Principal roots wear the radical; negatives need complexes
A square root of x is any y with y squared equal to x—so 4 and −4 both root 16. Nonnegative reals own a unique nonnegative principal root marked √ with a vinculum over the radicand. Positive numbers also carry a negative twin often bundled with a plus-minus sign.
People were computing roots nearly four thousand years ago. The Babylonian tablet YBC 7289, made between 1800 and 1600 BC, marks the diagonal of a square with a base-60 value for the root of two that is right to five decimal places. The Rhind Papyrus, copied around 1650 BC, shows Egyptians extracting roots by inverse proportion. India's Sulba Sutras, from about 800 to 500 BC, give close approximations for the roots of two and three, and Apastamba's version is also accurate to five places. In Han China, a text written between 202 and 186 BC used an excess-and-deficiency method, and Aryabhata later described how to handle numbers with many digits.
Greeks proved that the root of any positive whole number that is not a perfect square is irrational, a result in Euclid's tenth book almost certainly owed to Theaetetus around 380 BC. The discovery of irrationals is tied to the Pythagoreans, sometimes to Hippasus, though sources are too thin to be sure. The root of two is simply the diagonal of a unit square.
Notation took centuries to settle. Moroccan mathematician Ibn al-Yasamin, in the late twelfth century, put an Arabic jīm, the first letter of the word for root, over numbers, and its shape resembles the modern sign. Regiomontanus used an ornate R, as did Cardano, and the √ itself first appeared in print in Christoph Rudolff's Coss in 1525.
Geometrically, the function turns a square's area into its side length. It is continuous for all nonnegative inputs and differentiable for positive ones. Roots sit inside the Euclidean distance formula, standard deviation and the quadratic formula, and a whole number's root reduces to the roots of primes that appear to an odd power.
Source: Square root
