Square Root Calculator

The square root of a number is the value that, when multiplied by itself, gives that number back. It's one of the oldest operations in mathematics — Babylonian scribes were approximating square roots on clay tablets nearly 4,000 years ago, and the radical symbol (√) itself dates to 16th-century German algebra texts, likely derived from a stylized lowercase "r" for "radix," the Latin word for root.

Square roots aren't always tidy. The discovery that √2 cannot be written as a simple fraction — it's irrational — is credited to the ancient Greek Pythagoreans, and legend says it upset their belief that all numbers could be expressed as ratios of whole numbers. This calculator also shows whether your input is a perfect square and identifies the nearest perfect squares around it, which is exactly the technique used to estimate irrational roots by hand.

Square roots are closely related to other root operations — see the Cube Root Calculator for the equivalent third-root operation — and they're a core building block of the Quadratic Equation Calculator, since the quadratic formula's discriminant is evaluated under a square root.

Every non-negative number has a unique non-negative square root — finding it means asking "what number, multiplied by itself, produces this value?" Negative numbers have no real square root, since a negative times a negative is always positive, so no real number squared can ever produce a negative result.

Technically, every positive number has two square roots — a positive one and a negative one (for example, both 5 and −5 square to 25) — but by convention the radical symbol √ always refers to the non-negative "principal" root, which is the value this calculator returns. Before electronic calculators existed, people found square roots using manual methods like long division-style algorithms or Newton's iterative approximation method, repeatedly refining a guess until it converged on the answer.

If the result is a perfect square, the square root is an exact whole number. Otherwise, the decimal shown is an approximation — the true value is an irrational number with infinitely many non-repeating decimal digits, so the 4-decimal-place result is a close but not perfectly exact representation.

The "nearest squares" shown below the result tell you which two whole numbers your answer falls between — for example, a result between 7 and 8 means the true square root lies somewhere in that range, and how close it sits to either end depends on how close your input is to 49 or 64.