Mathematical symbols

2020 Mathematics Subject Classification: Primary: 03-XX Secondary: 01Axx [MSN][ZBL]

Conventional signs used for the written notation of mathematical notions and reasoning. For example, the notion "the square root of the number equal to the ratio of the length of the circumference of a circle to its diameter" is denoted briefly by $\sqrt{\pi}$, while the statement "the ratio of the length of the circumference of the circle to its diameter is greater than three and ten seventy-firsts and less than three and one seventh" is written as \[ 3 + \frac{10}{71} < \pi < 3 + \frac{1}{7}\, . \]

Early history

The development of mathematical notation was intimately bound up with the general evolution of mathematical concepts and methods.

The origins

The first mathematical symbols were signs for the depiction of numbers — ciphers, the appearance of which apparently preceded the introduction of written language. The most ancient systems of numbering (see Numbers, representations of) — the Babylonian and the Egyptian — date back to around 3500 B.C..

Ancient Greece

The first mathematical symbols for arbitrary quantities appeared much later (from the 5th-4th centuries B.C.) in Greece. Arbitrary quantities (areas, volumes, angles) were represented by the lengths of lines and the product of two such quantities was represented by a rectangle with sides representing the respective factors. In Euclid's Elements (3th century B.C.), quantities are denoted by two letters, the initial and final letters of the corresponding segment, and sometimes by one letter. Dating from Archimedes (287–213 B.C.), the latter device became standard. This mode of notation could potentially have developed into a calculus of letters. In the mathematics of classical Antiquity, however, no operations were carried out on letters and such a letter calculus did not materialize.

Post-Hellenistic era

The rudiments of letter notation and calculus appeared in the post-Hellenistic era, thanks to the liberation of algebra from its geometric setting. Diophantus (probably 3th century A.D.) denoted the unknown $x$ and its powers by the following symbols: \[ \begin{array}{llllll} x\quad & x^2\quad & x^3\quad & x^4\quad & x^5\quad & x^6\\ \varsigma' &\delta^{\tilde{\upsilon}} &\kappa^{\tilde{\upsilon}} & \delta \delta^{\tilde{\upsilon}}&\delta \kappa^{\tilde{\upsilon}}& \kappa \kappa^{\tilde{\upsilon}} \end{array} \] ($\delta^\tilde{\upsilon}$ — from the Greek term $\delta\upsilon'\nu\alpha\mu\iota\varsigma$, denoting the square of the unknown; $\kappa^{\tilde{\upsilon}}$ — from the Greek $\kappa\upsilon'\beta\omicron\varsigma$, cube). Diophantus wrote coefficients to the right of the unknown or its powers, e.g. $3 x^5$ was denoted by $\delta \kappa^{\tilde{\upsilon}} \bar{\gamma}$ (where $\bar{\gamma} = 3$). Terms to be added together were simply juxtaposed, while subtraction required the special symbol $\wedge$; equality was denoted by the letter $\iota$ (from the Greek $\iota\sigma\omicron\varsigma$, equal). For example, Diophantus would have written the equation \[ (x^3+8x)-(5x^2+1) = x \] as follows: \[ \kappa^{\tilde{\upsilon}}\;\bar{\alpha}\;\varsigma'\; \bar{\eta}\; \bigwedge\; \delta^{\tilde{\upsilon}}\; \bar{\epsilon}\; \mu^0\; \bar{\alpha}\; \iota\; \varsigma'\;\bar{\alpha} \] (here $\bar{\alpha} =1$, $\bar{\eta}=8$, $\bar{\epsilon}=5$ and $\mu^0\bar{\alpha}$ means that the unit $\bar{\alpha}$ is not to be multiplied by a power of the unknown).

Indian mathematics

Several centuries later, the Indians, who had developed a numerical algebra, introduced various mathematical symbols for several unknowns (abbreviations for the names of colours, which denoted the unknowns), the square, the square root, and the subtrahend. Thus, the equation \[ 3 x^2 + 10x - 8 = x^2 +1 \] was written in Brahmaputra's notation (7th century) as follows:

ya va $3$ ya $10$ ru $8$

ya va $1$ ya $0$ ru $1$

(ya — from yavat — tavat, unknown; va — from varga, squared number; ru — from rupa, a rupee coin — free term; a dot above a number denotes subtraction).

Renaissance

The creation of modern algebraic symbols dates to the 14th–15th centuries; it was conditioned by achievements in practical arithmetic and the study of equations. Symbols for various operations and for powers of an unknown quantity appeared spontaneously in different countries. Many decades — sometimes centuries — elapsed until a specific symbol became accepted as convenient for calculations. Thus, at the end of the 15th century N. Chuquet and L. Pacioli (Fra Luca Pacioli) were using the symbols $p$ and $m$ (from the Latin plus and minus) for addition and subtraction, respectively, while German mathematicians introduced the modern $+$ (probably an abbreviation for the Latin et) and $-$. As late as the 17th century, one could count about ten different symbols for multiplication.

The history of the radical sign is instructive. Following Leonardo Pisano (Leonardo da Pisa) (1220), and up to the 17th century, the symbol $RR$ (from the Latin "radix", i.e. root) was widely employed for "square root" . Chuquet denoted square, cube, etc., roots by $RR^2, RR^3$, etc. In a German manuscript of ca. 1480 the square root is denoted by a dot before the number, the cube root by three dots, and the fourth root by two dots. By 1525 one can already find the symbol $\sqrt{}$ (Ch. Rudolff, sometimes written as K. Rudolff). For higher-order roots, some scholars simply repeated this symbol; others wrote a suitable letter after the symbol (an abbreviation of the name of the exponent), and still others inscribed a suitable figure in a circle or between parentheses or square brackets in order to distinguish it from the number under the radical sign (the horizontal line over the radicand was introduced by R. Descartes, 1637). Only at the beginning of the 18th century did it become customary to write the exponent above the opening of the radical sign; the first appearance of this convention, though, was much earlier (A. Girard, 1629). Thus, the evolution of the radical sign extended over almost 500 years.

Modern algebraic notation

Mathematical symbols for an unknown quantity and its powers were highly diverse. During the 16th century and early 17th century, more than ten rival notations were current for just one square of an unknown; among these were ce (from census — the Latin term serving as translation for the Greek term $\delta \upsilon'\nu\alpha\mu \iota\varsigma$), Q (for quadratum), $zz$, $\frac{ii}{1}$, A , $1^2$, $A^{ii}$, aa, $a^2$, etc. G. Cardano (1545) would have written the equation \[ x^3 + 5x = 12 \] as follows: \[ 1 .\; {\rm cubus}\, \square.\;\varsigma\; . {\rm positionibus equantur}\; 12 \] (cubus $=$ cube, positio $=$ unknown, æquantur $=$ equals).

The same equation, written by M. Stifel (1544), would have been: \[ 1+5.\; {\rm aequ}.\; 12 \] by R. Bombelli (1572): \[ 1p\; .\; 5 {\rm eguale\; a\;} 12 \] by F. Viète (1591): \[ 1C + fN,\; {\rm aequatur}\; 12 \] (C $=$ cubus $=$ cube, N $=$ numerus $=$ number);

and by T. Harriot (1631): \[ aaa+5.a=12 \] The 16th century and early 17th century saw the first appearance and use of the equality sign and brackets; square brackets (Bombelli, 1550), parentheses (N. Tartaglia, 1556), and curly brackets (Viète, 1593).

Viète

A significant step forward in the development of mathematical notation was Viète's introduction (1591) of capital letters of the Latin alphabet to denote both arbitrary constant quantities and unknowns; consonants, such as B, D, ... were reserved for constants, and vowels A, E, ... for unknowns. This made it possible for the first time to write down algebraic equations with arbitrary coefficients and to operate with them. For example, Viète's equation \[ A\, {\rm cubus}\; +\; B\, {\rm plano}\; {\rm in}\; A3\;.\; {\rm aequatur}\;D\; {\rm solido} \] (cubus $=$ cube, planus $=$ plane, i.e. B is a two-dimensional constant; solidus $=$ solid (three-dimensional); the dimensionality was indicated to ensure homogeneity of the different terms) stands for the following equation in our notation: \[ x^3+3Bx=D\, . \] Viète, then, was the creator of algebraic formulas.

Descartes

Descartes (1637) gave algebraic notation its modern appearance, denoting unknowns by the last letters of the alphabet $x,y,z$, and arbitrary given quantities by the first letters $a,b,c$. Descartes is also to be credited with the modern notation for powers. As his notation offered considerable advantages over its predecessors, it rapidly gained universal recognition.

Infinitesimal calculus

The further development of mathematical symbols was intimately connected with the invention of infinitesimal calculus, though the basis had already been prepared to a considerable extent in algebra. I. Newton, in his method of fluxions and fluents (1666 and later), introduced symbols for successive fluxions (derivatives) of a quantity $x$: $\dot{x}$, $\ddot{x}$, and the symbol $o$ for an infinitesimal increment. Somewhat earlier J. Wallis (1655) had proposed the symbol $\infty$ for infinity.

Leibniz and the modern notation

The creator of the modern notation for the differential and integral calculus was G. Leibniz. In particular, it was he who invented the modern differentials $dx, d^2 x, d^3 x$ and the integral \[ \int y\, dx \] It is worth emphasizing the essential advantage of Leibniz' integral symbol over Newton's proposal, namely the incorporation of the $x$. Leibniz's notation $\int y\, dx$, while hinting at the actual process of constructing an integral sum, also includes explicit indication of the integrand and the variable of integration. As a result, the notation $\int y\, dx$ is also suited for writing formulas for transformation of variables and is readily used for multiple and line integrals. Newton's notation does not directly offer such possibilities. Similar remarks hold concerning Leibniz's differential signs as against Newton's signs for fluxions and infinitesimal increments.

Euler

L. Euler deserves the credit for a considerable proportion of modern mathematical notation. He introduced the first generally accepted symbol for a variable operation, the function symbol $f x$ (from the Latin functio $=$ function; 1734). Somewhat earlier, the symbol $\phi x$ had been used by J. Bernoulli (1718). After Euler, the symbols for many individual functions (including the trigonometric functions) became standard. Euler was also the first to use the notations $e$ (the base of the natural logarithms, 1736), to spread the notation $\pi$ (probably from the Greek $\pi\epsilon\rho\iota\phi\epsilon\rho\epsilon\iota\alpha$, i.e. circumference, 1736; the notation was borrowed by Euler from H. Jones.), and to introduce the imaginary unit $i$ (from the French "imaginaire" , 1777, published in 1794), which soon gained universal acceptance.

XIX Century

During the 19th century, the role of notation became even more important; as new fields of mathematics were opened up, scholars endeavored to standardize the basic symbols. Some widely employed modern symbols appeared only at that time: the absolute value $|x|$ (K. Weierstrass, 1841), the vector $\vec{v}$ (A. Cauchy, 1853), the determinant \[ \left| \begin{array}{ll} a_1 & a_2\\ b_1 & b_2 \end{array}\right| \] (A. Cayley, 1841), and others. Many of the new theories of the 19th century, such as the tensor calculus, could not have been developed without suitable notation. A characteristic phenomenon in this respect was the increase in the relative proportion of symbols denoting relations, such as the congruence $\equiv$ (C.F. Gauss, 1801), membership $\in$, isomorphism $\cong$, equivalence $\sim$, etc. Symbols for variable relations appeared with the advent of mathematical logic, which makes particularly extensive use of mathematical symbols.

Mathematical logic and classification of symbols

From the point of view of mathematical logic, mathematical symbols can be classified under the following main headings: A) symbols for objects, B) symbols for operations, C) symbols for relations. For example, the symbols 1, 2, 3, 4 denote numbers, i.e., the objects studied in arithmetic. The symbol for the addition operation, $+$, standing on its own, does not denote any object; it takes an objective content only when the numbers to be added are specified: $1+3$ denotes the number $4$. The symbol $>$ (greater) denotes a relation between numbers. A relation symbol assumes a definite content only when the objects that can stand in that specific relation are specified. One further, fourth, group of symbols may be added: D) auxiliary symbols, which determine the order in which the basic symbols are to be combined. A good example of this type of symbol is provided by parentheses, which indicate the order in which arithmetical operations are to be carried out.

The symbols of each of the three main groups A), B), C) are of two kinds: 1) individual symbols for definite objects, operations and relations; and 2) general symbols for "variable" or "unknown" objects, operations and relations. Examples of symbols of the first kind are the following (see also the table in this article):

$A_1$) The notation for the natural numbers 1, 2, 3, 4, 5, 6, 7, 8, 9; the transcendental numbers $e$ and $\pi$; the imaginary unit $i$; etc.

$B_1$) The signs for the arithmetical operations, $+,-,\times, :$; root extraction $\sqrt{}$, $(\cdot)^{1/n}$, differentiation $\frac{d}{dx}$, the Laplace operator \[ \Delta = \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2} \] This subgroup also contains the individual symbols $\sin$, $\tan$, $\log$, etc.

$C_1$) Equality and inequality signs, $=, >, <, \neq$, the symbols denoting parallel ($||$) and perpendicular ($\perp$), etc.

Symbols of the second kind denote arbitrary objects, operations and relations of a certain class, or objects, operations and relations resulting from some previously mentioned conditions. For example, in the written identity \[ (a+b)(a-b) = a^2 - b^2 \] the letters $a$ and $b$ denote arbitrary numbers; when one is studying the functional dependence \[ y=x^2 \] the letters $y$ and $x$ denote arbitrary numbers standing in the given relation; in the solution of the equation \[ x^2-1=0 \] $x$ denotes any number satisfying the equation (by solving the equation, one knows that there are only two numbers satisfying the condition: $+1$ and $-1$).

From a logical point of view it is quite legitimate to call all symbols of this kind variable symbols, as is customary in mathematical logic (the "domain of variation" of the variable may prove to consist of a single object; it may even be "empty" — e.g. in the case of equations with no solutions). Further examples of this kind of signs are:

$A_2$) Symbols for points, straight lines, planes, and more complex geometrical figures, denoted in geometry by letters.

$B_2$) Notations such as $f,F,\phi$ for functions and notations in operator calculus, when one letter <$L$ may be used to denote, say, an arbitrary operator of the form \[ L[y] = a_0 y + a_1 \frac{dy}{dx} + \ldots + a_n \frac{d^n y}{dx^n}\, . \] Symbols for "variable relations" are less common; they find application only in mathematical logic and in comparatively abstract, primarily axiomatic, branches of mathematics.

The origin of some mathematical symbols

Symbol	Meaning	Introduced by	Year
$\infty$	infinity	J. Wallis	1655
$e$	base of the natural logarithms	L. Euler	1736
$\pi$	ratio of the length of a circumference to the diameter	W. Jones	1706
$i$	square toot of $-1$	L. Euler	1777 (pubbl. 1794)
$i,j,k$	unit vectors	W. Hamilton	1853
$\Pi (\alpha)$	angle of parallelism	N.I. Lobachevskii	1835
$x,y,z$	Unknown or variable quantities	R. Descartes	1637
$\vec{v}$	vector	A.L. Cauchy	1853
$+, -$	addition, subtraction	German mathematicians	end of XV cent.
$\times$	multiplication	W. Oughtred	1631
$\cdot$	multiplication	G. Leibniz	1698
$:$	division	G. Leibniz	1684
$a^2, \ldots, a^n$	powers	R. Descartes	1637
$\sqrt{}$	square root	K. Rudolff	1525
$\sqrt[n]{}$	roots	A. Girard	1629
${\rm Log}$	logarithm	J. Kepler	1624
${\rm log}$	logarithm	B. Cavalieri	1632
$\sin$	sine	L. Euler	1748
$\cos$	cosine	L. Euler	1748
${\rm tg}$	tangent	L. Euler	1753
$\tan$	tangent	L. Euler	1753
$\arcsin$	arcsine	J. Lagrange	1772
${\rm Sh}$	hyperbolic sine	V. Riccati	1757
${\rm Ch}$	hyperbolic cosine	V. Riccati	1757
$dx, ddx, d^2 x, d^3 x, \ldots$	differentials	G. Leibniz	1675 (publ. 1684)
$\int y\, dx$	integral	G. Leibniz	1675 (publ. 1684)
$\frac{d}{dx}$	derivative	G. Leibniz	1675
$f', y', f'x$	derivative	J. Lagrange	1770-1779
$\Delta x$	difference, increment	L. Euler	1755
$\frac{\partial}{\partial x}$	partial derivative	A. Legendre	1786
$\int_a^b f(x)\, dx$	definite integral	J. Fourier	1819-1820
$\sum$	sum	L. Euler	1755
$\prod$	product	C.F. Gauss	1812
$!$	factorial	Ch. Kramp	1808
$|x|$	absolute value	K. Weierstrass	1841
$\lim$	limit	S. l'Huillier	1786
$\lim_{n=\infty}$	limit	W. Hamilton	1853
$\lim_{n\to\infty}$	limit	various mathematicians	beg. of 20th cent.
$\zeta$	zeta-function	B. Riemann	1857
$\Gamma$	gamma-function	A. Legendre	1808
$B$	beta-function	J. Binet	1839
$\Delta$	Laplace operator	R. Murphy	1833
$\nabla$	nabla, Hamilton operator	W. Hamilton	1853
$\phi x$	function	J. Bernoulli	1718
$f x$	function	L. Euler	1734
$=$	equality	R. Recorde	1557
$>, <$	greater than, smaller than	T. Harriot	1631
$\equiv$	congruence	C.F. Gauss	1801
$||$	parallel	W. Oughtred	1677 (post. publ.)
$\perp$	perpendicular	P. Hérigone	1634

References

[Bo]	C.B. Boyer, "A history of mathematics" , Wiley (1968)
[Ca]	F. Cajori, "A history of mathematical notations" , 1–2 , Open Court (1952–1974)
[Kl]	M. Kline, "Mathematical thought from ancient to modern times" , Oxford Univ. Press (1972)