having many digits. It was known to the ancient Greeks that square roots of positive integers that are not perfect squares are always irrational numbers: numbers not expressible as a ratio of two integers (that is, they cannot be written exactly as , where m and n are integers). This is the theorem Euclid X, 9, almost certainly due to Theaetetus dating back to circa 380 BC. The particular case of the square root of 2 is assumed to date back earlier to the Pythagoreans, and is traditionally attributed to Hippasus. It is exactly the length of the diagonal of a square with side length 1. In the Chinese mathematical work Writings on Reckoning, written between 202 BC and 186 BC during the early Han Dynasty, the square root is approximated by using an "excess and deficiency" method, which says to "...combine the excess and deficiency as the divisor; (taking) the deficiency numerator multiplied by the excess denominator and the excess numerator times the deficiency denominator, combine them as the dividend." A symbol for square roots, written as an elaborate R, was invented by Regiomontanus (1436–1476). An R was also used for radix to indicate square roots in Gerolamo Cardano's Ars Magna. According to historian of mathematics D.E. Smith, Aryabhata's method for finding the square root was first introduced in Europe by Cataneo—in 1546. According to Jeffrey A. Oaks, Arabs used the letter jīm/ĝīm (), the first letter of the word "" (variously transliterated as jaḏr, jiḏr, ǧaḏr or ǧiḏr, "root"), placed in its initial form () over a number to indicate its square root. The letter jīm resembles the present square root shape. Its usage goes as far as the end of the twelfth century in the works of the Moroccan mathematician Ibn al-Yasamin. The symbol "√" for the square root was first used in print in 1525, in Christoph Rudolff's Coss. Properties and uses The principal square root function (usually just referred to as the "square root function") is a function that maps the set of nonnegative real numbers onto itself. In geometrical terms, the square root function maps the area of a square to its side length. The square root of x is rational if and only if x is a rational number that can be represented as a ratio of two perfect squares. (See square root of 2 for proofs that this is an irrational number,