Square Root Calculator
Find the square root of a number, marked as exact only when it is a perfect square.
Square root
Worked example
Input: 9. Method: find the number that, multiplied by itself, gives 9. Result: √9 = 3, because 3 × 3 = 9.
How it works
√a is the principal, non-negative square root of a. For a > 0, the equation x² = a has two solutions: +√a and −√a. This tool shows √a, the principal root, and does not list the negative solution.
When the number is a perfect square, such as 9 or 0.25, the result is exact and is shown as a plain number. Most numbers are not perfect squares, and their square roots are irrational: they have no exact, finite decimal form. Those results are shown with a leading "≈" sign and a limited number of decimal places, so the value is never presented as more exact than it is. Negative numbers are not accepted, because they have no real square root; complex numbers are outside this tool's scope.
What this result means
The result is √a, the principal (non-negative) square root: the non-negative number that, multiplied by itself, gives the number you entered. For a > 0, the equation x² = a has two solutions, +√a and −√a; only +√a is shown. A plain number means the result is exact; a leading "≈" means it is a rounded approximation of a value with no finite decimal form.
Common mistakes
- Expecting a decimal such as 1.414213562 to be the exact square root of 2: it is only an approximation, because the true value never terminates or repeats.
Assumptions and limitations
- Negative numbers are rejected: they have no real square root, and complex numbers are outside this tool.
- A result is only marked exact for numbers of up to 15 significant digits, since longer numbers cannot be held exactly by ordinary computer arithmetic.