Square Root Calculator

Every positive number has two square roots

Both 4 and −4 square to 16. The radical symbol denotes the principal root, which is the non-negative one, so √16 is 4 rather than ±4. The plus-or-minus appears when solving an equation: x² = 16 has both solutions, because you are asking which numbers square to 16, not evaluating a function. Conflating these is why sign errors appear in quadratic work.

Simplified radical form

√72 is not 8.485 for most purposes; it is 6√2. Pull out the largest perfect-square factor: 72 = 36 × 2, and √36 is 6. This keeps the value exact rather than rounded, which matters when the result feeds into further calculation — three roundings compound in a way one does not. Most mathematics courses require this form, and it is also what computer algebra systems keep internally.

Irrational roots go on forever

The square root of any number that is not a perfect square is irrational: non-terminating and non-repeating. √2 = 1.41421356… has no exact decimal or fractional representation. The Pythagoreans' discovery of this is one of the oldest results in mathematics, and it is the reason exact radical form exists as a notation at all.

Negative numbers and imaginary roots

No real number squares to a negative, so √−9 has no real value. In the complex numbers it is 3i, where i is defined as the square root of −1. These are not a workaround: complex roots are essential in electrical engineering, quantum mechanics and signal processing. But a negative under a radical in a geometry or physics problem usually signals an error in setting it up.

Estimating without a calculator

Bracket between perfect squares. √50 sits between √49 = 7 and √64 = 8, and much nearer 7, so about 7.07. For more precision, the Babylonian method converges quickly: guess, divide the number by the guess, average the two, repeat. Starting at 7 for 50 gives 7.0714 after one pass, which is accurate to four decimals — the method is around 4,000 years old and is essentially what hardware still does.

Floating point breaks the identity

Squaring a computed square root does not reliably return the original number. In double precision, √2 squared gives 2.0000000000000004. Any code testing equality after a root will eventually fail; compare against a tolerance instead. The same caution applies to chained roots and powers, where the error compounds.