in Christoff Rudolff's algebra book Coss.
- Heron's method converges quadratically: each iteration roughly doubles the number of correct digits, so sqrt(10) is accurate to six digits after just two steps from a starting guess of 3.
What is the Square Root Calculator?
The square root of a number x is the value r such that r x r = x; the principal square root is the non-negative one, written with the radical sign as sqrt(x). Every positive number has two square roots (+r and -r), but the radical symbol denotes only the positive root: sqrt(25) = 5, even though (-5)^2 is also 25. Perfect squares such as 1, 4, 9, 16, and 144 have integer roots; every other positive integer has an irrational root whose decimal expansion never repeats. Negative numbers have no real square root - their roots are imaginary.
How does the Square Root Calculator work?
The calculator evaluates the root with IEEE 754 double-precision arithmetic, correct to roughly 15-17 significant digits. The classical hand method it mirrors is Heron's (Newton's) iteration: guess r, then repeatedly replace r with (r + x/r) / 2. For sqrt(10), starting at 3: (3 + 10/3) / 2 = 3.1667, then (3.1667 + 10/3.1667) / 2 = 3.16228 - already six correct digits, because the method roughly doubles the number of correct digits each pass. Convergence this fast is why square roots cost barely more than a division on modern hardware.
What is the Square Root Calculator formula?
- x – the input (≥ 0)
- r – the principal square root
Square roots of common numbers
| Number | Square root | Type |
|---|---|---|
| 2 | 1.414214 | Irrational |
| 3 | 1.732051 | Irrational |
| 4 | 2 | Perfect square |
| 9 | 3 | Perfect square |
| 10 | 3.162278 | Irrational |
| 25 | 5 | Perfect square |
| 50 | 7.071068 | Irrational |
| 100 | 10 | Perfect square |
| 144 | 12 | Perfect square |
| 200 | 14.142136 | Irrational |
How do you use the Square Root Calculator?
- Enter the number.
- Read the square root.
Worked example
| Scenario | √144 |
| Calculation | 12 × 12 = 144 |
| Result | 12. |
Common use cases
Tips & best practices
For non-perfect squares, the result is irrational — round only after final calculations.
Frequently asked questions
The square root of 144 is 12, because 12 x 12 = 144. Since 144 is a perfect square, the answer is a whole number with no decimal part. Strictly, 144 has two square roots, +12 and -12, but the radical symbol and calculators return the principal (positive) root. 144 is also 12 squared in the 12x12 multiplication table, which is why it is called a gross.
No - sqrt(2) = 1.41421356... is irrational, meaning it cannot be written as a fraction of two integers. The classic proof assumes sqrt(2) = a/b in lowest terms, shows both a and b would then have to be even, and reaches a contradiction. The discovery is attributed to the Pythagoreans around the 5th century BC, and the digits never terminate or repeat.
Use Heron's method: guess a value r, then repeatedly average it with x/r, i.e. replace r with (r + x/r) / 2. To find sqrt(10), start at 3: (3 + 10/3) / 2 = 3.1667; iterate again to get 3.16228, correct to six digits. Each pass roughly doubles the number of correct digits, so three or four iterations beat most practical needs.
Not within the real numbers - no real number multiplied by itself gives a negative result, since a positive times itself and a negative times itself are both positive. Mathematics extends to complex numbers to handle this: the imaginary unit i is defined by i^2 = -1, so sqrt(-9) = 3i. Everyday calculators, including this one, report negative inputs as invalid.
Because squaring destroys sign: both 5^2 and (-5)^2 equal 25, so both +5 and -5 are square roots of 25. The radical symbol resolves the ambiguity by convention, always denoting the principal (non-negative) root. That is why solving x^2 = 25 gives x = +/-5, while sqrt(25) alone is exactly 5 - the equation and the symbol ask subtly different questions.
The square root of 0 is exactly 0, because 0 x 0 = 0. Zero is the only number with a single square root rather than a positive/negative pair, and it is also a perfect square (0 = 0^2). It marks the boundary of the real square root function: sqrt(x) is defined for every x >= 0 and undefined (in the reals) below it.
The theorem states c^2 = a^2 + b^2 for a right triangle, so finding the hypotenuse requires a square root: c = sqrt(a^2 + b^2). With legs 3 and 4, c = sqrt(9 + 16) = sqrt(25) = 5. Distance formulas in 2D and 3D space, screen diagonals, and navigation calculations are all this same square root in disguise.