Numbers
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 the known languages have words for at least "one" and "two" (although this is disputed: see Piraha language), 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. 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. Even into the period of programmable calculators, Richard Feynman would unhesitatingly compute any steps which overflowed the memory of the calculators, by hand, just to learn the answer.
Read more about this topic: History Of Computing
Famous quotes containing the word numbers:
“What culture lacks is the taste for anonymous, innumerable germination. Culture is smitten with counting and measuring; it feels out of place and uncomfortable with the innumerable; its efforts tend, on the contrary, to limit the numbers in all domains; it tries to count on its fingers.”
—Jean Dubuffet (1901–1985)
“All experience teaches that, whenever there is a great national establishment, employing large numbers of officials, the public must be reconciled to support many incompetent men; for such is the favoritism and nepotism always prevailing in the purlieus of these establishments, that some incompetent persons are always admitted, to the exclusion of many of the worthy.”
—Herman Melville (1819–1891)
“He bundles every forkful in its place,
And tags and numbers it for future reference,
So he can find and easily dislodge it
In the unloading. Silas does that well.
He takes it out in bunches like birds’ nests.”
—Robert Frost (1874–1963)