History of computing
The history of computing is longer than the history of computing hardware and modern computing technology and includes the history of methods intended for pen and paper or for chalk and slate, with or without the aid of tables. Digital computing is intimately tied to the representation of numbers. But long before abstractions like the number arose, there were mathematical concepts to serve the purposes of civilization. These concepts are implicit in concrete practices such as: One-to-one correspondence, a rule to count how many items, e.g. on a tally stick, eventually abstracted into numbers. Comparison to a standard, a method for assuming reproducibility in a measurement, for example, the number of coins. The 3-4-5 right triangle was a device for assuring a right angle, using ropes with 12 evenly spaced knots, for example. Eventually, the concept of numbers became concrete and familiar enough for counting to arise, at times with sing-song mnemonics to teach sequences to others. All known human languages, except the Piraha language, have words for at least "one" and "two", and even some animals like the blackbird can distinguish a surprising number of items. Advances in the numeral system and mathematical notation eventually led to the discovery of mathematical operations such as addition, subtraction, multiplication, division, squaring, square root, and so forth. Eventually the operations were formalized, and concepts about the operations became understood well enough to be stated formally, and even proven. See, for example, Euclid's algorithm for finding the greatest common divisor of two numbers. By the High Middle Ages, the positional Hindu–Arabic numeral system had reached Europe, which allowed for systematic computation of numbers. During this period, the representation of a calculation on paper actually allowed calculation of mathematical expressions, and the tabulation of mathematical functions such as the square root and the common logarithm (for use in multiplication and division) and the trigonometric functions.