Find the square root of any number, including negative numbers (as imaginary).
√x is the value that, multiplied by itself, gives x. Negative numbers have no real square root — the result is expressed as an imaginary number (i√|x|).
Finding a square root without a lookup table requires an iterative approximation method, and the one still conceptually used today — known as the Babylonian method — dates back roughly 3,500 years, to Babylonian mathematicians working around 1500 BC. The method is a simple guess-divide-average loop: start with a reasonable guess for the square root, divide the original number by that guess, then average the guess and the result — each repetition of this loop produces a more accurate approximation, converging on the true square root within just a handful of iterations. A clay tablet known as YBC 7289 (roughly 1800-1600 BCE) shows Babylonian scribes had already computed the square root of 2 accurate to six decimal digits using exactly this kind of iterative approach.
What the Babylonians didn't grasp, despite this computational sophistication, is that some square roots — like the square root of 2 — are irrational numbers: they cannot be written as an exact fraction and their decimal expansion never terminates or repeats, no matter how many digits are calculated. That discovery is credited to the school of Pythagoras in ancient Greece (roughly 570-495 BCE), with legend attributing it specifically to Hippasus, a student who reportedly realized √2 could not be expressed as a ratio of two whole numbers — a genuinely unsettling discovery at the time, since it broke the prevailing assumption that all numbers could be expressed as ratios. This is exactly why square roots of non-perfect-square numbers are given as decimal approximations rather than exact values — the exact value has infinitely many non-repeating digits, so any decimal shown is necessarily a rounded approximation, however many digits are displayed.
Most methods trace back conceptually to the Babylonian method, an iterative guess-divide-average approach dating to roughly 1500 BC: start with an estimate, divide the target number by that estimate, then average the estimate and result. Repeating this a few times converges rapidly on an accurate square root, and modern computing methods are refined descendants of this same core idea.
Because it cannot be expressed as an exact fraction (ratio of two whole numbers), and its decimal expansion continues forever without repeating. This was discovered by the school of Pythagoras in ancient Greece, credited by legend to a student named Hippasus, and was considered a genuinely unsettling mathematical discovery at the time.
Because for any number that isn't a perfect square, the true square root is irrational — its decimal representation has infinitely many non-repeating digits. Whatever decimal value is displayed is necessarily a rounded approximation of the true value, not the exact answer, regardless of how many digits are shown.
A perfect square is a number that results from squaring a whole number (4, 9, 16, 25, and so on, from 2², 3², 4², 5²). Their square roots are exact whole numbers by definition. Any number that isn't a perfect square has an irrational square root, expressible only as an approximate decimal.