Table of mathematical symbols by introduction date

The following table lists many specialized symbols commonly used in mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, ordered by their introduction date.

	Name	Date of earliest use	First author to use
−	plus and minus signs Plus and minus signs The plus and minus signs are mathematical symbols used to represent the notions of positive and negative as well as the operations of addition and subtraction. Their use has been extended to many other meanings, more or less analogous...	ca. 1360 (abbreviation for Latin et resembling the plus sign)	Nicole Oresme
−		1489 (first appearance of plus and minus signs in print)	Johannes Widmann Johannes Widmann Johannes Widmann was a German mathematician who invented the addition and the subtraction signs.Born at in Eger, Bohemia, Widmann attended the University of Leipzig in the 1480s, and published Behende und hubsche Rechenung auff allen Kauffmanschafft Johannes Widmann (c. 1460 - after 1498) was a...
	radical symbol (for square root Square root In mathematics, a square root of a number x is a number r such that r2 = x, or, in other words, a number r whose square is x... )	1525 (without the vinculum above the radicand)	Christoff Rudolff
	parentheses (for precedence grouping)	1544 (in handwritten notes)	Michael Stifel Michael Stifel Michael Stifel or Styfel was a German monk and mathematician. He was an Augustinian who became an early supporter of Martin Luther. Stifel was later appointed professor of mathematics at Jena University...
	parentheses (for precedence grouping)	1556	Nicolo Tartaglia
	equals sign Equals sign The equality sign, equals sign, or "=" is a mathematical symbol used to indicate equality. It was invented in 1557 by Robert Recorde. The equals sign is placed between the things stated to have the same value, as in an equation...	1557	Robert Recorde Robert Recorde Robert Recorde was a Welsh physician and mathematician. He introduced the "equals" sign in 1557.-Biography:A member of a respectable family of Tenby, Wales, he entered the University of Oxford about 1525, and was elected a fellow of All Souls College in 1531...
	multiplication sign	1618	William Oughtred William Oughtred William Oughtred was an English mathematician.After John Napier invented logarithms, and Edmund Gunter created the logarithmic scales upon which slide rules are based, it was Oughtred who first used two such scales sliding by one another to perform direct multiplication and division; and he is...
	plus-minus sign Plus-minus sign The plus-minus sign is a mathematical symbol commonly used eitherto indicate the precision of an approximation, orto indicate a value that can be of either sign....	1628
	proportion sign	1628
	radical symbol (for nth root Nth root In mathematics, the nth root of a number x is a number r which, when raised to the power of n, equals xr^n = x,where n is the degree of the root... )	1629	Albert Girard Albert Girard Albert Girard was a French-born mathematician. He studied at the University of Leiden. He "had early thoughts on the fundamental theorem of algebra" and gave the inductive definition for the Fibonacci numbers....
>	strict inequality signs (less-than sign and greater-than sign)	1631	Thomas Harriot Thomas Harriot Thomas Harriot was an English astronomer, mathematician, ethnographer, and translator. Some sources give his surname as Harriott or Hariot or Heriot. He is sometimes credited with the introduction of the potato to Great Britain and Ireland...
	superscript notation (for exponentiation Exponentiation Exponentiation is a mathematical operation, written as an, involving two numbers, the base a and the exponent n... )	1636 (using Roman numerals Roman numerals The numeral system of ancient Rome, or Roman numerals, uses combinations of letters from the Latin alphabet to signify values. The numbers 1 to 10 can be expressed in Roman numerals as:... as superscripts)	James Hume
		1637 (in the modern form)	René Descartes René Descartes René Descartes ; was a French philosopher and writer who spent most of his adult life in the Dutch Republic. He has been dubbed the 'Father of Modern Philosophy', and much subsequent Western philosophy is a response to his writings, which are studied closely to this day...
	radical symbol (for square root Square root In mathematics, a square root of a number x is a number r such that r2 = x, or, in other words, a number r whose square is x... )	1637 (with the vinculum above the radicand)	René Descartes René Descartes René Descartes ; was a French philosopher and writer who spent most of his adult life in the Dutch Republic. He has been dubbed the 'Father of Modern Philosophy', and much subsequent Western philosophy is a response to his writings, which are studied closely to this day...
	percent sign Percent sign The percent sign is the symbol used to indicate a percentage .Related signs include the permille sign ‰ and the permyriad sign , which indicate that a number is divided by one thousand or ten thousand respectively...	ca. 1650	unknown
	division sign (a.k.a. obelus Obelus An obelus is a symbol consisting of a short horizontal line with a dot above and below. It is mainly used to represent the mathematical operation of division. It is therefore commonly referred to as the division sign.- History :The word "obelus" comes from the Greek word for a sharpened stick,... )	1659	Johann Rahn Johann Rahn Johann Rahn was a Swiss mathematician who is credited with the first use of the division symbol, ÷ and the therefore sign, ∴. The symbol is used in Teutsche Algebra, published in 1659.-References:...
	infinity Infinity Infinity is a concept in many fields, most predominantly mathematics and physics, that refers to a quantity without bound or end. People have developed various ideas throughout history about the nature of infinity... sign	1655	John Wallis
≥	unstrict inequality signs (less-than or equals to sign and greater-than or equals to sign)	1670 (with the horizontal bar over the inequality sign, rather than below it)	John Wallis
≥		1734 (with double horizontal bar below the inequality sign)	Pierre Bouguer Pierre Bouguer Pierre Bouguer was a French mathematician, geophysicist, geodesist, and astronomer. He is also known as "the father of naval architecture"....
	differential Differential (calculus) In calculus, a differential is traditionally an infinitesimally small change in a variable. For example, if x is a variable, then a change in the value of x is often denoted Δx . The differential dx represents such a change, but is infinitely small... sign	1675	Gottfried Leibniz Gottfried Leibniz Gottfried Wilhelm Leibniz was a German philosopher and mathematician. He wrote in different languages, primarily in Latin , French and German ....
	integral sign	1675
	colon Colon (punctuation) The colon is a punctuation mark consisting of two equally sized dots centered on the same vertical line.-Usage:A colon informs the reader that what follows the mark proves, explains, or lists elements of what preceded the mark.... (for division Division (mathematics) right\|thumb\|200px\|20 \div 4=5In mathematics, especially in elementary arithmetic, division is an arithmetic operation.Specifically, if c times b equals a, written:c \times b = a\,... )	1684 (deriving from use of colon to denote fractions, dating back to 1633)
	middle dot (for multiplication Multiplication Multiplication is the mathematical operation of scaling one number by another. It is one of the four basic operations in elementary arithmetic .... )	1698 (perhaps deriving from a much earlier use of middle dot to separate juxtaposed numbers)
	division slash Slash (punctuation) The slash is a sign used as a punctuation mark and for various other purposes. It is now often called a forward slash , and many other alternative names.-History:... (a.k.a. solidus)	1718 (deriving from horizontal fraction bar, invented by Arabs in 12th century)	Thomas Twining Thomas Twining Thomas Twining may refer to:Thomas Twining , English merchant and founder of the Twinings tea companyThomas Twining , English scholar and classicist, grandson of the above...
	inequality sign (not equal to)	unknown	Leonhard Euler Leonhard Euler Leonhard Euler was a pioneering Swiss mathematician and physicist. He made important discoveries in fields as diverse as infinitesimal calculus and graph theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion...
	summation Summation Summation is the operation of adding a sequence of numbers; the result is their sum or total. If numbers are added sequentially from left to right, any intermediate result is a partial sum, prefix sum, or running total of the summation. The numbers to be summed may be integers, rational numbers,... symbol	1755
	proportionality Proportionality (mathematics) In mathematics, two variable quantities are proportional if one of them is always the product of the other and a constant quantity, called the coefficient of proportionality or proportionality constant. In other words, are proportional if the ratio \tfrac yx is constant. We also say that one... sign	1768	William Emerson William Emerson (mathematician) William Emerson , English mathematician, was born at Hurworth, near Darlington, where his father, Dudley Emerson, also a mathematician, taught a school...
	partial differential sign (a.k.a curly d or Jacobi's delta)	1770	Marquis de Condorcet Marquis de Condorcet Marie Jean Antoine Nicolas de Caritat, marquis de Condorcet , known as Nicolas de Condorcet, was a French philosopher, mathematician, and early political scientist whose Condorcet method in voting tally selects the candidate who would beat each of the other candidates in a run-off election...
	prime symbol (for derivative Derivative In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a... )	1770	Joseph Louis Lagrange Joseph Louis Lagrange Joseph-Louis Lagrange , born Giuseppe Lodovico Lagrangia, was a mathematician and astronomer, who was born in Turin, Piedmont, lived part of his life in Prussia and part in France, making significant contributions to all fields of analysis, to number theory, and to classical and celestial mechanics...
	identity Identity (mathematics) In mathematics, the term identity has several different important meanings:*An identity is a relation which is tautologically true. This means that whatever the number or value may be, the answer stays the same. For example, algebraically, this occurs if an equation is satisfied for all values of... sign (for congruence relation Congruence relation In abstract algebra, a congruence relation is an equivalence relation on an algebraic structure that is compatible with the structure... )	1801 (first appearance in print; used previously in personal writings of Gauss)	Carl Friedrich Gauss Carl Friedrich Gauss Johann Carl Friedrich Gauss was a German mathematician and scientist who contributed significantly to many fields, including number theory, statistics, analysis, differential geometry, geodesy, geophysics, electrostatics, astronomy and optics.Sometimes referred to as the Princeps mathematicorum...
	integral part (a.k.a. floor)	1808
	product Multiplication Multiplication is the mathematical operation of scaling one number by another. It is one of the four basic operations in elementary arithmetic .... symbol	1812
	factorial Factorial In mathematics, the factorial of a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n...	1808	Christian Kramp Christian Kramp Christian Kramp was a French mathematician, who worked primarily with factorials.Christian Kramp's father was his teacher at grammar school in Strasbourg. Kramp studied medicine and graduated, however, his interests certainly ranged outside medicine for, in addition to a number of medical...
⊃	set inclusion signs (subset of, superset of)	1817	Joseph Gergonne
⊃	set inclusion signs (subset of, superset of)	1890	Ernst Schröder Ernst Schröder Ernst Schröder was a German mathematician mainly known for his work on algebraic logic. He is a major figure in the history of mathematical logic , by virtue of summarizing and extending the work of George Boole, Augustus De Morgan, Hugh MacColl, and especially Charles Peirce...
	absolute value Absolute value In mathematics, the absolute value \|a\| of a real number a is the numerical value of a without regard to its sign. So, for example, the absolute value of 3 is 3, and the absolute value of -3 is also 3... notation	1841	Karl Weierstrass Karl Weierstrass Karl Theodor Wilhelm Weierstrass was a German mathematician who is often cited as the "father of modern analysis".- Biography :Weierstrass was born in Ostenfelde, part of Ennigerloh, Province of Westphalia....
	determinant Determinant In linear algebra, the determinant is a value associated with a square matrix. It can be computed from the entries of the matrix by a specific arithmetic expression, while other ways to determine its value exist as well... of a matrix	1841	Arthur Cayley Arthur Cayley Arthur Cayley F.R.S. was a British mathematician. He helped found the modern British school of pure mathematics....
	matrices Matrix (mathematics) In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element... notation	1843
	nabla symbol Nabla symbol right\|thumb\|256px\|The [[harp]], the instrument after which the nabla symbol is namedNabla is the symbol \nabla . The name comes from the Greek word for a Hebrew harp, which had a similar shape. Related words also exist in Aramaic and Hebrew. The symbol was first used by William Rowan Hamilton in... (for vector differential)	1846 (previously used by Hamilton as a general-purpose operator sign)	William Rowan Hamilton William Rowan Hamilton Sir William Rowan Hamilton was an Irish physicist, astronomer, and mathematician, who made important contributions to classical mechanics, optics, and algebra. His studies of mechanical and optical systems led him to discover new mathematical concepts and techniques...
∪	intersection Intersection (set theory) In mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B , but no other elements.... and union Union (set theory) In set theory, the union of a collection of sets is the set of all distinct elements in the collection. The union of a collection of sets S_1, S_2, S_3, \dots , S_n\,\! gives a set S_1 \cup S_2 \cup S_3 \cup \dots \cup S_n.- Definition :... signs	1888	Giuseppe Peano Giuseppe Peano Giuseppe Peano was an Italian mathematician, whose work was of philosophical value. The author of over 200 books and papers, he was a founder of mathematical logic and set theory, to which he contributed much notation. The standard axiomatization of the natural numbers is named the Peano axioms in...
	membership sign (is an element of)	1894
	existential quantifier (there exists)	1897
	aleph Aleph number In set theory, a discipline within mathematics, the aleph numbers are a sequence of numbers used to represent the cardinality of infinite sets. They are named after the symbol used to denote them, the Hebrew letter aleph... symbol (for cardinal number Cardinal number In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality of sets. The cardinality of a finite set is a natural number – the number of elements in the set. The transfinite cardinal numbers describe the sizes of infinite... s of transfinite sets)	1893	Georg Cantor Georg Cantor Georg Ferdinand Ludwig Philipp Cantor was a German mathematician, best known as the inventor of set theory, which has become a fundamental theory in mathematics. Cantor established the importance of one-to-one correspondence between the members of two sets, defined infinite and well-ordered sets,...
	braces, a.k.a. curly brackets (for set notation)	1895
	double-struck capital N (for natural number Natural number In mathematics, the natural numbers are the ordinary whole numbers used for counting and ordering . These purposes are related to the linguistic notions of cardinal and ordinal numbers, respectively... s set)	1895	Giuseppe Peano Giuseppe Peano Giuseppe Peano was an Italian mathematician, whose work was of philosophical value. The author of over 200 books and papers, he was a founder of mathematical logic and set theory, to which he contributed much notation. The standard axiomatization of the natural numbers is named the Peano axioms in...
	middle dot (for dot product Dot product In mathematics, the dot product or scalar product is an algebraic operation that takes two equal-length sequences of numbers and returns a single number obtained by multiplying corresponding entries and then summing those products... )	1902	J. Willard Gibbs?
	multiplication sign (for cross product Cross product In mathematics, the cross product, vector product, or Gibbs vector product is a binary operation on two vectors in three-dimensional space. It results in a vector which is perpendicular to both of the vectors being multiplied and normal to the plane containing them... )	1902	J. Willard Gibbs?
	logical disjunction Logical disjunction In logic and mathematics, a two-place logical connective or, is a logical disjunction, also known as inclusive disjunction or alternation, that results in true whenever one or more of its operands are true. E.g. in this context, "A or B" is true if A is true, or if B is true, or if both A and B are... (a.k.a. OR)	1906	Bertrand Russell Bertrand Russell Bertrand Arthur William Russell, 3rd Earl Russell, OM, FRS was a British philosopher, logician, mathematician, historian, and social critic. At various points in his life he considered himself a liberal, a socialist, and a pacifist, but he also admitted that he had never been any of these things...
	matrices Matrix (mathematics) In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element... notation	1909	Gerhard Kowalewski Gerhard Kowalewski Dr. Gerhard Kowalewski was a German mathematician and member of the Nazi party who introduced the matrices notation.-Early life:...
		1913	Cuthbert Edmund Cullis
	contour integral Line integral In mathematics, a line integral is an integral where the function to be integrated is evaluated along a curve.The function to be integrated may be a scalar field or a vector field... sign	1917	Arnold Sommerfeld Arnold Sommerfeld Arnold Johannes Wilhelm Sommerfeld was a German theoretical physicist who pioneered developments in atomic and quantum physics, and also educated and groomed a large number of students for the new era of theoretical physics...
	double-struck capital Z (for integer Integer The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively... numbers set)	1930	Edmund Landau Edmund Landau Edmund Georg Hermann Landau was a German Jewish mathematician who worked in the fields of number theory and complex analysis.-Biography:...
		1930s	Nicolas Bourbaki Nicolas Bourbaki Nicolas Bourbaki is the collective pseudonym under which a group of 20th-century mathematicians wrote a series of books presenting an exposition of modern advanced mathematics, beginning in 1935. With the goal of founding all of mathematics on set theory, the group strove for rigour and generality...
	double-struck capital Q (for rational number Rational number In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number... s set)
	universal quantifier (for all)	1935	Gerhard Gentzen Gerhard Gentzen Gerhard Karl Erich Gentzen was a German mathematician and logician. He had his major contributions in the foundations of mathematics, proof theory, especially on natural deduction and sequent calculus...
	empty set Empty set In mathematics, and more specifically set theory, the empty set is the unique set having no elements; its size or cardinality is zero. Some axiomatic set theories assure that the empty set exists by including an axiom of empty set; in other theories, its existence can be deduced... sign	1939	André Weil André Weil André Weil was an influential mathematician of the 20th century, renowned for the breadth and quality of his research output, its influence on future work, and the elegance of his exposition. He is especially known for his foundational work in number theory and algebraic geometry... / Nicolas Bourbaki Nicolas Bourbaki Nicolas Bourbaki is the collective pseudonym under which a group of 20th-century mathematicians wrote a series of books presenting an exposition of modern advanced mathematics, beginning in 1935. With the goal of founding all of mathematics on set theory, the group strove for rigour and generality...
	double-struck capital C (for complex number Complex number A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part... s set)	1939	Nathan Jacobson Nathan Jacobson Nathan Jacobson was an American mathematician....
	arrow Arrow An arrow is a shafted projectile that is shot with a bow. It predates recorded history and is common to most cultures.An arrow usually consists of a shaft with an arrowhead attached to the front end, with fletchings and a nock at the other.- History:... (for function Function (mathematics) In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can... notation)	1936 (to denote images of specific elements)	Øystein Ore Øystein Ore Øystein Ore was a Norwegian mathematician.-Life:Ore was graduated from the University of Oslo in 1922, with a Cand.Scient. degree in mathematics. In 1924, the University of Oslo awarded him the Ph.D. for a thesis titled Zur Theorie der algebraischen Körper, supervised by Thoralf Skolem...
		1940 (in the present form of f: X → Y)	Witold Hurewicz Witold Hurewicz Witold Hurewicz was a Polish mathematician.- Early life and education :Witold Hurewicz was born to a Jewish family in Łódź, Russian Empire ....
	integral part (a.k.a. floor)	1962	Kenneth E. Iverson Kenneth E. Iverson Kenneth Eugene Iverson was a Canadian computer scientist noted for the development of the APL programming language in 1962. He was honored with the Turing Award in 1979 for his contributions to mathematical notation and programming language theory...
	end of proof sign (a.k.a. tombstone Tombstone (typography) The tombstone, halmos, or end of proof mark "" is used in mathematics to denote the end of a proof, in place of the traditional abbreviation "QED" for the Latin phrase "quod erat demonstrandum" .... )	unknown	Paul Halmos Paul Halmos Paul Richard Halmos was a Hungarian-born American mathematician who made fundamental advances in the areas of probability theory, statistics, operator theory, ergodic theory, and functional analysis . He was also recognized as a great mathematical expositor.-Career:Halmos obtained his B.A...

Source

Jeff Miller: Earliest Uses of Various Mathematical Symbols

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