Earliest Uses of Various Mathematical Symbols

Earliest Uses of Various Mathematical Symbols

These pages show the names of the individuals who first used various common mathematical symbols, and the dates the symbols first appeared. The most important written source is the definitive A History of Mathematical Notations by Florian Cajori.

Symbols of operation, including +, -, X, division, exponents, radical symbol, dot and vector product

Grouping symbols, including (), [], {}, vinculum

Symbols of relation, including =, >, <

Fractions, including decimals

Symbols for various constants, such as pi, i, e, 0

Symbols for variables

Symbols to represent various functions, such as log, ln, gamma, absolute value; also the f(x) notation

Symbols used in geometry

Symbols used in trigonometry; also symbols for hyperbolic functions

Symbols used in calculus

Set notation and logic

Symbols used in number theory

Symbols used in statistics

Written sources for these pages

ADDITION AND SUBTRACTION SYMBOLS

Plus (+) and minus (-).Nicole d' Oresme (1323-1382) may have used a figure which looks like a plus symbol as an abbreviation for the Latin et (meaning "and") in Algorismus proportionum, believed to have been written between 1356 and 1361. The symbol appears in a manuscript of this work believed to have been written in the fourteenth century, but perhaps by a copyist and not Oresme himself. The symbol appears, for example, in the sentence: "Primi numeri sesquiterti sunt .4. et .3., et primi numeri sev termini sesquialtere sunt .3. et .2." [Dic Sonneveld].

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 hpsche 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).

Here is an image of the first use in print of the + and - signs, from Widman's Behennde vnd hpsche Rechnung. This image is taken from the Augsburg edition of 1526.

Widman wrote, "Was - ist / das ist minus ... vnd das + das ist mer." He also wrote, "4 centner + 5 pfund" and "5 centner - 17 pfund," thus showing the excess or deficiency in the weight of boxes or bales (Smith vol. 2, page 399).

Smith (vol. 2, page 398) explains the origin of the + sign by connecting it to the Latin word for "and":

In a manuscript of 1456, written in Germany, the word et is used for addition and is generally written so that it closely resembles the symbol +. The et is also found in many other manuscripts, as in "5 et 7" for 5 + 7, written in the same contracted form, as when we write the ligature & rapidly. There seems, therefore, little doubt that this sign is merely a ligature for et.

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).

However, Scott (page 128) writes that Harriot was "in the habit of using the dot to denote multiplication." And Eves (page 231) writes, "Although Harriot on occasion used the dot for multiplication, this symbol was not prominently used until Leibniz adopted it."

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). Juxtaposition was used by al-Qalasadi in the fifteenth century (Cajori vol. 1, page 230).

According to Lucas, Michael Stifel (1487 or 1486 - 1567) first showed multiplication by juxtaposition in 1544 in Arithmetica integra. In 1553, Michael Stifel brought out a revised edition of Rudolff's Coss, in which he showed multiplication by juxtaposition and repeating a letter to designate powers (Cajori vol. 1, pages 145-147).

DIVISION SYMBOLS

Close parenthesis. The arrangement 8)24 was used by Michael Stifel (1487-1567 or 1486-1567) in Arithmetica integra, which was completed in 1540 and published in 1544 in Nuernberg (Cajori vol. 1, page 269; DSB).

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).

The obelus () was first used as a division symbol by Johann Rahn (or Rhonius) (1622-1676) in 1659 in Teutsche Algebra (Cajori vol. 2, page 211). Here is the page in which the division symbol first appears in print, as reproduced in Cajori.

Rahn's book was translated into English and published, with additions by John Pell, in London in 1668, with the division symbol retained. According to some recent sources, John Pell was a major influence on Rahn and he may in fact be responsible for the invention of the symbol. However, according to Cajori there is no evidence to support this claim. The division symbol was used by many writers before Rahn as a minus sign.

Recent symbolism. In nineteenth century U. S. textbooks, long division is typically shown with the divisor, dividend, and quotient on the same line, separated by parentheses, as 36)116(3. This notation is used, for example, in 1866 in Primary Elements of Algebra for Common Schools and Academies by Joseph Ray.

In 1882 in Complete Graded Arithmetic by James B. Thomson, the 36)116(3 notation is used for long division. However, in examples for short division, a vinculum is placed under the dividend and the vinculum is almost attached to the bottom of the close parenthesis. The quotient is written under the vinculum, as shown below.

The symbol is not mentioned by Cajori. An early use of this symbol is in 1888 in The Elements of Algebra by G. A. Wentworth. The symbol was seen in the teacher's edition but presumably is also in the student edition. [I welcome earlier uses of this symbol that may be found by readers of this page.]

In 1901, the second edition of Robinson's Complete Arithmetic by Daniel W. Fish uses the same notations for short and long division as Thomson (1882) above, except that the vinculum under the dividend is actually attached to the close parenthesis. This notation may appar in the earlier 1873 edition, which has not been seen.

David E. Smith writes, "It is impossible to fix an exact date for the origin of our present arrangement of figures in long division, partly because it developed gradually" (Smith vol. 2).

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 Hrigone (or Herigonus) (1580-1643) wrote a, a2, a3, etc., in Cursus mathematicus, which was published in several volumes from 1634 to 1637; the numerals were not raised, however (Cajori vol. 1, page 202, and Ball). In 1636 James Hume used Roman numerals as exponents in L'Algbre de Vite 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 pour multiplier par soimme; et a3 pour le multiplier encore une fois par a, et ainsi 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 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 to represent 91/3. According to Cajori (1919), this notation remained unnoticed.

Simon Stevin (1548-1620) "had no occasion to use the fractional index notation," but "he clearly stated that 1/2 in a circle would mean square root and 3/2 in a circle would indicate the square root of the cube" (Boyer, page 356). 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).

Scientific notation. The earliest use of scientific notation is not known. However, some physicists working with electricity in in the decade or so up to 1873, when our modern volt, ohm, etc., were standardized, used scientific notation. James A. Landau has found only two usages of scientific notation in Maxwell's collected papers, and could find no other physicists of mid-century using scientific notation. In 1863, in The Annual Encyclopedia and Register of Important Events of the Year 1862 the article on "Electricity" has on page 404:

The aim should be to make this standard [of electrical resistance] correspond to a current force equal to 10,000,000,000 times the value given by the quotient of 1 metre by 1 second of time, that is, 1010 mtre/seconds.

In 1868 Rep. Brit. Assoc. 1867 has: 105 EMF, acting on a circuit of 1013, will pass in one second 10-8 absolute units of quantity; and similarly, 105 EMF will charge a condenser of absolute capacity equal to 10-13 absolute units with 10-8 absolute units of quantity... Mr. Clark calls the unit of quantity thus defined (10-8) one Farad, and similarly says that the unit of capacity has a capacity of one Farad, it being understood that this is the capacity when charged with unit electromotive force (105).

The above quotation was taken from the OED2. In 1885 Johann Jakob Balmer in "Notiz uber die Spectrallinien des Wasserstoffs" (Annalen der Physik and Chemie, Vol. 25, p. 80, 1885) wrote in English translation:

From the formula we obtained for a fifth hydrogen line

49/45 3645.6 = 3969.65 10-7 mm

In 1887 Albert Abraham Michelson and Edward Williams Morley wrote in Philosophical Magazine, Series 5, December, 1887:

Considering the motion of the earth in its orbit only, this displacement should be

2 D v2/V2 = 2D x 10-8

The distance D was about eleven metres, or 2 x 107 wave-lengths of yellow light. Both of these citations were taken from A Source Book in Physics by William Francis Magie. An even earlier possible use of scientific notation is by Robert Whillhelm Bunsen in 1857 in Philosophical Transactions, where these formulae appear on page 357:

I = I0 10-h alpha

I = I1 10-h alpha + I2 10-h alpha + ...

However the objection is that Bunsen was measuring the intensity of light before and after going through a tube of chlorine, and the alpha above is defined as "The value of 1/alpha, which signifies...the depth of chlorine to which the chemical rays must penetrate in order to be reduced to one-tenth of their original amount..." Therefore the 10 is not necessarily part of scientific notation but comes from the fact that Bunsen elected to measure a reduction of light to one-tenth of the original. This entry was largely contributed by James A. Landau.

ORDER OF OPERATIONS

The convention that multiplication precedes addition and subtraction was in use in the earliest books employing symbolic algebra in the 16th century. The convention that exponentiation precedes multiplication was used in the earliest books in which exponents appeared.

In 1892 in Mental Arithmetic, M. A. Bailey advises avoiding expressions containing both and .

In 1898 in Text-Book of Algebra by G. E. Fisher and I. J. Schwatt, abb is interpreted as (ab)b.

In 1907 in High School Algebra, Elementary Course by Slaught and Lennes, it is recommended that multiplications in any order be performed first, then divisions as they occur from left to right.

In 1910 in First Course of Algebra by Hawkes, Luby, and Touton, the authors write that and should be taken in the order in which they occur.

In 1912, First Year Algebra by Webster Wells and Walter W. Hart has: "Indicated operations are to be performed in the following order: first, all multiplications and divisions in their order from left to right; then all additions and subtractions from left to right."

In 1913, Second Course in Algebra by Webster Wells and Walter W. Hart has: "Order of operations. In a sequence of the fundamental operations on numbers, it is agreed that operations under radical signs or within symbols of grouping shall be performed before all others; that, otherwise, all multiplications and divisions shall be performed first, proceeding from left to right, and afterwards all additions and subtractions, proceeding again from left to right."

In 1917, "The Report of the Committee on the Teaching of Arithmetic in Public Schools," Mathematical Gazette 8, p. 238, recommended the use of brackets to avoid ambiguity in such cases.

In A History of Mathematical Notations (1928-1929) Florian Cajori writes (vol. 1, page 274), "If an arithmetical or algebraical term contains and , there is at present no agreement as to which sign shall be used first."

Modern textbooks seem to agree that all multiplications and divisions should be performed in order from left to right.

OTHER SYMBOLS OF OPERATION

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).

Postfix notation or RPN began as prefix notation, a mathematical notation which did away with grouping symbols. It was proposed by Jan Lukasiewicz (1878-1956). "Prefix" meant that the operators ( * + - etc.) preceded the operands or variables they were meant to operate on. Then it was discovered that it is much more convenient to place the operands first and operators last, so "Postfix" or "reverse Lukasiewicz" or "reverse Polish" notation was created. With postfix notation the operators themselves become delimiters between operations. Thus the sequence U*(V^(W + 3))/(X - Y) becomes

U V W 3 + ^ * X Y - /

RPN is used in the computer language Forth. [Axel Harvey]

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.

It is often suggested that the origin of the modern radical symbol is that it is an altered letter r, the first letter in the word radix. This is the opinion of Leonhard Euler in his Institutiones calculi differentialis (1775). However, Florian Cajori, author of A History of Mathematical Notations, argues against this theory.

In 1637 Rene Descartes used , adding 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 in 1629 by Albert Girard (1595-1632) in Invention nouvelle. He suggested this notation for the cube root (DSB; Cajori vol. 1, page 371).

According to Cajori (vol. 1, page 372) the first person to adopt Girard's suggestion and place the index within the opening of the radical sign was Michel Rolle (1652-1719) in 1690 in Trait d' Algbre.

However, a history note in a high school textbook states that the symbol was first used by Girard "around 1633" (UCSMP Advanced Algebra, 2nd ed., 1996, page 496).

In the Mathematical Gazette of Feb. 1895, G. Heppel wrote, "Following Chrystal, Todhunter, Hall and Knight, and the majority of writers [sqrt]a should be considered a quantity having one and not two values, although the algebra of C. Smith and the article by Professor Kelland in the Encyclopedia Britannica make [sqrt]a have two values."

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 Institutiones calculi differentialis (St. Petersburg, 1755), Cap. I, para. 26, p. 27. This symbol was used by Lagrange, but otherwise received little attention during the eighteenth century (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).

Matrices. In 1841, Arthur Cayley (1821-1895) used the modern notation for the determinant of a matrix, a single vertical line on both sides of the entries. The notation appeared in the Cambridge Mathematical Journal, Vol. II (1841), p. 267-271. However, Cayley 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 "Mmoire sur les hyperdterminants" by Arthur Cayley in Crelle's Journal (Cajori vol. 2, page 93).

Cajori (vol. 2, p. 103) writes that round parentheses were used for matrices by many, including Maxine Bocher in 1919 in Introduction to Higher Algebra and G. Kowalewski in 1909 in Determinantentheorie (although Kowalewski also used double vertical lines and a single brace).

Cajori also shows a use of brackets for matrices (and no commas within) by C. E. Cullis in Matrices and Determinoids in 1913.

Arrow notation. In 1936 in L'Agebre Abstraite Oystein Ore wrote:

Nous dirons que deux systemes algebriques S et S' sont homomorphes (par rapport a l'addition et a la multiplication) s'il existe une correspondance a --> a' entre les elements de S et S' donnant a chaque element a de S une image unique a' dans S' telle que chaque element de S' soit l'image d'au moins un element de S et en outre telle que de a --> a', b --> b' on puisse conclure

a + b --> a' + b' , ab --> a'b' .

As early as 1939 Bourbaki used the arrow in element-to-element notation ["la application x --> f(x)"].

Saunders Mac Lane wrote:

At first the vivid arrow notation f : X ---> Y for a map was not available, and homomorphisms of homology groups (or rings) were always expressed in terms of the corresponding quotient group or rings. Thus the familiar long exact sequence of the homotopy groups of a fibration was originally described in terms of subgroups and quotient groups; this is the style used by all three discoveries of the sequence and of the covering homotopy theorem [...] The occurrence of exact sequences of homology groups (though not the name 'exact') was first noted by W. Hurewicz in 1941 [...] The practice of using an arrow to represent a map f : X ---> Y arose at the same time. I have not been able to determine who first introduced this convenient notation; it may well have appeared first on the blackboard, perhaps in lectures by Hurewicz and it is used in the Hurewicz-Steenrod paper, submitted November 1940 ...

The above quotation is from Saunders Mac Lane, "Concepts and Categories in Perspective," A Century of Mathematics in America, Part I, AMS, vol 1, 1988 [Julio Gonzlez Cabilln].

Earliest Uses of Grouping Symbols.

Last revision: June 24, 1999

Vinculum below. The first use of the vinculum was in 1484 by Nicolas Chuquet (1445?-1500?) in his Le Triparty en la Science des Nombres. The bar was placed under the parts affected (Cajori vol. 1, pages 101 and 385). Chuquet wrote:

The above expression in modern notation is . This use of a vinculum appears to be the earliest use of a grouping symbol of any kind mentioned by Cajori.

Vinculum above. According to Cajori, the first use of the vinculum above the parts affected was by Frans van Schooten (c. 1615-1660), who "in editing Vieta's collected works, discarded the parentheses and placed a horizontal bar above the parts affected." In Van Schooten's 1646 edition of Vieta, is used to represent B(D2 + BD). Ball (page 242) says the vinculum was introduced by Francois Vieta (1540-1603) in 1591. This information may be incorrect.

Grouping expressed by letters. In the late fifteenth century and in the sixteenth century various writers used letters or words to indicate grouping. The earliest use of such a device mentioned by Cajori (vol. 1, page 385) is the use of the letter v for vniversale by Luca Paciolo (or Pacioli) (c. 1445 - prob. after 1509) in his Summa of 1494 and 1523.

Parentheses. Parentheses ( ) are "found in rare instances as early as the sixteenth century" (Cajori vol. 1, page 390). Apparently the earliest work Cajori names in which round parentheses are found is General trattato di numeri e misure by Nicolo Tartaglia (c. 1506-1557) in 1556. Round parentheses occur once in Ars magna by Cardan, as printed in Opera (1663) (Cajori vol. 1, page 392; Cajori does not indicate whether the parentheses occur in the original 1545 edition). Cajori (vol. 1, page 391) says that Michael Stifel (1487 or 1486 - 1567) does not use parentheses as signs of aggregation in his printed works, but that they are found in one of his handwritten marginal notes. Cajori expresses the opinion that these parentheses are actually punctuation marks rather than mathematical symbols. Kline says parentheses appear in 1544. He presumably refers to Arithmetica integra by Michael Stifel.

Brackets. Brackets [ ] are found in the manuscript edition of Algebra by Rafael Bombelli (1526-1573) from about 1550 (Cajori vol. 1, page 391). Ball (page 242) and Lucas say brackets were introduced by Albert Girard (1595-1632) in 1629. This information appears to be inaccurate. Kline says square brackets were introduced by Vieta (1540-1603). He presumably refers to the 1593 edition of Zetetica, which according to Cajori uses both braces and brackets.

Braces. Braces { } are found in the 1593 edition of Francois Vieta's Zetetica (Cajori vol. 1, page 391).

Grouping symbols in numeration. In the writing of large numbers, various methods have been used to separate numerals into groups, including dots, vertical bars, commas, arcs, colons, and accent marks.

In 1202, Leonardo of Pisa in Liber Abaci directs that the hundreds, hundred thousands, hundred millions, etc., be marked with an accent mark above, and that thousands, millions, thousands of millions, etc., be marked with an accent below (Cajori vol. 1, page 58). The earliest example of the modern system of simply separating the numeral into groups of three with commas shown by Cajori is in 1795 in the article "Numeration" in Mathematical and Philosophical Dictionary by Charles Hutton.

Earliest Uses of Symbols of Relation

Last updated: July 29, 2001

Equality. In printed books before the modern equal sign, equality was usually expressed with a word, such as aequales, aequantur, esgale, faciunt, ghelijck, or gleich, and sometimes by the abbreviated form aeq (Cajori vol. 1, page 297).

The equal symbol (=) was first used by Robert Recorde (c. 1510-1558) in 1557 in The Whetstone of Witte. He wrote, "I will sette as I doe often in woorke use, a paire of parralles, or Gemowe lines of one lengthe, thus : ==, bicause noe 2, thynges, can be moare equalle." Recorde used an elongated form of the present symbol. He proposed no other algebraic symbol (Cajori vol. 1, page 306).

Here is an image of the page of The Whetstone of Witte on which the equal sign is introduced.

The equal symbol did not appear in print again until 1618, when it appeared in an anonymous Appendix, very probably due to Oughtred, printed in Edward Wright's English translation of Napier's Descriptio. It reappeared 1631, when it was used by Thomas Harriot and William Oughtred (Cajori vol. 1, page 298).

Cajori states (vol. 1, page 126):

A manuscript, kept in the Library of the University of Bologna, contains data regarding the sign of equality (=). These data have been communicated to me by Professor E. Bortolotti and tend to show that (=) as a sign of equality was developed at Bologna independently of Robert Recorde and perhaps earlier. Cajori elsewhere writes that the manuscript was probably written between 1550 and 1568.

Less than and greater than. The symbols < and > first appear in Artis Analyticae Praxis ad Aequationes Algebraicas Resolvendas (The Analytical Arts Applied to Solving Algebraic Equations) by Thomas Harriot (1560-1621), which was published posthumously in 1631: "Signum majoritatis ut a > b significet a majorem quam b" and "Signum minoritatis ut a < b significet a minorem quam b."

According to Johnson (page 144), while Harriot was surveying North America, he saw a native American with this symbol on his arm: . Johnson says it is likely he developed the two symbols from this symbol.

However, Seltman and Mizzy say that Harriot himself did not use the symbols which appear in the work, which was published after his death:

We now know that Harriot was not directly responsible for the Praxis, which was put together after his death from papers which are no longer extant by Walter Warner (and, perhaps, one or two others), when Nathaniel Torporley had failed to complete the task which had been assigned to him in Harriot's will. Torporley was a respected mathematician of the day, reputed to have been associated with Vite himself. The manuscripts that we do have (in the British Library and Petworth) cannot have been the origin of the Praxis, not only on account of their disorder and incoherence, but also because there are significant differences between them and the published work. Notably, the inequality signs associated with his name are never found in his handwriting in the manuscripts but appear throughout as and . Similarity, equality is denoted in the manuscripts by II and not by = (the sign introduced by Robert Recorde), as in the Praxis. The significance of the inequality signs lies in the fact that this is the first time that such signs were used and accorded the same status as the equality sign.

Less than or equal to, greater than or equal to. Pierre Bouguer (1698-1758) used and in 1734 (Ball). In 1670, John Wallis used similar symbols each with a single horizontal bar, but the bar was above the < and > rather than below it (Cajori vol. 2, page 118). Cajori apparently does not show a use of the modern symbols with the single horizontal bar.

Not equal to, not greater than, not less than. These symbols were "employed, if not invented by, Euler" (Ball, page 242). Ball shows the symbol rather than .Is nearly equal to. The symbol was used in 1875 by Anton Steinhauser in Lehrbuch der Mathematik, "Algebra" (Cajori vol. 2, page 256). The same symbol was used in 1832 by Wolfgang Bolyai to signify absolute equality (Cajori vol. 1, page 307).

Proportion. The symbol :: was introduced by William Oughtred (1574-1660) in Clavis Mathematicae, composed about 1628 and published in London in 1631. He wrote a proportion as a.b::c.d (Gullberg). The astronomer Vincent Wing (1619-1668) used colons to write a proportion in the modern notation, as A:B::C:D, in 1651 in Harmonicon Celeste (Cajori vol. 1, page 286). The symbol for variation (an eight lying on its side with a piece removed) was introduced in 1768 by W. Emerson in Doctrine of Fluxions (3d ed., London) (Cajori vol. 1, page 297).

Earliest Uses of Symbols for Fractions

Last revision: Dec. 30, 2000

Earliest notations for fractions. The Babylonians wrote numbers in a system which was almost a place-value (positional) system, using base 60 rather than base 10. Their place value system of notation made it easy to write fractions. The numeral

has been found on an old Babylonian tablet from the Yale collection. It is an approximation for the square root of two. The symbols are 1, 24, 51, and 10. Because the Babylonians used a base 60, or sexagesimal, system, this number is 1 x 600 + 24 x 60-1 + 51 x 60-2 + 10 x 60-3, or about 1.414222.

The Babylonian system of numeration was not a pure positional system because of the absence of a symbol for zero. In the older tablets, a space was placed in the appropriate place in the numeral; in some later tablets, a symbol for zero does appear but in the tablets which have been discovered, this symbol only used between other symbols and never in a terminal position.

The earliest Egyptian and Greek fractions were usually unit fractions (having a numerator of 1), so that the fraction was shown simply by writing a numeral with a mark above or to the right indicating that the numeral was the denominator of a fraction.

Ancient Rome. The Romans did not use numerals to indicate fractions, but instead used words to indicate parts of a whole. A unit of weight was the as and the uncia (from which we have the word "ounce") was a twelfth part of the as. The following words were used to indicate parts of the as or, more generally, parts of any quantity:

11/12 deunx for de uncia, 1/12 taken away

10/12 dextans for de sextans, 1/6 taken away

9/12 dodrans for de quadrans, 1/4 taken away

8/12 bes bi as for duae partes, 2/3

7/12 septunx for septem unciae

6/12 semis

5/12 quincunx for quinque unciae

4/12 triens

3/12 quadrans

2/12 sextans

1/12 uncia

1/24 semuncia

1/48 sicilicus

1/72 scriptulum

1/144 scripulum

1/288 scrupulum

Multiples of the as were indicated using the following scheme, in which a denarius represents 16 asses. Denarii semuncia sicilicus represented 1/24 + 1/48 of a denarius or 1/16 denarius, or 1 as. Denarii uncia semuncia represented 1/12 + 1/24 of a denarius or 1/8 denarius, or 2 asses. Denarii sextans sicilicus represented 1/6 + 1/48 of a denarius, or 3/16 denarius, or 3 asses. Denarii deunx sicilicus represented 11/12 + 1/48 of a denarius, or 15/16 denarius, or 15 asses [Smith vol. 2, pages 208-209].

Ordinary fractions without the horizontal bar. According to Smith (vol. 2, page 215), it is probable that our method of writing common fractions is due essentially to the Hindus, although they did not use the bar. Brahmagupta (c. 628) and Bhaskara (c. 1150) wrote fractions as we do today but without the bar.

The horizontal fraction bar was introduced by the Arabs. "The Arabs at first copied the Hindu notation, but later improved on it by inserting a horizontal bar between the two numbers" (Burton).

Several sources attribute the horizontal fraction bar to al-Hassar around 1200.

When Rabbi ben Ezra (c. 1140) adopted the Moorish forms he generally omitted the bar.

Fibonacci (c.1175-1250) was the first European mathematician to use the fraction bar as it is used today. He followed the Arab practice of placing the fraction to the left of the integer (Cajori vol. 1, page 311).

The bar is generally found in Latin manuscripts of the late Middle Ages, but when printing was introduced it was frequently omitted, doubtless owing to typographical difficulties. This inference is confirmed by such books as Rudolff's Kunstliche rechnung (1526), where the bar is omitted in all ordinary fractions but is inserted in fractions printed in larger type and those having large numbers (Smith vol. 2, page 216).

Michael Closs points out that if we define a horizontal fraction bar to be a horizontal line that separates the numerator from the denominator and demarcates them as such, then this type of notation was used with exactly that purpose more than a millennium before al-Hassar. In Demotic Mathematical Papyri, (Brown University Press, London, 1972, pages 8-9) Richard A. Parker writes that in three papyri dating from the third century B. C. to the Roman period, "the numerator is written first, and the denominator follows on the same line. In problems 2, 3, 10, and 13 (the Cairo papyrus) the numerator is underlined. In problems 51 and 72 the denominator is underlined."

Some writers use the term vinculum for the horizontal fraction bar. This term originally applied to the mark when used as a grouping symbol. Fibonacci used the Latin word virga for the horizontal fraction bar.

The diagonal fraction bar (also called a solidus or virgule) was introduced because the horizontal fraction bar was difficult typographically, requiring three terraces of type.

An early handwritten document with forward slashes in lieu of fraction bars is Thomas Twining's Ledger of 1718, where quantities of tea and coffee transactions are listed, e.g. 1/4 pound green tea. This usage of the horizontal fraction bar was found by Hans Lausch, who believes there are likely even earlier occurrences.

Lausch has also found the horizontal fraction bar in Allgemeine Deutsche Bibliothek, a Berlin review journal which was started in 1765. A precise reference may be forthcoming.

The earliest instance of a diagonal fraction bar shown by Cajori (vol. 1, page 313) is in 1784, when a curved line resembling the sign of integration was used in the Gazetas de Mexico by Manuel Antonio Valdes.

In 1843, a curved line was used by Henri Cambuston in Definicion de las principales operaciones de arismetica (Cajori vol. 1, page 313)

In 1845, the use of the solidus was recommended by De Morgan in an article "The Calculus of Functions" published in the Encyclopaedia Metropolitana of 1845 (Cajori vol. 1, page 313).

In 1852, the solidus was used by Antonio Serra Y Oliveres in Manuel de la Tipografia Espaola (Cajori vol. 1, page 313).

Decimal fractions. Abu'l Hasan Ahmad ibn Ibrahim Al-Uqlidisi (c. 920-c. 980) wrote the earliest known text offering a direct treatment of decimal fractions. "Al-Uqlidisi uses decimal fractions as such, appreciates the importance of a decimal sign, and suggests a good one," according to A. S. Saidan, "The earliest extant Arabic arithmetic," Isis 57 (1966), 475-490.

The idea of decimal fractions had been present in the work of several mathematicians of al-Karaji's school, in particular Ibn Yahya al-Maghribi Al-Samawal (c. 1130-c. 1180), according to the University of St. Andrews website.

In The Key to Arithmetic, Ghiyath al-Din Jamshid Mas'ud al-Kashi (c. 1380-1429) gave a clear description of decimal fractions, according to P. Luckey, Die Rechnenkunst bei Gamsid b. Masud al-Kasi (1951).

Al-Kashi in his al-Risali al-mohitije (Treatise on the circumference) wrote the value of pi using Arabic characters as follows:

sah-hah 3 1415926535898732

The word sah-hah meant complete, correct, integral. (The modern Turkish form is sahih.) Thus the part at the right is the decimal, although there is no decimal point. According to Smith (vol. 2, page 240), "Manifestly it is, therefore, a clear case of a decimal fraction, and it seems to be earlier than any similar one to be found in Europe."

Yang Hui (c. 1238-c. 1298) was a minor Chinese official who wrote two books, dated 1261 and 1275, which use decimal fractions (in the modern form). The 1275 work is called Cheng Chu Tong Bian Ben Mo [University of St. Andrews website]. Smith (vol. 1, page 255) writes, "Francesco Pellos or Pellizzati, a native of Nice, published a commercial arithmetic at Turin in 1492 in which, as will be shown in Volume II, use is made of a decimal point to denote the division of a number by a power of ten." In vol. 2 (page 138) Smith says Pellos "unwittingly made use of the decimal point for the first time in a printed work" and that he "did not recognize the significance of the decimal point." Cajori (vol. 1, page 315) says Pellos "used a point and came near the invention of decimal fractions."

In 1530, Christoff Rudolff (1499?-1545?) used a vertical bar exactly as we use a decimal point today in setting up a compound interest table in the Exempel Bchlin (Cajori vol. 1, page 316).

Smith (vol. 2, page 240) writes:

The first man who gave evidence of having fully comprehended the significance of all this preliminary work seems to have been Christoff Rudolff, whose Exempel Bchlin appeared at Augsburg in 1530. In this work he solved an example in compound interest, and used the bar precisely as we should use a decimal point today. If any particular individual were to be named as having the best reason to be called the inventor of decimal fractions, Rudolff would seem to be the man, because he apparently knew how to operate with these forms as well as merely to write them, as various predecessors had done. His work, however, was not appreciated, and apparently was not understood, and it was not until 1585 that a book upon the subject appeared.

In 1579 Francois Vieta (1540-1603) published a work which included a systematic use of decimal fractions, using a vertical stroke as a separator; "from the vertical stroke to the actual comma there is no great change" (Cajori vol. 1, page 316).

In 1585 Simon Stevin (or Stevinus) (1548-1620) published La Thiende ("The Tenth") and La Disme ("The Decimal"), both of which explained the use of decimal fractions. He is credited with introducing decimal fractions into common use, although he did not use the notation we use today. He wrote 5.912 as or .

Boyer writes (on page 340):

The use of a decimal point separatrix generally is attributed either to G. A. Magini (1555-1617), a map-making friend of Kepler and rival of Galileo for a chair at Bologna, in his De planis triangulis of 1592, or to Christoph Clavius (1537-1612), a Jesuit friend of Kepler, in a table of sines of 1593. But the decimal point did not become popular until Napier used it more than twenty years later.

Jobst Brgi (1552-1632) "was not clear as to the best method of representing these fractions, however, and in his manuscript of 1592 he used both a period and a comma for the decimal point" (Smith vol. 2, page 243-244). He also used instead a small circle placed above or below the units digit (Smith vol. 2, page 244 and Cajori (vol. 1, page 317).

In 1593 Christopher Clavius (1537-1612) used a period to separate the units and tenths digits in a table of sines in Astrolabe. However, he used the period for other reasons in his works, and his purpose in using the period in this case is not clear (Cajori vol. 1, page 322). Carl Boyer says Clavius was the first person to use the decimal point with a clear idea of its significance.

William Oughtred (1574-1660) did not use a decimal point, but instead wrote 0.56 as 0/56, with the 56 underlined.

The dot as a separator occurs in 1616 in E. Wright's translation of John Napier's Descriptio. Boyer refers to this as the first appearance of a decimal point separating the whole number part from the decimal part, in the notation we use today. However Cajori (vol. 1, page 323) says "no evidence has been advanced, thus far, to show that the sign was intended as a separator of units and tenths, and not as a more general separator as in Pitiscus." According to Scott (p. 128), "Wright's translation of his treatise on logarithms, which was published in 1616 shows the decimal point on the first page."

In 1617 in his Latin Rabdologia, Napier used both the comma and the period as separators of units and tenths. Before 1617, he used the period in his Constructio, which was not published until 1619 (Cajori vol. 1, page 324).

The percent symbol is believed to have evolved from a symbol introduced in an anonymous Italian manuscript of about 1425, according to D. E. Smith in Rara arithmetica in 1898.

Earliest Uses of Symbols for Constants

Last revision: March 24, 2001

for 3.14159... Early writers indicated this constant as a ratio of two values. William Oughtred (1574-1660) designated the ratio by the fraction lower case pi over lower case delta in Clavis mathematicae. The symbolism appears in the editions of this book of 1647, 1648, 1652, 1667, 1693, and 1694 (Cajori vol. 2, page 9).

Cajori writes that "perhaps the earliest use of a single letter to represent the ratio of the length of a circle to its diameter" occurs in 1689 in Mathesis enucleata by J. Christoph Sturm, who used e for 3.14159....

si diameter alicuius circuli ponatur a, circumferentiam appellari posse ea (quaecumque enim inter eas fuerit ratio, illius nomen potest designari littera e).

Cajori cites a note by A. Krazer in Euleri opera omnia as a reference for the above.

The first person to use to represent the ratio of the circumference to the diameter (3.14159...) was William Jones (1675-1749) in 1706 in Synopsis palmariorum mathesios. It is believed he used the Greek letter pi because it is the first letter in perimetron (= perimeter). From Cajori (vol. 2, page 9):

The modern notation for 3.14159 .... was introduced in 1706. It was in that year that William Jones made himself noted, without being aware that he was doing anything noteworthy, through his designation of the ratio of the length of the circle to its diameter by the letter . He took this step without ostentation. No lengthy introduction prepares the reader for the bringing upon the stage of mathematical history this distinguished visitor from the field of Greek letters. It simply came, unheralded, in the following prosaic statement (p. 263):

"There are various other ways of finding the Lengths or Areas of particular Curve Lines, or Planes, which may very much facilitate the Practice; as for instance, in the Circle, the Diameter is to the Circumference as 1 to , &c. = 3.14159, &c. = . This series (among others for the same purpose, and drawn from the same Principle) I received from the Excellent Analyst, and my much esteem'd Friend Mr. John Machin; and by means thereof, Van Ceulen's Number, or that in Art. 64.38 may be Examin'd with all desirable Ease and Dispatch."

In 1734 Leonhard Euler (1707-1783) employed p instead of in "De summis serierum reciprocarum."

In a letter of April 16, 1738, from Stirling to Euler, as well as in Euler's reply, the letter p is used.

In 1736 in Mechanica sive motus scientia analytice exposita, Euler used 1 : and "thus either consciously adopted the notation of Jones or independently fell upon it" (Cajori vol. 2, page 10). Euler wrote, "Si enim est m = 1/2 terminus respondens inuenitur /2 denotante 1 : rationem diametri ad peripheriam." But the letter is not restricted to this use in his Mechanica, and the definition of is repeated when it is taken for 3.14159...

In 1737 Euler used for 3.14159... in a letter, and again in various letters in 1737, 1738, and 1739.

Johann Bernoulli used c in 1739, in his correspondence with Euler, but in a letter of 1740 he began to use .

In 1741 was used in H. Sherwin's Mathematical Tables.

Nikolaus Bernoulli employed in his letters to Euler of 1742.

Euler popularized the use of by employing it in 1748 in Introductio in Analysin Infinitorum:

Satis liquet Peripheriam hujus Circuli in numeris rationalibus exacte exprimi non posse, per approximationes autem inventa est .. esse = 3,14159 [etc., to 128 places], pro quo numero, brevitatis ergo, scribam , ita ut sit = Semicircumferentiae Circuli, cujus Radius = 1, seu erit longitudo Arcus 180 graduum.

The base of natural logarithms. This constant, 2.71828..., was referred to in Edward Wright's English translation of Napier's work on logarithms, published in 1618.

The first symbol used for the constant mentioned by Cajori is the letter b used by Leibniz in letters to Huygens in 1690 and 1691.

Leonhard Euler (1707-1783) introduced e for this constant in a manuscript, Meditatio in Experimenta explosione tormentorum nuper instituta (Meditation on experiments made recently on the firing of cannon), written at the end of 1727 or the beginning of 1728 (when Euler was just 21 years old). The manuscript was first printed in 1862 in Euler's Opera postuma mathematica et physica, Petropoli, edited by P. H. Fuss and N. Fuss (vol ii, pp. 800-804). The manuscript describes seven experiments performed between August 21 and September 2, 1727:

For the number whose logarithm is unity, let e be written, which is 2,7182817... [sic] whose logarithm according to Vlacq is 0,4342944... [translated from Latin by Florian Cajori].

Euler next used e in a letter addressed to Goldbach on November 25, 1731, writing that e "denotes that number whose hyperbolic logarithm is = 1."

The earliest appearance of e in a published work was in Euler's Mechanica (1736), in which he laid the foundations of analytical mechanics (Maor, p. 156).

Maor writes (page 156):

Why did he choose the letter e? There is no general consensus. According to one view, Euler chose it because it is the first letter of the word exponential. More likely, the choice came to him naturally as the first "unused" letter of the alphabet, since the letters a, b, c, and d frequently appear elsewhere in mathematics. It seems unlikely that Euler chose the letter because it is the initial of his own name, as occasionally been suggested: he was an extremely modest man and often delayed publication of his own work so that a colleague or student of his would get due credit. In any event, his choice of the symbol e, like so many other symbols of his, became universally accepted.

Ball says: "It is probable that the choice of e for a particular base was determined by its being the vowel consecutive to a."

According to Boyer (page 494), this notation was "suggested perhaps by the first letter of the word 'exponential.'"

In a post in sci.math in 1995, Wei-hwa Huang wrote: "I believe that e was not named because it was the first letter in Euler's name, but rather because he was using vowels for constants in a proof of his and e happened to be the second one."

In a post to a history of mathematics list in 1999, Olivier Gerard wrote: "The hypothesis made by my friend Etienne Delacroix de La Valette was that e was for 'ein' (one in German) or 'Einheit' (unity), which would be matching the sentence Euler uses to define it (whose logarithm is unity). As always, many explanations may be true at the same time."

Several textbooks claim that the letter e was chosen to honor Euler. Cajori has no information to support this claim, and in fact the earliest uses of e were by Euler himself.

The early uses of symbols for 2.718... mentioned by Cajori are as follows:

1690 b Leibniz Letter to Huygens

1691 b Leibniz Letter to Huygens

1703 a A reviewer Acta eruditorum

1727/8 e Euler Meditatio in Experimenta explosione tormentorum nuper instituta

1736 e Euler Mechanica sive motus scientia analytice exposita

1747 c D'Alembert Histoire de l'Acadmie

1747 e Euler various articles

1751 e Euler various articles

1760 e Daniel Bernoulli Histoire de l'Acadmie r. d. sciences

1763 e J. A. Segner Cursus mathematici

1764 c D'Alembert Histoire de l'Acadmie

1764 e J. H. Lambert Histoire de l'Acadmie r. d. sciences et d. belles lettres

1771 e Condorcet Histoire de l'Acadmie

1774 e Abb Sauri Cours de mathmatiques

1775 e J. A. Fas Inleiding tot de Kennisse en het gebruyk der Oneindig Kleinen

1782 e P. Frisi Operum tomus primus

1787 c Daniel Melandri Nova Acta Helvetica physico-mathematica

The term Napier's constant has been suggested for 2.718... The name Euler's constant may be inappropriate for this number, as the number was known before Euler's birth and Euler's constant more frequently is used to refer to 0.577....

Benjamin Peirce suggested the innovative notation for and e shown below:

From J. D. Runkin's Mathematical Monthly, vol. I, No. 5, Feb. 1859A somewhat modified notation, shown below, appears in the Century Dictionary (1889-1897) in the entry notation:

Euler-Mascheroni constant. Leonhard Euler (1707-1783) used C in De progressionibus harmonicis observationes, Commentarii academiae scientiarum petropolitanae 7 (1734-35), published in 1740, pp.150-161. Reprinted in Opera omnia (1) 14, pp. 87-100.

According to Cajori (vol. 2, page 32), Mascheroni used A in Adnotationes ad calculum integralem Euleri (1790-1792).

According to a new book by William Dunham, Euler, the Master of Us All (1999), Mascheroni introduced the symbol for the Euler-Mascheroni constant. Dunham has kindly supplied this website with a copy of the paper, "On the History of Euler's Constant," which is the source of this information. [The paper, by J. W. L. Glaisher, appeared in 1872 in The Messenger of Mathematics.] In the paper, Glaisher does not specify where Mascheroni used the symbol, but seems to imply it is in Adnotationes ad Euleri Calculum Integralem, which Glaisher indicates in a footnote is a work he has not seen but which is referred to in volume 3 of Lacroix's Differential and Integral Calculus.

Gauss used the Greek letter psi.

Julio Gonzlez Cabilln has found in "Theoriae logarithmi integralis lineamenta nova," an essay submitted by Carl Anton Bretschneider (1808-1878) on October 13, 1835, to Crelle's Journal. The article was published in volume 17, pp. 257-285, 1837. The symbol itself can be found on page 260.

DeMorgan used , according to J. W. L. Glaisher in "On the History of Euler's Constant" (1872) in The Messenger of Mathematics.

W. Shanks used "E. or Eul. constant" in Proc. Royl. Soc. of London, Vol XV (1867).

The letter E was adopted by J. W. L. Glaisher in 1871 and J. C. Adams in 1878.

Ernst Pascal retained Mascheroni's notation A in 1900 (Cajori vol. 2, page 32).

In vol. VII of "L'Intermediaire des mathematiciens" (1900), G. Vacca (from Turin) asks who introduced . His question #1998 reads as follows:

In the German "Encyclopaedie" (1900, vol. II, p. 171) it says that Mascheroni has denoted the Euler's constant 0.577... by . According to my research, this author designated it by letter "A".

In "Synopsis" of Mr. Hagen (1891, vol. I, p. 86), it is said that Euler has introduced this symbol in "Acta Petr." (1769, vol. XIV).

Mr. E. Pascal, in his "Repertorium" (1900, vol. I, p. 478 of the German edition) reproduces that suggestion. But, in the quoted volume, and in many memoirs of Euler, I have found that this author has just used the symbol "C", 'et parfois' "O".

Who is the first mathematician that has introduced the symbol for the Euler's constant?

In his Adnotationes ad calculum integrale Euleri (1790), Lorenzo Mascheroni (1750-1800) calculated the constant to 32 decimal places:

.57721 56649 01532 86061 81120 90082 39...

In 1809 Johann von Soldner (1766-1833) published his "Thorie d'un nouvelle fonction transcendante," in which his value of the constant given on page 13 is:

.57721 56649 01532 86060 6065...

which differs from Mascheroni's value at the twentieth decimal place. In 1812 Gauss asked F. G. B. Nicolai (1793-1846), "juvenem in calculo indefessum," to check their results, and he agreed with von Soldner. A note in a memoir by Gauss which contained the results of Nicolai's calculation apparently attracted little attention and Mascheroni's value was repeatedly quoted thereafter (Glaisher).

for the golden ratio. According to The Curves of Life: Being an Account of Spiral Formations and Their Application to Growth in Nature, to Science, and to Art: With Special Reference to the Manuscripts of Leonardo da Vinci (1914) by Sir Theodore Andrea Cook (1867-1928), page 420:

Mr. Mark Barr . . . suggested . . . that this ratio should be called the phi proportion for reasons given below . . . The symbol phi was given to this proportion partly because it has a familiar sound to those who wrestle constantly with pi and partly because it is the 1st letter of the name of Pheidias, in whose sculpture this sculpture is seen to prevail when the distance between salient points are measured.

The above quotation and citation were provided by Samuel S. Kutler and Julio Gonzlez Cabilln. Barr was an American mathematician.

According to Gardner (1961) and Huntley, the letter phi was chosen because it is the first letter in the name of Phidias who is believed to have used the golden proportion frequently in his sculpture. However, Schwartzman (page 164) implies the letter stands for Fibonacci.

The Greek letter tau is also used for this constant. Tau is found in 1948 in Regular Polytopes by Harold Scott MacDonald Coxeter, according to John Conway, who believes Coxeter may have used the symbol in his papers of the 1920s and 1930s. Ball and Coxeter (1987, page 57) write, "The symbol [tau] is appropriate because it is the initial of tomh\ ("section") [Antreas P. Hatzipolakis].

H. v. Baravalle used G for 0.618... in "The Geometry of the Pentagon and the Golden Section," which appeared in The Mathematics Teacher in January 1948. He may have used the same symbol in his "Die Geometrie des Pentagrammes und der Goldene Schnitt" in 1932.

In The Shape of the Great Pyramid (1999), Roger Herz-Fischler uses G for 1.618... and g for .618....

i for the imaginary unit was first used by Leonhard Euler (1707-1783) in a memoir presented in 1777 but not published until 1794 in his "Institutionum calculi integralis."

On May 5, 1777, Euler addressed to the 'Academiae' the paper "De Formulis Differentialibus Angularibus maxime irrationalibus quas tamen per logarithmos et arcus circulares integrare licet," which was published posthumously in his "Institutionum calculi integralis," second ed., vol. 4, pp. 183-194, Impensis Academiae Imperialis Scientiarum, Petropoli, 1794.

Quoniam mihi quidem alia adhuc via non patet istud praestandi nisi per imaginaria procedendo, formulam littera i in posterum designabo, ita ut sit ii = -1 ideoque 1/i = -i.

According to Cajori, the next appearance of i in print is by Gauss in 1801 in the Disquisitiones Arithmeticae. Carl Boyer believes that Gauss' adoption of i made it the standard. By 1821, when Cauchy published Cours d'Analyse, the use of i was rather standard, and Cauchy defines i as "as if was a real quantity whose square is equal to -1."

Throughout his Introductio, Euler consistently writes , denoting by i the "numerus infinite magnus" [namely, an infinitely large number]. Nonetheless, there are very few occasions where Euler chose i with a different meaning. Thus, chapter XXI (volume 2) of Euler's Introductio contains the first appearance of i as quantitas imaginaria:

Cum enim numerorum negativorum Logarithmi sint imaginarii (...) erit log(-n) quantitas imaginaria, quae sit = i.

The citation above is from "Introductio in analysin infinitorum," Lausannae, Apud Marcum-Michaelem Bousquet & socios, M.DCC.XLVIII (1748).

Please note that, in this fascinating passage about logarithms, Euler does not introduce the symbol i such that i2 = -1.

[This entry was contributed by Julio Gonzlez Cabilln.]

Older symbols for zero. The following is taken from a paper "Africa, Cradle of Mathematics" by Beatrice Lumpkin:

It is well known that a zero placeholder was not used or needed in Egyptian numerals, a system of numerals without place value. Still historians such as Boyer and Gillings have found examples of the use of the zero concept in ancient Egypt. But Gillings added, "Of course zero, which had not yet been invented, was not written down by the scribe or clerk; in the papyri, a blank space indicates zero." However, some Egyptologists did know that the ancient Egyptians used a zero symbol, but it may have been missed by historians of mathematics because the symbol did not appear in the surviving mathematical papyri.

The Egyptian zero symbol was a triliteral hieroglyph, with consonant sounds [symbol]. This was the same hieroglyph used to represent beauty, goodness, or completion. There are two major sources of evidence for an Egyptian zero symbol:

1. Zero reference level for construction guidelines. Massive stone structures such as the ancient Egyptian pyramids required deep foundations and careful leveling of the courses of stone. Horizontal leveling lines were used to guide the construction. One of these lines, often at pavement level, was used as a reference and was labeled [symbol], or zero. Other horizontal leveling lines were spaced 1 cubit apart and labeled as 1 cubit above [symbol], 2 cubits above [symbol], or 1 cubit, 2 cubits, 3 cubits, and so forth, below [symbol]. Here zero was used as a reference for directed or signed numbers.

In 1931, George Reisner described the terms used to label the leveling lines at the Mycerinus (Menkure) pyramid at Giza, built c. 2600 BCE. He gave the following list collected earlier by Borchardt and Petrie from their study of Old Kingdom pyramids. [...]

2. Bookkeeping, zero remainders. A bookkeeper's record from the 13th dynasty c 1700 BCE shows a monthly balance sheet for items received and disbursed by the royal court during its travels. On subtracting total disbursements from total income, a zero remainder was left in many columns. This zero remainder was represented with the same symbol, [symbol], as used for the zero reference line in construction.

These practical applications of a zero symbol in ancient Egypt, a society which conventional wisdom believed did not have a zero, may encourage historians to reexamine the everyday records of ancient cultures for mathematical ideas that have been overlooked.

According to Milo Gardner, Mesoamericans used a fully positional base 4, 5 system, with zero as a place holder, counting 0-19, as early as 1,000 BC.

3 As early as the fourth century BC, the Chinese represented zero as a blank space on a counting board (Johnson, page 160).

Babylonians in the Seleucid period (300 BC onward) used a symbol for zero, mainly in astronomical texts, and never in final positions. According to Neugebauer in The Exact Sciences in Antiquity, Dover, 2nd edition, 1969, p. 20:

The Babylonian place value notation shows in its earlier development two disadvantages which are due to the lack of a symbol for zero. The first difficulty consists in the possibility of misreading a number 1 20 as 1,20 = 80 when actually 1,0,20 = 3620 was meant. Occasionally this ambiguity is overcome by separating the two numbers very clearly if a whole sexagesimal place is missing. But this method is by no means strictly applied and we have many cases where numbers are spaced widely without any significance. In the latest period, however, when astronomical texts were computed, a special symbol for "zero" was used. This symbol also occurs earlier as a separation mark between sentences, and I therefore transcribe it by a "period". Thus we find in Seleucid astronomical texts many instances of numbers like 1, . , 20 or even 1, . , . ,20 which apply exactly the same principle as, e.g.,our 201 or 2001.

But even in the final phase of Babylonian writing we do not find any examples of zero signs at the end of numbers. Though there are many instances of cases like . ,20 there is no safe example of a writing like 20, . known to me. In others words, in all periods the context alone decides the absolute value of a sexagesimal written number.

However, Georges Iffrah (Histoire Universelle des Chiffres, Seghers, Paris, 1981, p. 400-401) writes that Babylonian astronomers used the zero not only in the intermediate positions but also in initial or final positions, and he gives Neugebauer as the source of this information.

According to Boyer (p. 29-30), "The Babylonians seem at first to have had no clear way in which to indicate an "empty" position--that is, they did not have a zero symbol, although they sometimes left a space where a zero was intended. ... By about the time of the conquest by Alexander the Great, however, a special sign, consisting of two small wedges placed obliquely, was invented to serve as a placeholder where a numeral was missing. ... The Babylonian zero symbol apparently did not end all ambiguity, for the sign seems to have been used for intermediate empty positions only. There are no extant tablets in which the zero sign appears in a terminal position. This means that the Babylonians in antiquity never achieved an absolute positional system."

Burton says that about A. D. 150, the Alexandrian astronomer Ptolemy began using the omicron [which looks something like a zero] in the manner of our zero, not only in a medial but also in a terminal position. He says there is no evidence that Ptolemy regarded the symbol as a number by itself that could enter into computations with other numbers. Omicron is the first letter of the Greek word for "nothing." However, Len Berggren says, "Ptolemy probably did not use omicron to denote 0. Papyri from the period when Ptolemy lived show a small 'o' with a bar over it as the symbol for 0, and the small 'o' alone doesn't come in until the Byzantine period. Even in that period Neugebauer considers it unlikely that the small 'o' stood for the Greek word ouden (= nothing). See the discussion in the second edition of Neugebauer's Exact Sciences in Antiquity, esp. pp. 13 - 14."

The oldest Maya artifact employing both positional notation and a zero is Pestac, Stela 1, with a contemporaneous date of Feb. 8, AD 665. The oldest Maya artifacts employing a zero but not positional notation are Uaxactun, Stelae 18 and 19, with a contemporaneous date of AD 357. The oldest Maya artifact employing the same chronological system as in the previous cases but without a zero and without positional notation is Tikal, Stela 29, with a contemporaneous date of July 8, AD 292 (Michael Closs).

[This website previously showed what seemed to be an occurrence of a symbol for zero on a tablet from the Old Babylonian period, about 1800 B. C. However, it is now believed that this sole appearance was not real and may have been due to an error of the scribe who made the tablet.]

The Hindu zero symbol. The Encyclopaedia Britannica says, "Hindu literature gives evidence that the zero may have been known before the birth of Christ, but no inscription has been found with such a symbol before the 9th century."

According to Johnson (page 160), by the third century A.D., Hindu mathematicians were using a heavy dot to mark its place in calculations and the dot was eventually replaced by an empty circle.

The earliest date Bell has found to be fairly proven for the use of zero was A. D. 505 in a Pancasiddhantika by Varahamihira.

According to Iffrah (op. cit., p. 468), a heavy point for zero appears in a Khmer inscription of 683 found at Trapeang Bay, Sambor Province, Cambodia. A small circle for zero appears in 683 in an inscription found in Kedukan Bukit, Palebang, Sumatra. A small circle is also found in 684 at Talang Tuwo, Palebang, Sumatra, and in 686 at Kota Kapur, Banka Island.

According to Menninger (p. 400):

This long journey begins with the Indian inscription which contains the earliest true zero known thus far (Fig. 226). This famous text, inscribed on the wall of a small temple in the vicinity of Gvalior (near Lashkar in Central India) first gives the date 933 (A.D. 870 in our reckoning) in words and in Brahmi numerals. Then it goes on to list four gifts to a temple, including a tract of land "270 royal hastas long and 187 wide, for a flower-garden." Here, in the number 270 the zero first appears as a small circle (fourth line in the Figure); in the twentieth line of the inscription it appears once more in the expression "50 wreaths of flowers" which the gardeners promise to give in perpetuity to honor the divinity.

Manoel Almeida reports that, following Iffrah, the date of the Gvalior inscription is A. D. 876. According to Johnson (page 160), the date of the inscription is A. D. 840. Johnson says this inscription is the oldest surviving use of an empty circle for zero.

Michael Closs writes that the Gvalior inscription is not the "earliest known written zero" from India:

There are several more ancient inscriptions which also exhibit written zeros within the context of positional notation. Rabindra Nath Mukherjee, an Indian historian of mathematics, has noted several such artifacts. Unfortunately, he has provided neither photographs of the artifacts nor descriptions of the texts. I am hoping to collect this information. The oldest hard evidence which Mukherjee gives is an inscription from AD 672 in which the zero is written as a small dot. Next is an inscription from AD 683 in which the zero is written as a large dot. This is followed by an inscription from AD 684 in which the zero is written as a small circle. This is the earliest known (by me!) antecedent having the same form as our modern zero symbol.

Iffrah (op. cit., p. 464-465) mentions two cooper's letters, dated from VIII AD, with small circles to represent zeros. Both are earlier than Gwalior inscriptions, but consensus about its authenticity is lacking.

Manoel Almeida, who contributed much of the information for this article, says he suspects, due to the puzzling coincidence of dates and of the use of the same symbols for zero, that the AD 683 and AD 684 inscriptions mentioned by Mukherjee can be the Trapeang Bay and the Talang Tuwo inscriptions, quoted by Iffrah. He is interested in additional information from Mukherjee's articles.

Another source says the first zero in the Hindu system was represented by a dot and was found in a text written by Bakhshali, the date of which is unknown. The Bakhshali manuscript may have been written in the 8th or 9th century, but may have been written later and may not even be of Hindu origin (Smith, vol. 1, page 164).

According to Georges Iffrah in Histoire Universelle des Chiffres, Seghers, Paris, 1981, p.489-490, the first European to advocate the use of the Hindu zero was Abraham ben Meir ibn Ezra (1092-1167), who wrote Sfer H Mispar (The Book of the Number), in which he used the circle to represent zero. He preferred to use the first nine letters of the Hebrew alphabet rather than the nine Hindu-Arabic numbers. He called the zero Galgal (Hebrew for wheel) or Sifra (apud the arab Sifr, certainly). He also changed the old Hebrew alphabetic numeration system into a new decimal positional system like ours.

Leonardo of Pisa (1180-1250) (or Fibonacci) also advocated the use of zero, using the term zephirum in Liber Abaci.

The zero symbol first appears in print in the 1200s, according to Burton. Smith says that Ch'in Kiu-shao (or Tsin Kiu tschaou or Ts'in K'ieou-Chao) of China used 0 in 1247 or 1257 in The Nine Sections of Mathematics.

Rida A. K. Irani, in a paper in Centaurus, vol. 4 (1955), pp. 1-12 gives forms of the Hindu-Arabic characters as they occur in dated Arabic manuscripts, and shows that the medieval Arabs used a small 'o' for zero. This tends to degenerate to a dot in late (e.g. 17th century) manuscripts.

Earliest Uses of Symbols for Variables

Last revision: April 8, 2000

Greek letters. The use of letters to represent general numbers goes back to Greek antiquity. Aristotle frequently used single capital letters or two letters for the designation of magnitude or number (Cajori vol. 2, page 1).

Diophantus (fl. about 250-275) used a Greek letter with an accent to represent an unknown. G. H. F. Nesselmann takes this symbol to be the final sigma and remarks that probably its selection was prompted by the fact that it was the only letter in the Greek alphabet which was not used in writing numbers. However, differing opinions exist (Cajori vol. 1, page 71).

In 1463, Benedetto of Florence used the Greek letter rho for an unknown in Trattato di praticha d'arismetrica. (Franci and Rigatelli, p. 314)

Roman letters. In Leonardo of Pisa's Liber abbaci (1202) the representation of given numbers by small letters is found (Cajori vol. 2, page 2). The Boncompagni edition, page 455, has:

diuidatur aliquis numerus .a. in duas partes, que sint .b.g.; et diuidatur .a. per .b., et ueniet .e.; et .a. per .g. ueniet .d.: dico quod multiplicatio .d. in .e.est sicut agregatio .d.cum .e. [divide some number .a. in two parts which are .b.g.; and divide .a. by .b. to obtain .e.; and .a. by .g. to obtain .d.: I say that the product of .d. in .e. is as the sum of .d. with .e.]The dots were used to bring into prominence letters occurring in the running text, a practice common in manuscripts of that time [Barnabas Hughes; Cajori vol. 2, page 2]. Jordanus Nemorarius (1225-1260) used letters to replace numbers.

Christoff Rudolff used the letters a, c, and d to represent numbers, although not in algebraic equations, in Behend vnnd Hubsch Rechnung (1525) (Cajori vol. 1, page 136).

Michael Stifel used q (abbreviation for quantita (which Cardan had already done) but he also used A, B, C, D, and F, for unknowns in 1544 in Arithmetica integra (Cajori vol. 1, page 140).

Girolamo Cardan (1501-1576) used the letters a and b to designate known numbers in De regula aliza (1570) (Cajori vol. 1, page 120).

In 1575 Guilielmus Xylander translated the Arithmetica of Diophantus from Greek into Latin and used N (numerus) for unknowns in equations (Cajori vol. 1, page 380).

In 1591 Francois Vieta (1540-1603) was the first person to use letters for unknowns and constants in algebraic equations. He used vowels for unknowns and consonants for given numbers (all capital letters) in In artem analyticem isogoge. Vieta wrote:

Quod oopus, ut arte aliqua juventur, symbolo constanti et perpetuo ac bene conspicuo date magnitudines ab incertis quaesititiis distinguantur ut [illegible in Cajori] magnitudines quaesititias elemento A aliave litera volcali, E, I, O, V, Y [illegible in Cajori] elementis B, G, D, aliisve consonis designando. [As one needs, in order that one may be aided by a particular device, some unvarying, fixed and clear symbol, the given magnitudes shall be distinguished from the unknown magnitudes with the letter A or with another vowel E, I, O, U, Y, the given ones with the letters B, G, D or other consonants.]

(Cajori vol. 1, page 183, and vol. 2, page 5).

Thomas Harriot (1560-1621) in Artis Analyticae Praxis, ad Aequationes Algebraicas used lower case vowels for unknowns and lower case consonants for known quantities.

Descartes' use of z, y, x. The following is from Cajori (vol 1, page 381):

The use of z, y, x . . . to represent unknowns is due to Ren Descartes, in his La gometrie (1637). Without comment, he introduces the use of the first letters of the alphabet to signify known quantities and the use of the last letters to signify unknown quantities. His own langauge is: "...l'autre, LN, est (1/2)a la moiti de l'autre quantit connue, qui estoit multiplie par z, que ie suppose estre la ligne inconnue." Again: "...ie considere ... Que le segment de la ligne AB, qui est entre les poins A et B, soit nomm x, et quie BC soit nomm y; ... la proportion qui est entre les costs AB et BR est aussy donne, et ie la pose comme de z a b; de faon qu' AB estant x, RB sera bx/z, et la toute CR sera y = bx/z. ..." Later he says: "et pour ce que CB et BA sont deux quantits indetermines et inconnus, ie les nomme, l'une y; et l'autre x. Mais, affin de trouver le rapport de l'une a l'autre, ie considere aussy les quantits connus qui determinent la description de cete ligne courbe: comme GA que je nomme a, KL que je nomme b, et NL, parallele a GA, que ie nomme C." As co-ordinates he uses later only x and y. In equations, in the third book of the Gomtrie, x predominates. In manuscripts written in the interval 1629-40, the unknown z occurs only once. In the other places x and y occur. In a paper on Cartesian ovals, prepared before 1629, x alone occurs as unknown, y being used as a parameter. This is the earliest place in which Descartes used one of the last letters of the alphabet to represent an unknown. A little later he used x, y, z again as known quantities.

Some historical writers have focused their attention upon the x, disregarding the y and z, and the other changes in notation made by Descartes; these wrtiers have endeavored to connect this x with older symbols or with Arabic words. Thus, J. Tropfke, P. Treutlein, and M. Curtze advanced the view that the symbol for the unknown used by early German writers, looked so much like an x that it could easily have been taken as such, and that Descartes actually did interpret and use it as an x. But Descartes' mode of introducing the knowns a, b, c, etc., and the unknowns z, y, x makes this hypothesis improbable. Moreover, G. Enestrm has shown that in a letter of March 26, 1619, addressed to Isaac Beeckman, Descartes used the symbol as a symbol in form distinct from x, hence later could not have mistaken it for an . At one time, before 1637, Descartes used x along the side of ; at that time x, y, z are still used by him as symbols for known quantities. German symbols including the for x, as they are found in the algebra of Clavius, occur regularly in a manuscript due to Descartes, the Opuscules de 1619-1621.

All these facts caused Tropfke in 1921 to abandon his old view on the origin of x, but he now argues with force that the resemblance of x and , and Descartes' familiarity with , may account for the fact that in the latter part of Descartes' Gomtrie the x occurs more frequently than z and y. Enestrm, on the other hand, inclines to the view that the predominance of x over y and z is due to typographical reasons, type for x being more plentiful because of the more frequent occurrence of the letter x, to y and z, in the French and Latin languages.

Descartes introduced the equation ax + by = c, which is still used to describe the equation of a line (Johnson, page 145).

Johnson says (on page 145):

The predominant use of the letter x to represent an unknown value came about in an interesting way. During the printing of La Geometrie and its appendix, Discours de La Methode, which introduced coordinate geometry, the printer reached a dilemma. While the text was being typeset, the printer began to run short of the last letters of the alphabet. He asked Descartes if it mattered whether x, y, or z was used in each of the book's many equations. Descartes replied that it made no difference which of the three letters was used to designate an unknown quantity. The printer selected x for most of the unknowns, since the letters y and z are used in the French language more frequently than is x.

There are, however, other explanations for Descartes' use of x, y, and z for unknowns. For example, the in the definition of x in Webster's New International Dictionary (1909-1916) and the subsequent second edition of the same dictionary, it is claimed that "X was used as an abbreviation for Arabic shei a thing, something, which, in the Middle Ages, was used to designate the unknown, and was then prevailingly transcribed as xei." Cajori says there is no evidence for this.

According to the Oxford English Dictionary (2nd ed.):

The introduction of x, y, z as symbols of unknown quantities is due to Descartes (Gomtrie, 1637), who, in order to provide symbols of unknowns corresponding to the symbols a, b, c of knowns, took the last letter of the alphabet, z, for the first unknown and proceeded backwards to y and x for the second and third respectively. There is no evidence in support of the hypothesis that x is derived ultimately from the mediaeval transliteration xei of shei "thing", used by the Arabs to denote the unknown quantity, or from the compendium for L. res "thing" or radix "root" (resembling a loosely-written x), used by mediaeval mathematicians.

Descartes used letters to represent only positive numbers; a negative number could be represented as -b (Cajori vol. 2, page 5).

John Hudde (1633-1704) was first to allow a letter to represent a positive or negative number, in 1657 in De reductione aequationum, published at the end of the first volume of F. Van Schooten's second Latin edition of Ren Descartes' Gomtrie (Cajori vol. 2, page 5).

Jonas Moore wrote in Arithmetic (1660): "Note alwayes the given quantities or numbers with Consonants, and those which are sought with Vowels, or else the given quantities with the former letters in the Alphabet, and the sought with the last sort of letters, as z y x, &c. lest you make a confusion in your work."

Complex numbers. The a + bi notation was introduced by Leonhard Euler (1707-1783).

Earliest Uses of Function Symbols

Last revision: April 18, 2001

The function symbol f(x) was first used by Leonhard Euler (1707-1783) in 1734 in Commentarii Academiae Scientiarum Petropolitanae (Cajori, vol. 2, page 268).

Absolute value function. Karl Weierstrass (1815-1897) used | | in an 1841 essay "Zur Theorie der Potenzreihen," in which the symbol appears on page 67. He also used the symbol in 1859 in "Neuer Beweis des Fundamentalsatzes der Algebra," in which the symbol appears on page 252. This latter essay was submitted to the Berlin Academy of Sciences on December 12, 1859. These are the two reference shown by Cajori (vol. 2, page 123).

Cajori says that the first essay was not printed at the time, and Julio Gonzlez Cabilln believes neither paper was published until 1894, "when the welcome Erster Band [vol. I] of Karl Weierstrass "Mathematische Werke" [Berlin: Mayer & Mueller], saw the light. I do not know to what extent the editors could have interfered with Weierstrass manuscripts. In both papers the notation under discussion does not appear with a definition or with a further comment; thus I am speculating that their subsequent published typesetting might differ from that of Weierstrass original."

The memoir "Zur Theorie der eindeutigen analytischen Functionen," which appeared in Abhandlungen der Koeniglich Akademie der Wissenschaften [pp. 11-60, Berlin 1876, and was reprinted in Zweiter Band (volume II) of Weierstrass "Mathematische Werke" (1895)] has a footnote on page 78 in which Weierstrass remarks:

Ich bezeichne den absoluten Betrag einer complex Groesse x mit |x|. [I denote the absolute value of complex number x by |x|] In this memoir, Weierstrass applied the absolute value symbolism to complex numbers.

Beta function. The use of the beta symbol (for the function created by Euler) was introduced by Jacques P. M. Binet (1786-1856) in 1839 (Cajori, vol. 2, page 272).

Julio Gonzlez Cabilln says the capital letter B is a common one in the Greek and Latin alphabets. If, after Le Gendre, the second Eulerian integral was known as the Gamma function, why Binet could not choose the initial of his name to denote the first Eulerian integral (Beta function), conventionally written as B(p,q). And the precise citation?... "Memoire sur les intgrales dfinies euleriennes, et sur leur application a la theorie des suites, ansi qu'a l'evaluation des fonctions des grands nombres," Journal de L'Ecole Royale Polytchnique, Tome XVI, pp. 123-343, Paris, 1839.

On page 131 of his "Memoire...", Binet states:

Je designerai la premiere de ces fonctions par B(p,q), et pour la seconde j'adoptarai la notacion Gamma(p) proposee par M. Legendre.

Gamma function. The use of (for the function created by Euler) was introduced by Adrien-Marie Legendre (1752-1833) (Cajori vol. 2, page 271). On page 277 of his "Exercices de Calcul integral sur divers ordres de transcendantes et sur les quadrantes," Tome Premier, Paris, 1811, Legendre states:

... Cette quantit tant simplement fonction de a, nous la designerons par (a), et nous ferons (a) = Integral[dx(log 1/x)(a-1)].

It is unknown why Legendre chose that letter, but Julio Gonzlez Cabilln says compare capital letter L (Le Gendre) and the upside-down L. Or the relation between G (in Gendre) and G in Gamma. And there is also a nice relation between the gamma function and the contant C (= 0.577...). Letter C (the one that Euler actually used in his De progressionibus harmonicis observationes) is third in our alphabet; gamma is also third in the Greek alphabet. Please mind that Legendre also used capital C to represent the famous Euler-Mascheroni constant (= 0.577...): On page 295 (ibidem) Legendre says:

C tant une constant dont la valeur calcule avec prcision par une autre voie est C = 0,5772156649015325 donc enfin on aura, k tant trs-petit log = -log k - Ck."

Riemann's zeta function. The use of the small letter zeta for this function was introduced by Bernhard Riemann (1826-1866) as early as 1857 (Cajori vol. 2, page 278).

Bessel functions. P. A. Hansen used the letter J for this function in 1843 in Ermittelung der absoluten Strungen, although the designation of the index and argument has varied since then. Bessel himself used the letter I (Cajori vol. 2, page 279).

Logarithm function. Log. (with a period, capital "L") was used by Johannes Kepler (1571-1630) in 1624 in Chilias logarithmorum (Cajori vol. 2, page 105)

log. (with a period, lower case "l") was used by Bonaventura Cavalieri (1598-1647) in Directorium generale Vranometricum in 1632 (Cajori vol. 2, page 106).

log (without a period, lower case "l") appears in the 1647 edition of Clavis mathematicae by William Oughtred (1574-1660) (Cajori vol. 1, page 193).

Kline (page 378) says Leibniz introduced the notation log x (showing no period), but he does not give a source.

was introduced by Edmund Gunter (1581-1626) according to an Internet source. [I do not see a reference for this in Cajori.]

ln (for natural logarithm) was used in 1893 by Irving Stringham (1847-1909) in Uniplanar Algebra (Cajori vol. 2, page 107).

William Oughtred (1574-1660) used a minus sign over the characteristic of a logarithm in the Clavis Mathematicae (Key to Mathematics), "except in the 1631 edition which does not consider logarithms" (Cajori vol. 2, page 110). The Clavis Mathematicae was composed around 1628 and published in 1631 (Smith 1958, page 393). Cajori shows a use from the 1652 edition.

Greatest integer function. Although [x] is commonly used for this function, the notation was introduced by Kenneth E. Iverson in 1962, according to the website of the University of Tennessee at Martin. The function is also called the floor function.

According to Grinstein (1970), "The use of the bracket notation, which has led some authors to term this the bracket function, stems back to the work of Gauss (1808) in number theory. The function is also referred to by Legendre who used the now obsolete notation E(x)."

Use of arrows. Saunders Mc Lane, in Categories for the working mathematician (Springer-Verlag, 1971, p. 29), says: "The fundamental idea of representing a function by an arrow first appeared in topology about 1940, probably in papers or lectures by W. Hurewicz on relative homotopy groups. (Hurewicz, W.: "On duality theorems," Bull. Am. Math. Soc. 47, 562-563) His initiative immediately attracted the attention of R. H. Fox and N. E. Steenrod, whose ... paper used arrows and (implicitly) functors... The arrow f: : X (arrow) Y rapidly displaced the occasional notation f(X) (subset of )