Earliest Uses of Symbols of Operation

Last revision: Dec. 28, 1998


ADDITION AND SUBTRACTION SYMBOLS

Plus (+) and minus (-). The plus symbol as an abbreviation for the Latin et (and), though appearing with the downward stroke not quite vertical, was found in a manuscript dated 1417 (Cajori).

The + and - symbols first appeared in print in Mercantile Arithmetic or Behende und hubsche Rechenung auff allen Kauffmanschafft, by Johannes Widmann (born c. 1460), published in Leipzig in 1489. However, they referred not to addition or subtraction or to positive or negative numbers, but to surpluses and deficits in business problems (Cajori vol. 1, page 128).

Cajori says, "There is clear evidence that, as a lecturer at the University of Leipzig, Widmann had studied manuscripts in the Dresden library in which + and - signify operations, some of these having been written as early as 1486." Johnson (page 144) says a series of notes from 1481, annotated by Widmann, contain the + and - symbols, and he asks whether Widman could have copied these symbols from some unknown professor at the University of Leipzig. Johnson also says that a student's notes from one of Widmann's 1486 lectures show the + and - signs.

Giel Vander Hoecke used + and - as symbols of operation in Een sonderlinghe boeck in dye edel conste Arithmetica, published at Antwerp in 1514 (Smith 1958, page 341). Burton (page 335) says Vander Hoecke was the first person to use + and - in writing algebraic expressions, but Smith (page 341) says he followed Grammateus.

Henricus Grammateus (also known as Henricus Scriptor and Heinrich Schreyber or Schreiber) published an arithmetic and algebra, entitled Ayn new Kunstlich Buech, printed in 1518, in which he used + and - in a technical sense for addition and subtraction (Cajori vol. 1, page 131).

The plus and minus symbols only came into general use in England after they were used by Robert Recorde in in 1557 in The Whetstone of Witte. Recorde wrote, "There be other 2 signes in often use of which the first is made thus + and betokeneth more: the other is thus made - and betokeneth lesse."

The plus and minus symbols were in use before they appeared in print. For example, they were painted on barrels to indicate whether or not the barrels were full. Some have attempted to trace the minus symbol as far back as Heron and Diophantus.


MULTIPLICATION SYMBOLS

X was used by William Oughtred (1574-1660) in the Clavis Mathematicae (Key to Mathematics), composed about 1628 and published in London in 1631 (Smith). Cajori calls X St. Andrew's Cross.

X actually appears earlier, in 1618 in an anonymous appendix to Edward Wright's translation of John Napier's Descriptio (Cajori vol. 1, page 197). However, this appendix is believed to have been written by Oughtred.

The raised dot (·) was advocated by Gottfried Wilhelm Leibniz (1646-1716). According to Cajori (vol. 1, page 267):

The dot was introduced as a symbol for multiplication by G. W. Leibniz. On July 29, 1698, he wrote in a letter to John Bernoulli: "I do not like X as a symbol for multiplication, as it is easily confounded with x; ... often I simply relate two quantities by an interposed dot and indicate multiplication by ZC · LM. Hence, in designating ratio I use not one point but two points, which I use at the same time for division."

The raised dot was used earlier by Thomas Harriot (1560-1621) in Analyticae Praxis ad Aequationes Algebraicas Resolvendas, which was published posthumously in 1631, and by Thomas Gibson in 1655 in Syntaxis mathematica. However Cajori says, "it is doubtful whether Harriot or Gibson meant these dots for multiplication. They are introduced without explanation. It is much more probable that these dots, which were placed after numerical coefficients, are survivals of the dots habitually used in old manuscripts and in early printed books to separate or mark off numbers appearing in the running text" (Cajori vol. 1, page 268).

The asterisk (*) was used by Johann Rahn (1622-1676) in 1659 in Teutsche Algebra (Cajori vol. 1, page 211).

By juxtaposition. In a manuscript found buried in the earth near the village of Bakhshali, India, and dating to the eighth, ninth, or tenth century, multiplication is normally indicated by placing numbers side-by-side (Cajori vol. 1, page 78).

Multiplication by juxtaposition is also indicated in "some fifteenth-century manuscripts" (Cajori vol. 1, page 250).

According to Lucas, Michael Stifel (1487 or 1486 - 1567) first showed multiplication by juxtaposition in 1544 in Arithmetica integra.

According to Ball, Rene Descartes (1596-1650) first showed multiplication by juxtaposition in 1637.

Before Descartes, however, Thomas Harriot (1560-1621) used aa for a2, aaa for a3, etc.


DIVISION SYMBOLS

a close parenthesis attached to a
vinculum was first used by Michael Stifel (1487-1567 or 1486-1567) in 1544 in Arithmetica integra.

The obelus (÷) was first used by Johann Rahn (or Rhonius) (1622-1676) in 1659 in Teutsche Algebra (Burton). According to recent research, John Pell, who edited Rahn's algebra, was a major influence on Rahn and he may in fact be responsible for the invention of the symbol.

The colon (:) was used in 1633 in a text entitled Johnson Arithmetik; In two Bookes (2nd ed.: London, 1633). However Johnson only used the symbol to indicate fractions (for example three-fourths was written 3:4); he did not use the symbol for division "dissociated from the idea of a fraction" (Cajori vol. 1, page 276).

Gottfried Wilhelm Leibniz (1646-1716) used : for both ratio and division in 1684 in the Acta eruditorum (Cajori vol. 1, page 295).


EXPONENTS

Positive integers as exponents. Nicole Oresme (c. 1323-1382) used numbers to indicate powering in the fourteenth century, although he did not use raised numbers.

Nicolas Chuquet (1445?-1500?) used raised numbers in Le Triparty en la Science des Nombres in 1484. However, in Chuquet's notation, 123 actually meant 12x3 (Cajori vol. 1, page 102).

In 1634, Pierre Hérigone (or Herigonus) (1580-1643) wrote a, a2, a3, etc., in Cursus mathematicus, which was published in five volumes from 1634 to 1637; the numerals were not raised, however (Ball).

In 1636 James Hume used Roman numerals as exponents in L'Algèbre de Viète d'vne methode novelle, claire, et Facile. Cajori writes (vol. 1, pages 345-346):

In 1636 James Hume brought out an edition of the algebra of Vieta, in which he introduced a superior notation, writing down the base and elevating the exponent to a position above the regular line and a little to the right. The exponent was expressed in Roman numerals. Thus, he wrote Aiii for A3. Except for the use of Roman numerals, one has here our modern notation. Thus, this Scotsman, residing in Paris, had almost hit upon the exponential symbolism which has become universal through the writings of Descartes.
In 1637 exponents in the modern notation (although with positive integers only) were used by Rene Descartes (1596-1650) in Geometrie. Descartes tended not to use 2 as an exponent, however, usually writing aa rather than a2, perhaps because aa occupies no less space than a2.

Descartes wrote: "aa ou a2 por multiplier à par soiméme; et a3 pour le multiplier encore une fois par a, et ainsi à l'infini" (Cajori 1919, page 178).

Negative integers as exponents were used by Nicolas Chuquet (1445?-1500?) in 1484 in Le Triparty en la Science des Nombres. Chuquet wrote 12 on the baseline and a superscript of 1m with a bar over the
m to indicate 12x-1 (Cajori vol. 1, page 102).

Negative integers as exponents were first used with the modern notation by Isaac Newton in June 1676 in a letter to Henry Oldenburg, secretary of the Royal Society, in which he described his discovery of the general binomial theorem twelve years earlier (Cajori 1919, page 178).

Before Newton, John Wallis suggested the use of negative exponents but did not actually use them (Cajori vol. 1, page 216).

Fractions as exponents. The first use of fractional exponents (although not with the modern notation) is by Nicole Oresme (c. 1323-1382) in Algorismus proportionum. Oresme used p as the superscript to the
right of the base and the fractional power written with normal size
and to the left of the base to represent 91/3.

Simon Stevin (1548-1620) considered fractional powers and wrote that the fraction 2/3 circled would mean x2/3, but he did not actually use this notation. This notation was advanced earlier by Oresme, but it had remained unnoticed (Cajori 1919).

John Wallis (1616-1703), in his Arithmetica infinitorum which was published in 1656, speaks of fractional "indices" but does not actually write them (Cajori vol. 1, page 354).

Fractional exponents in the modern notation were first used by Isaac Newton in the 1676 letter referred to above (Cajori 1919, page 178).


OTHER SYMBOLS OF OPERATION

Factorial. The notation n! was introduced by Christian Kramp (1760-1826) in 1808 as a convenience to the printer. In the Preface [Avertissement, pp. xi-xii of his "Éléments d'arithmétique universelle," Hansen, Cologne, 1808] Kramp remarks:
...je leur avais donné le nom de facultés. Arbogast lui avait substitué la nomination plus nette et plus fran&#231aise de factorielles; j'ai reconnu l'avantage de cette nouvelle dénomination; et en adoptant son idée, je me suis félicité de pouvoir rendre hommage à la mémoire de mon ami. [...I've named them facultes. Arbogast has proposed the denomination factorial, clearer and more French. I've recognised the advantage of this new term, and adopting its philosophy I congratulate myself of paying homage to the memory of my friend.]
(Please recall that Louis François Antoine Arbogast died, in Strasbourg (France), in 1803.) In his well known "Mémoire sur les facultés numériques," published in Gergonne's Annales [vol. III, 1812/1813], Kramp says:
1. [...] Je donne le nom de Facultés aux produits dont les facteurs constituent une progression arithmétique, tels que

a(a + r)(a + 2r)...[a + (m-1)r];

et, pour désigner un pareil produit, j'ai proposé la notation

                                   m|r
                                  a    .

Les facultés forment une classe de fontions très-élementaires, tant que leur exposant est un nombre entier, soit positif soit négatif; mais, dans tous les autres cas, ces mêmes fonctions deviennent absolument transcendantes. [page 1]

2. J'observe que toute faculté numérique quelconque est constamment réductible ô la forme trés-simple

                         m|1
                        1   = 1 . 2 . 3 ... m

ou à cette autre forme plus simple [page 2]

m!,

si l'on veut adopter la notation dont j'ai fait usage dans mes Éléments d'arithmétique universelle, no. 289. [page 3]

Another widely-used factorial symbol, in which the argument is placed inside an L-shaped symbol, was suggested by Rev. Thomas Jarrett (1805-1882) in 1827. It occurs in a paper "On Algebraic Notation" that was printed in 1830 in the Transactions of the Cambridge Philosophical Society and it appears in 1831 in An Essay on Algebraic Development containing the Principal Expansions in Common Algebra, in the Differential and Integral Calculus and in the Calculus of Finite Differences (Cajori vol. 2, pages 69, 75).

(This entry was contributed by Julio González Cabillón.)

Dot for scalar product was used in 1902 in J. W. Gibbs's Vector Analysis by E. B. Wilson. However the dot was written at the baseline and was not a "raised dot."

X for vector product was used in 1902 in J. W. Gibbs's Vector Analysis by E. B. Wilson.

Plus-or-minus symbol (±) was used by William Oughtred (1574-1660) in Clavis Mathematicae, published in 1631 (Cajori vol. 1, page 245).

Polish notation was invented by Jan Lukasiewicz (1878-1956).

The product symbol (using the
capital pi) was introduced by Rene Descartes, according to Gullberg.

Cajori says this symbol was introduced by Gauss in 1812 (vol. 2, page 78).

Square root. The first use of a capital
R with a diagonal line was in 1220 by Leonardo of Pisa in Practica geometriae, where the symbol meant "square root" (Cajori vol. 1, page 90).

The radical symbol first appeared in 1525 in Die Coss by Christoff Rudolff (1499-1545). He used the
symbol (without the vinculum) for square roots. He did not use indices to indicate higher roots, but instead modified the appearance of the radical symbol for higher roots.

In 1637 Rene Descartes added the vinculum to the radical symbol La Geometrie (Cajori vol. 1, page 375).

Placement of the index within the opening of the radical sign was suggested as early as 1629 by Albert Girard (Cajori vol. 1, page 371). This notation first appears in 1690 in Traité d' Algébre by Michel Rolle (1652-1719) (Cajori vol 1., page 372).

Summation. The summation symbol (the Greek letter sigma) was first used by Leonhard Euler (1707-1783) in 1755:

Quemadmodum ad differentiam denotandam vsi sumus signo [capital delta], ita summam indicabimus signo (the Greek letter sigma).
The citation above is from Institutiones calculi differentialis (St. Petersburg, 1755), Cap. I, para. 26, p. 27 (Cajori vol. 2, pages 61 and 265.)

Absolute value of a difference. The tilde was introduced for this purpose by William Oughtred (1574-1660) in the Clavis Mathematicae (Key to Mathematics), composed about 1628 and published in London in 1631, according to Smith, who shows a reversed tilde (Smith 1958, page 394).

Binomial coefficients (or combinations). Leonhard Euler (1707-1783) designated the binomial coefficients by n over r within parentheses and using a horizontal fraction bar in a paper written in 1778 but not published until 1806. He used used the same device except with brackets in a paper written in 1781 and published in 1784 (Cajori vol. 2, page 62).

The modern notation, using parentheses and no fraction bar, was introduced in 1827 by Andreas von Ettingshausen in Vorlesungen über höhere Mathematik, Vol. I (Cajori vol. 2, page 63).

Matrices. In 1841, Arthur Cayley (1821-1895) used a single vertical line on either side of the entries to indicate the determinant of a matrix. He used commas to separate entries within rows (Cajori vol. 2, page 92).

The double vertical line notation was introduced by Cayley in 1843 (Cajori vol. 2, page 95).

In 1846, the first occurrence of both the single vertical line notation for determinants and double vertical lines for matrices is found in "Mémoire sur les hyperdéterminants" by Arthur Cayley in Crelle's Journal (Cajori vol. 2, page 93).

In 1845 brackets and braces are used in place of the vertical lines in articles by Cayley in Liouville's Journal, vol. 10. [ ] is used on page 105 and { } is used on page 383 (this seems unclear--JM) (Cajori vol. 2, page 93).


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