Square Root Calculator
Calculate the square root of any positive number. Enter a number to get its square root instantly.
The square root of a number is a value that, when multiplied by itself, gives the original number.
For example, the square root of 25 is 5 because 5 × 5 = 25.
Key points:
• Square roots of negative numbers are not real numbers
• The square root of 0 is 0
• For most numbers, square roots are irrational numbers with infinite decimal places
• This calculator provides results to 8 decimal places for precision
How to use Square Root Calculator
- 1
Enter a number
Type a positive number, such as 25 or 144.5, into the input field.
- 2
Set decimal places
Choose how many decimal places you want, up to 8, to control the precision of the result.
- 3
Calculate the root
Click Calculate Square Root to compute the value instantly in your browser.
- 4
Copy the answer
Use the Copy Result button to save the square root to your clipboard for use elsewhere.
A Closer Look at Square Roots
What a square root is
The square root of a number is the value that, multiplied by itself, gives that number. Since 5 times 5 is 25, the square root of 25 is 5. The operation undoes squaring, which is why it is the natural inverse of raising a number to the power of two. The radical symbol marks it, so the square root of 25 is written as the radical of 25.
Strictly speaking, 25 has two square roots, positive 5 and negative 5, because a negative times a negative is also positive. By convention the radical symbol returns only the non-negative answer, called the principal square root. That is the value this calculator reports, which matches the standard you will see in school and in most software.
Perfect squares
A perfect square is a number whose square root is a whole number. The first several are 1, 4, 9, 16, 25, 36, 49, 64, 81, and 100, with roots 1 through 10. Recognizing these on sight is worth doing, because they anchor every estimate you make for the numbers in between.
Perfect squares appear constantly in geometry and number work. A square tile floor that holds 144 tiles is 12 by 12, since the root of 144 is 12. When a calculation lands on a clean whole-number root, that is usually a sign you are dealing with a perfect square, and the result needs no rounding at all.
Estimating between perfect squares
Most numbers are not perfect squares, but you can pin their roots between two that are. Take 50. It sits between 49 and 64, so its root sits between 7 and 8. Because 50 is much closer to 49 than to 64, the root is just above 7, and indeed the root of 50 is about 7.07.
This bracketing method gives a quick sanity check before you trust any tool. If you ask for the root of 90 and see a result near 9.5, that is reasonable, since 90 lies between 81 and 100, so its root lies between 9 and 10. An answer wildly outside that band would signal a typo in your input.
Irrational results and rounding
When a whole number is not a perfect square, its square root is irrational: its decimal expansion runs forever without repeating. The root of 2 begins 1.41421356 and never settles into a pattern, so it can only be written exactly with the radical symbol. Any decimal form is necessarily an approximation.
That is why this calculator shows the result rounded to eight decimal places. More places would mean a closer approximation, not a different true value, but eight is already far more than enough for homework, and for a rough estimate you can simply read the first two or three digits. The tool rounds the unending decimal to a fixed eight places so the answer stays precise without running on forever.
How square roots are computed
One classic by-hand method is prime factorization. Break the number into prime pairs and pull one out of each pair. For 144, the factorization is 2 times 2 times 2 times 2 times 3 times 3; pairing gives 2 times 2 times 3, which is 12. This works cleanly only for perfect squares but is excellent for verifying them.
For everything else, computers use an iterative method such as the Babylonian, or Newton's, method. To approximate the root of 10, start with a guess like 3, then average it with 10 divided by the guess: 3 plus 3.333 is 6.333, halved is about 3.167. Repeat once more and you are already near the true 3.16228. Each pass roughly doubles the accuracy.
Negative numbers and imaginary roots
No real number squared gives a negative result, so a negative number has no real square root. That is why this calculator accepts only zero and positive values and returns an error for negative input. Asking for the root of -9 has no answer within the real numbers you use day to day.
Mathematics extends the idea with imaginary numbers, defining the root of -1 as the unit i. Then the root of -9 is 3i. This is essential in advanced fields like electrical engineering and signal processing, but it lives outside the real-number scope of this tool, which is built for the everyday positive case.
Where square roots get used
Geometry leans on square roots through the Pythagorean theorem. For a right triangle with legs 3 and 4, the hypotenuse is the root of 3 squared plus 4 squared, which is the root of 25, or exactly 5. Any time you compute a straight-line distance from horizontal and vertical components, a square root is involved.
Statistics relies on them too. The standard deviation, the most common measure of spread in data, is the square root of the variance. Finance uses roots to annualize volatility, and physics uses them in formulas from free-fall time to wave speed. Far from being abstract, the square root is one of the most quietly useful operations in applied math.
Frequently asked questions
How precise are the square root results?
Can it find the square root of a negative number?
Does it work for decimals and very large numbers?
Why is the square root of a number like 2 not exact?
Is the calculator free and private?
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