Editing History of computing (section)

==Numbers==
Eventually, the concept of numbers became concrete and familiar enough for counting to arise, at times with sing-song [[mnemonic]]s to teach [[sequence]]s to others. All known human languages, except the [[Piraha language]], have words for at least the [[numeral (linguistics)|numerals]] "one" and "two", and even some animals like the [[common blackbird|blackbird]] can distinguish a surprising number of items.<ref>{{cite book|first1=Konrad|last1=Lorenz|author-link1=Konrad Lorenz|year=1961|title=King Solomon's Ring|translator1=Marjorie Kerr Wilson|publisher=Methuen|location=London|isbn=0-416-53860-6}}</ref>

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 [[theorem|stated formally]], and even [[mathematical proof|proven]]. See, for example, [[Euclidean algorithm|Euclid's algorithm]] for finding the greatest common divisor of two numbers.

By the High Middle Ages, the [[positional notation|positional]] [[Hindu–Arabic numeral system]] had reached [[Europe]], which allowed for the systematic computation of numbers. During this period, the representation of a calculation on [[paper]] allowed the calculation of [[mathematical expression]]s, and the tabulation of [[mathematical function]]s such as the [[square root]] and the [[common logarithm]] (for use in multiplication and division), and the [[trigonometric function]]s. By the time of [[Isaac Newton]]'s research, paper or vellum was an important [[computing resource]], and even in our present time, researchers like [[Enrico Fermi]] would cover random scraps of paper with calculation, to satisfy their curiosity about an equation.<ref>{{cite web|title=DIY: Enrico Fermi's Back of the Envelope Calculations|url=http://www.knowqout.com/science-technology/diy-enrico-fermis-back-of-the-envelope-calculations/}}</ref> Even into the period of programmable calculators, [[Richard Feynman]] would unhesitatingly compute any steps that overflowed the [[Computer memory|memory]] of the calculators, by hand, just to learn the answer; by 1976 Feynman had purchased an [[HP-25]] calculator with a 49 program-step capacity; if a differential equation required more than 49 steps to solve, he could just continue his computation by hand.<ref>"Try numbers" was one of Feynman's problem solving techniques.</ref>