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.

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 withx; ... often I simply relate two quantities by an interposed dot and indicate multiplication byZC·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 *a*^{2}, *aaa* for *a*^{3}, etc.

**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).

Nicolas Chuquet (1445?-1500?) used raised numbers in *Le Triparty en
la Science des Nombres* in 1484. However, in Chuquet's notation,
12^{3} actually meant 12*x*^{3} (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 wroteIn 1637 exponents in the modern notation (although with positive integers only) were used by Rene Descartes (1596-1650) inAfor^{iii}A^{3}. 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.

Descartes wrote: "*aa* ou *a*^{2} por multiplier
à par soiméme; et *a*^{3} 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 to indicate 12*x*^{-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 to represent 9^{1/3}.

Simon Stevin (1548-1620) considered fractional powers and wrote that
the fraction 2/3 circled would mean *x*^{2/3}, 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).

...je leur avais donné le nom de(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:facultés.Arbogast lui avait substitué la nomination plus nette et plus française defactorielles;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 themfacultes.Arbogast has proposed the denominationfactorial,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.]

1. [...] Je donne le nom deAnother 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 theFacultésaux 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 ... mou 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]

(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** () 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 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 (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 () was first used by Leonhard Euler
(1707-1783) in 1755:

Quemadmodum ad differentiam denotandam vsi sumus signo [capital delta], ita summam indicabimus signo ().The citation above is from

**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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