Square Root Calculator
Square Root Calculator
Enter any number to find its square root. This calculates the principal (positive) square root, flags whether your number is a perfect square, and — for negative numbers — returns the imaginary result, since no real number squared gives a negative value.
Perfect squares like 4, 9, 16, and 25 have whole-number square roots, but most numbers don't — the square root of 2 or 10, for example, is an irrational number that never terminates or repeats, which is why calculators show a truncated decimal approximation rather than an exact value. Square roots of negative numbers have no real-number answer at all, since squaring any real number (positive or negative) always produces a non-negative result — that's exactly the gap imaginary numbers were invented to fill.
- Definition: √x is the non-negative value that, multiplied by itself, equals x.
- Perfect squares have whole-number roots: numbers like 4, 9, 16, and 25 have square roots that are themselves whole numbers, while most others (like 2 or 10) have irrational, never-ending decimal roots.
- Negative numbers give imaginary results: since no real number squared is negative, the square root of a negative number is expressed using i (the imaginary unit), e.g. √−25 = 5i.
Why does every positive number have two square roots?
Both a positive and negative value square to the same result (5² = 25 and (−5)² = 25 too) — this calculator returns the principal (positive) root, which is the standard convention.
What does the imaginary result "5i" mean?
It means the square root of a negative number isn't a real number at all — i is defined as the square root of −1, so √−25 = √25 × √−1 = 5i, a concept used in advanced math, physics, and engineering.
Square Root Calculator
Enter any number to find its square root. This calculates the principal (positive) square root, flags whether your number is a perfect square, and — for negative numbers — returns the imaginary result, since no real number squared gives a negative value.
Perfect squares like 4, 9, 16, and 25 have whole-number square roots, but most numbers don't — the square root of 2 or 10, for example, is an irrational number that never terminates or repeats, which is why calculators show a truncated decimal approximation rather than an exact value. Square roots of negative numbers have no real-number answer at all, since squaring any real number (positive or negative) always produces a non-negative result — that's exactly the gap imaginary numbers were invented to fill.
- Definition: √x is the non-negative value that, multiplied by itself, equals x.
- Perfect squares have whole-number roots: numbers like 4, 9, 16, and 25 have square roots that are themselves whole numbers, while most others (like 2 or 10) have irrational, never-ending decimal roots.
- Negative numbers give imaginary results: since no real number squared is negative, the square root of a negative number is expressed using i (the imaginary unit), e.g. √−25 = 5i.
Why does every positive number have two square roots?
Both a positive and negative value square to the same result (5² = 25 and (−5)² = 25 too) — this calculator returns the principal (positive) root, which is the standard convention.
What does the imaginary result "5i" mean?
It means the square root of a negative number isn't a real number at all — i is defined as the square root of −1, so √−25 = √25 × √−1 = 5i, a concept used in advanced math, physics, and engineering.