History of numbers

The Ishango bone on exhibition at the Royal Belgian Institute of Natural Sciences

Counting

Numbers that answer the question "How many?" are 0, 1, 2, 3 and so on. These are

cardinal numbers. When used to indicate position in a sequence they are ordinal numbers.

To the Pythagoreans and Greek mathematician Euclid, the numbers were 2, 3, 4, 5, . . . Euclid did not consider 1 to be a number.

Fractions and rational numbers

Numbers like $3+{\frac {1}{7}}={\frac {22}{7}}$ , expressible as fractions in which the numerator and denominator are whole numbers, are rational numbers. These make it possible to measure such quantities as two and a quarter gallons and six and a half miles.

Incommensurable magnitudes in geometry and irrational numbers

In the fifth century BC, one of the ancient Pythagoreans showed that some quantities arising in geometry, including the length of the diagonal of a square, when the unit of measurement is the length of the side of the square, cannot be expressed as rational numbers. If the side of a square were divided into five segments of equal lengths, and if the length of the diagonal of the square were equal to that of exactly seven such short segments (which is in fact a reasonable approximation, but not exact), then those short segments would be what Euclid called a "common measure" of the side and the diagonal. What we today would consider a proof that a number is irrational Euclid called a proof that two lengths arising in geometry have no common measure, or are "incommensurable". Euclid included proofs of incommensurability of lengths arising in geometry in his Elements.

Negative numbers

In the Rhind Mathematical Papyrus, a pair of legs walking forward marked addition, and walking away subtraction. They were the first known civilization to use negative numbers.

Negative numbers came into widespread use as a result of their utility in accounting. They were used by late medieval Italian bankers.

Some particular numbers

Zero

By 1740 BC, the Egyptians had a symbol for zero in accounting texts. In Maya civilization zero was a numeral with a shell shape as a symbol, with the plastron uppermost).

One

The ancient Egyptians represented all fractions (with the exception of 2/3) in terms of sums of fractions with numerator 1 and distinct denominators. For example, 2/5 = 1/3 + 1/15. Such representations are popularly known as Egyptian Fractions or Unit Fractions.

$π$

The earliest written approximations of $π$ are found in Egypt and Babylon, both within one percent of the true value. In Babylon, a clay tablet dated 1900–1600 BC has a geometrical statement that, by implication, treats $π$ as 25/8 = 3.1250.^[1] In Egypt, the Rhind Papyrus, dated around 1650 BC but copied from a document dated to 1850 BC, has a formula for the area of a circle that treats $π$ as (16/9)² ≈ 3.1605.^[1]

Astronomical calculations in the Shatapatha Brahmana (ca. 4th century BC) use a fractional approximation of 339/108 ≈ 3.139 (an accuracy of 9×10⁻⁴).^[2] Other Indian sources by about 150 BC treat $π$ as $\sqrt 10$ ≈ 3.1622^[3]

e

The first references to the constant e were published in 1618 in the table of an appendix of a work on logarithms by John Napier. However, this did not contain the constant itself, but simply a list of logarithms calculated from the constant. It is assumed that the table was written by William Oughtred. The discovery of the constant itself is credited to Jacob Bernoulli,^[4]^[5] who attempted to find the value of the following expression (which is in fact $e$ ):

\lim _{n\to \infty }\left(1+{\frac {1}{n}}\right)^{n}.

The first known use of the constant, represented by the letter $b$ , was in correspondence from Gottfried Leibniz to Christiaan Huygens in 1690 and 1691. Leonhard Euler introduced the letter $e$ as the base for natural logarithms, writing in a letter to Christian Goldbach of 25 November 1731.^[6]^[7] Euler started to use the letter $e$ for the constant in 1727 or 1728, in an unpublished paper on explosive forces in cannons,^[8] and the first appearance of $e$ in a publication was Euler's Mechanica (1736).^[9] While in the subsequent years some researchers used the letter $c$ , $e$ was more common and eventually became the standard.

Numeral systems

The first numeral system known is Babylonian numeric system, that has a 60 base, it was introduced in 3100 B.C. and is the first Positional numeral system known.

The first known system with place value was the Mesopotamian base 60 system (ca. 3400 BC) and the earliest known base 10 system dates to 3100 BC in Egypt.^[10]

Roman numerals evolved primitives system of cutting notches.^[11] It was once believed that they came from alphabetic symbols or from pictographs, but these theories have been disproved.^[12]^[13]

References

1 2 Arndt & Haenel 2006, p. 167
↑ Chaitanya, Krishna. A profile of Indian culture. Indian Book Company (1975). p.133.
↑ Arndt & Haenel 2006, p. 169
↑ Jacob Bernoulli considered the problem of continuous compounding of interest, which led to a series expression for $e$ . See: Jacob Bernoulli (1690) "Quæstiones nonnullæ de usuris, cum solutione problematis de sorte alearum, propositi in Ephem. Gall. A. 1685" (Some questions about interest, with a solution of a problem about games of chance, proposed in the Journal des Savants (Ephemerides Eruditorum Gallicanæ), in the year (anno) 1685.**), Acta eruditorum, pp. 219–223. On page 222, Bernoulli poses the question: "Alterius naturæ hoc Problema est: Quæritur, si creditor aliquis pecuniæ summam fænori exponat, ea lege, ut singulis momentis pars proportionalis usuræ annuæ sorti annumeretur; quantum ipsi finito anno debeatur?" (This is a problem of another kind: The question is, if some lender were to invest [a] sum of money [at] interest, let it accumulate, so that [at] every moment [it] were to receive [a] proportional part of [its] annual interest; how much would he be owed [at the] end of [the] year?) Bernoulli constructs a power series to calculate the answer, and then writes: " … quæ nostra serie [mathematical expression for a geometric series] &c. major est. … si a = b, debebitur plu quam 2 ¹⁄₂a & minus quam 3a." ( … which our series [a geometric series] is larger [than]. … if $a = b$ , [the lender] will be owed more than $21 ⁄ 2 a$ and less than $3 a$ .) If $a = b$ , the geometric series reduces to the series for $a \times e$ , so $2.5 < e < 3$ . (** The reference is to a problem which Jacob Bernoulli posed and which appears in the Journal des Sçavans of 1685 at the bottom of page 314.)
↑ Carl Boyer; Uta Merzbach (1991). A History of Mathematics (2nd ed.). Wiley. p. 419.
↑ Lettre XV. Euler à Goldbach, dated November 25, 1731 in: P. H. Fuss, ed., Correspondance Mathématique et Physique de Quelques Célèbres Géomètres du XVIIIeme Siècle … (Mathematical and physical correspondence of some famous geometers of the 18th century), vol. 1, (St. Petersburg, Russia: 1843), pp. 56–60 ; see especially page 58. From page 58: " … ( e denotat hic numerum, cujus logarithmus hyperbolicus est = 1), … " ( … ( $e$ denotes that number whose hyperbolic [i.e., natural] logarithm is equal to 1) … )
↑ Remmert, Reinhold (1991). Theory of Complex Functions. Springer-Verlag. p. 136. ISBN 0-387-97195-5.
↑ Euler, Meditatio in experimenta explosione tormentorum nuper instituta.
↑ Leonhard Euler, Mechanica, sive Motus scientia analytice exposita (St. Petersburg (Petropoli), Russia: Academy of Sciences, 1736), vol. 1, Chapter 2, Corollary 11, paragraph 171, p. 68. From page 68: Erit enim $dc / c = dy ds / rdx$ seu $c = e \int dy ds / rdx$ ubi e denotat numerum, cuius logarithmus hyperbolicus est 1. (So it [i.e., $c$ , the speed] will be $dc / c = dy ds / r dx$ or $c = e \int dy ds / rdx$ , where $e$ denotes the number whose hyperbolic [i.e., natural] logarithm is 1.)
↑ "Egyptian Mathematical Papyri – Mathematicians of the African Diaspora". Math.buffalo.edu. Retrieved 2012-01-30.
↑ Ifrah, Georges (2000) The Universal History of Numbers: From Prehistory to the Invention of the Computer, Wiley. pp. 191–194. ISBN 0-471-37568-3.
↑ Keyser, Paul (1988). "The origin of the Latin numerals 1 to 1000". American Journal of Archeology. 92: 529–546. JSTOR 505248.
↑ Chrisomalis, Stephen (2010) Numerical Notation: A Comparative History.

Cited sources

Arndt, Jörg; Haenel, Christoph (2006). Pi Unleashed. Springer-Verlag. ISBN 978-3-540-66572-4.

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