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:
“... there are persons who seem to have overcome obstacles and by character and perseverance to have risen to the top. But we have no record of the numbers of able persons who fall by the wayside, persons who, with enough encouragement and opportunity, might make great contributions.”
—Mary Barnett Gilson (1877–?)
“I had but three chairs in my house; one for solitude, two for friendship; three for society. When visitors came in larger and unexpected numbers there was but the third chair for them all, but they generally economized the room by standing up.”
—Henry David Thoreau (1817–1862)
“The principle of majority rule is the mildest form in which the force of numbers can be exercised. It is a pacific substitute for civil war in which the opposing armies are counted and the victory is awarded to the larger before any blood is shed. Except in the sacred tests of democracy and in the incantations of the orators, we hardly take the trouble to pretend that the rule of the majority is not at bottom a rule of force.”
—Walter Lippmann (1889–1974)