Enter a decimal or exact numeric form
Choose the operation, precision, simplification, and result settings.
Batch, comparison, history, and reference tools
Process many values, compare roots, review saved work, or inspect examples.
Batch actions
| # | Input | Fraction | Exact root | Approximation | Status |
|---|---|---|---|---|---|
| No batch calculations yet. | |||||
Recent calculations
History is stored in this browser only.
| Decimal | Square root | Exact form | Notes | |
|---|---|---|---|---|
| 0.25 | 0.5 | 1/2 | Perfect rational square | |
| 0.5 | 0.70710678 | √2/2 | Irrational result | |
| 0.72 | 0.84852814 | 3√2/5 | Fraction simplifies first | |
| 2 | 1.41421356 | √2 | Classic irrational root | |
| 6.25 | 2.5 | 5/2 | Exact terminating root | |
| 12.5 | 3.53553391 | 5√2/2 | Rationalized denominator | |
| -9 | 3i | 3i | Complex principal root | |
| 0.(3) | 0.57735027 | √3/3 | Repeating decimal input |
Number line and square-area visualization
The drawings update after each successful real-number calculation.
Root on a number line
Square with matching area
Square-root and radical conversion formulas
The principal square root is the nonnegative value whose square equals the input. When solving an equation such as x² = a, both positive and negative roots must be considered.
A terminating decimal can be written as an integer over a power of ten. The fraction is reduced first. Perfect-square factors are then extracted from its numerator and denominator. Any remaining radical in the denominator is rationalized.
A nonnegative decimal can also be represented as the square root of its square. A negative decimal requires a leading negative sign because the principal square root itself is never negative.
How to use this calculator
Enter a decimal, fraction, mixed number, repeating decimal, or scientific-notation value. Select whether you want the principal root, both equation roots, a radical representation, or an approximation-focused answer. Choose precision and rounding settings. Enable complex mode when negative values are allowed. Press the calculation button to generate exact and approximate results.
Review the fraction conversion, radical simplification, prime factors, nearest perfect squares, and detailed steps. Select Newton or Babylonian mode to inspect iterative estimates. Use the batch tab for many inputs. Use the comparison tab to measure differences between two roots. Results may be copied, printed, shared, saved, or exported.
Understanding decimal square roots
A square root reverses squaring. For example, 3.5 squared equals 12.25. Therefore, the principal square root of 12.25 is 3.5. Some decimals produce rational roots. Others produce irrational values that never terminate or repeat. Exact radical notation preserves mathematical accuracy without forcing premature rounding.
Converting the decimal to a reduced fraction is often the best first step. Consider 0.72. It equals 72/100, which reduces to 18/25. Its square root becomes √18/5. Extracting the perfect-square factor gives 3√2/5. The decimal approximation is useful, but the radical form remains exact.
Principal roots and equation roots
The radical symbol indicates the principal root. It returns one nonnegative real answer when the input is nonnegative. However, an equation has two roots in most nonzero cases. For x² = 6.25, the solutions are x = 2.5 and x = -2.5. The calculator separates these ideas to prevent a common mistake.
Negative decimals and complex numbers
No real number squares to a negative value. Complex-number mode uses the imaginary unit i, where i² = -1. Thus, √-9 equals 3i. The calculator reports the principal complex root and can also show the corresponding negative solution for an equation.
Precision, rounding, and exactness
Decimal places control digits after the decimal point. Significant figures control overall meaningful digits. Scientific notation is useful for extremely large or small values. Engineering notation uses exponents divisible by three. Exact radicals should be preferred when symbolic accuracy matters. Rounded values remain useful for measurement, design, and numerical comparison.
Frequently asked questions
Can the calculator simplify irrational square roots?
Yes. It extracts perfect-square factors and returns a reduced radical with a rationalized denominator whenever exact integer factorization is available.
What repeating-decimal format is supported?
Place repeating digits inside parentheses. Examples include 0.(3), 1.2(34), and -0.1(6).
Why does √6.25 show only 2.5?
The radical symbol means the principal nonnegative root. Select equation mode to solve x² = 6.25 and show ±2.5.
Can negative decimals be calculated?
Yes. Enable complex-number mode. The calculator then uses i and returns the principal complex square root.
Does the calculator support fractions?
Yes. Enter forms such as 7/8 or mixed numbers such as 2 1/3. Denominators cannot be zero.
What is Newton's method?
It repeatedly averages an estimate with the input divided by that estimate. The values quickly approach the square root.
Where is calculation history stored?
History is stored locally in your browser. It is not uploaded by this page and can be cleared anytime.
How are PDF reports created?
The page uses a browser-side PDF library. The exported report includes the input, exact answer, approximation, and solution steps.