Mathematical notation

# Mathematical notation

Discussion

Encyclopedia
Mathematical notation is a system of symbol
Symbol
A symbol is something which represents an idea, a physical entity or a process but is distinct from it. The purpose of a symbol is to communicate meaning. For example, a red octagon may be a symbol for "STOP". On a map, a picture of a tent might represent a campsite. Numerals are symbols for...

ic representations of mathematical objects and ideas. Mathematical notations are 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...

, the physical sciences, engineering
Engineering
Engineering is the discipline, art, skill and profession of acquiring and applying scientific, mathematical, economic, social, and practical knowledge, in order to design and build structures, machines, devices, systems, materials and processes that safely realize improvements to the lives of...

, and economics
Economics
Economics is the social science that analyzes the production, distribution, and consumption of goods and services. The term economics comes from the Ancient Greek from + , hence "rules of the house"...

. Mathematical notations include relatively simple symbolic representations, such as the numbers 0, 1 and 2, 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...

symbols sin
Sine
In mathematics, the sine function is a function of an angle. In a right triangle, sine gives the ratio of the length of the side opposite to an angle to the length of the hypotenuse.Sine is usually listed first amongst the trigonometric functions....

and +
Addition is a mathematical operation that represents combining collections of objects together into a larger collection. It is signified by the plus sign . For example, in the picture on the right, there are 3 + 2 apples—meaning three apples and two other apples—which is the same as five apples....

; conceptual symbols, such as lim
Limit (mathematics)
In mathematics, the concept of a "limit" is used to describe the value that a function or sequence "approaches" as the input or index approaches some value. The concept of limit allows mathematicians to define a new point from a Cauchy sequence of previously defined points within a complete metric...

, dy/dx
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...

, equation
Equation
An equation is a mathematical statement that asserts the equality of two expressions. In modern notation, this is written by placing the expressions on either side of an equals sign , for examplex + 3 = 5\,asserts that x+3 is equal to 5...

s and variables
Variable (mathematics)
In mathematics, a variable is a value that may change within the scope of a given problem or set of operations. In contrast, a constant is a value that remains unchanged, though often unknown or undetermined. The concepts of constants and variables are fundamental to many areas of mathematics and...

; and complex diagrammatic notations such as Penrose graphical notation
Penrose graphical notation
In mathematics and physics, Penrose graphical notation or tensor diagram notation is a visual depiction of multilinear functions or tensors proposed by Roger Penrose. A diagram in the notation consists of several shapes linked together by lines, much like tinker toys...

and Coxeter-Dynkin diagram
Coxeter-Dynkin diagram
In geometry, a Coxeter–Dynkin diagram is a graph with numerically labeled edges representing the spatial relations between a collection of mirrors...

s.

## Definition

A mathematical notation is a writing system
Writing system
A writing system is a symbolic system used to represent elements or statements expressible in language.-General properties:Writing systems are distinguished from other possible symbolic communication systems in that the reader must usually understand something of the associated spoken language to...

used for recording concepts in mathematics.
• The notation uses symbols or symbolic expressions
Expression (mathematics)
In mathematics, an expression is a finite combination of symbols that is well-formed according to rules that depend on the context. Symbols can designate numbers , variables, operations, functions, and other mathematical symbols, as well as punctuation, symbols of grouping, and other syntactic...

which are intended to have a precise semantic meaning.
• In the history of mathematics
History of mathematics
The area of study known as the history of mathematics is primarily an investigation into the origin of discoveries in mathematics and, to a lesser extent, an investigation into the mathematical methods and notation of the past....

, these symbols have denoted numbers, shapes, patterns, and change. The notation can also include symbols for parts of the conventional discourse between mathematicians, when viewing mathematics as a language
Mathematics as a language
The language of mathematics is the system used by mathematicians to communicate mathematical ideas among themselves. This language consists of a substrate of some natural language using technical terms and grammatical conventions that are peculiar to mathematical discourse , supplemented by a...

.

The media used for writing are recounted below, but common materials currently include paper and pencil, board and chalk (or dry-erase marker), and electronic media. Systematic adherence to mathematical concepts is a fundamental concept of mathematical notation. (See also some related concepts: Logical argument, Mathematical logic
Mathematical logic
Mathematical logic is a subfield of mathematics with close connections to foundations of mathematics, theoretical computer science and philosophical logic. The field includes both the mathematical study of logic and the applications of formal logic to other areas of mathematics...

, and Model theory
Model theory
In mathematics, model theory is the study of mathematical structures using tools from mathematical logic....

.)

## Expressions

A mathematical expression
Expression (mathematics)
In mathematics, an expression is a finite combination of symbols that is well-formed according to rules that depend on the context. Symbols can designate numbers , variables, operations, functions, and other mathematical symbols, as well as punctuation, symbols of grouping, and other syntactic...

is a sequence of symbols which can be evaluated. For example, if the symbols represent numbers, the expressions are evaluated according to a conventional order of operations
Order of operations
In mathematics and computer programming, the order of operations is a rule used to clarify unambiguously which procedures should be performed first in a given mathematical expression....

which provides for calculation, if possible, of any expressions within parentheses, followed by any exponents and roots, then multiplications and divisions and finally any additions or subtractions, all done from left to right. In a computer language, these rules are implemented by the compiler
Compiler
A compiler is a computer program that transforms source code written in a programming language into another computer language...

s. For more on expression evaluation, see the computer science
Computer science
Computer science or computing science is the study of the theoretical foundations of information and computation and of practical techniques for their implementation and application in computer systems...

topics: eager evaluation
Eager evaluation
In computer programming, eager evaluation or greedy evaluation is the evaluation strategy in most traditional programming languages. In eager evaluation an expression is evaluated as soon as it gets bound to a variable. The term is typically used to contrast lazy evaluation, where expressions are...

, lazy evaluation
Lazy evaluation
In programming language theory, lazy evaluation or call-by-need is an evaluation strategy which delays the evaluation of an expression until the value of this is actually required and which also avoids repeated evaluations...

, and evaluation operator.

## Precise semantic meaning

Modern mathematics needs to be precise, because ambiguous notations do not allow formal proofs. Suppose that we have statements, denoted by some formal sequence
Sequence
In mathematics, a sequence is an ordered list of objects . Like a set, it contains members , and the number of terms is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence...

of symbols, about some objects (for example, numbers, shapes, patterns). Until the statements can be shown to be valid, their meaning is not yet resolved. While reasoning, we might let the symbols refer to those denoted objects, perhaps in a model. The semantics
Semantics
Semantics is the study of meaning. It focuses on the relation between signifiers, such as words, phrases, signs and symbols, and what they stand for, their denotata....

of that object has a heuristic
Heuristic
Heuristic refers to experience-based techniques for problem solving, learning, and discovery. Heuristic methods are used to speed up the process of finding a satisfactory solution, where an exhaustive search is impractical...

side and a deductive side. In either case, we might want to know the properties of that object, which we might then list in an intensional definition
Intensional definition
In logic and mathematics, an intensional definition gives the meaning of a term by specifying all the properties required to come to that definition, that is, the necessary and sufficient conditions for belonging to the set being defined....

.

Those properties might then be expressed by some well-known and agreed-upon symbols from a table of mathematical symbols
Table of mathematical symbols
This is a listing of common symbols found within all branches of mathematics. Each symbol is listed in both HTML, which depends on appropriate fonts being installed, and in , as an image.-Symbols:-Variations:...

. This mathematical notation might include annotation such as
• "All x", "No x", "There is an x" (or its equivalent, "Some x"), "A set", "A function"
• "A mapping from the real numbers to the complex numbers"

In different contexts, the same symbol or notation can be used to represent different concepts. Therefore, to fully understand a piece of mathematical writing, it is important to first check the definitions that an author gives for the notations that are being used. This may be problematic if the author assumes the reader is already familiar with the notation in use.

### Counting

It is believed that a mathematical notation to represent counting
Counting
Counting is the action of finding the number of elements of a finite set of objects. The traditional way of counting consists of continually increasing a counter by a unit for every element of the set, in some order, while marking those elements to avoid visiting the same element more than once,...

was first developed at least 50,000 years ago — early mathematical ideas such as finger counting
Finger counting
Finger counting, or dactylonomy, is the art of counting along one's fingers. Though marginalized in modern societies by Arabic numerals, formerly different systems flourished in many cultures, including educated methods far more sophisticated than the one-by-one finger count taught today in...

have also been represented by collections of rocks, sticks, bone, clay, stone, wood carvings, and knotted ropes. The tally stick
Tally stick
A tally was an ancient memory aid device to record and document numbers, quantities, or even messages. Tally sticks first appear as notches carved on animal bones, in the Upper Paleolithic. A notable example is the Ishango Bone...

is a timeless way of counting. Perhaps the oldest known mathematical texts are those of ancient Sumer
Sumer
Sumer was a civilization and historical region in southern Mesopotamia, modern Iraq during the Chalcolithic and Early Bronze Age....

. The Census Quipu of the Andes and the Ishango Bone
Ishango bone
The Ishango bone is a bone tool, dated to the Upper Paleolithic era. It is a dark brown length of bone, the fibula of a baboon, with a sharp piece of quartz affixed to one end, perhaps for engraving...

from Africa both used the tally mark method of accounting for numerical concepts.

The development of zero as a number is one of the most important developments in early mathematics. It was used as a placeholder by the Babylonians
Babylonian numerals
Babylonian numerals were written in cuneiform, using a wedge-tipped reed stylus to make a mark on a soft clay tablet which would be exposed in the sun to harden to create a permanent record....

and Greek Egyptians
Greek numerals
Greek numerals are a system of representing numbers using letters of the Greek alphabet. They are also known by the names Ionian numerals, Milesian numerals , Alexandrian numerals, or alphabetic numerals...

, and then as an integer by the Mayans
Maya numerals
Maya Numerals were a vigesimal numeral system used by the Pre-Columbian Maya civilization.The numerals are made up of three symbols; zero , one and five...

, Indians
Indian numerals
Most of the positional base 10 numeral systems in the world have originated from India, where the concept of positional numeration was first developed...

and Arabs
Arabic numerals
Arabic numerals or Hindu numerals or Hindu-Arabic numerals or Indo-Arabic numerals are the ten digits . They are descended from the Hindu-Arabic numeral system developed by Indian mathematicians, in which a sequence of digits such as "975" is read as a numeral...

### Geometry becomes analytic

The mathematical viewpoints in geometry
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

did not lend themselves well to counting. The 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, their relationship to fraction
Fraction (mathematics)
A fraction represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, we specify how many parts of a certain size there are, for example, one-half, five-eighths and three-quarters.A common or "vulgar" fraction, such as 1/2, 5/8, 3/4, etc., consists...

s, and the identification of continuous
Continuum (theory)
Continuum theories or models explain variation as involving a gradual quantitative transition without abrupt changes or discontinuities. It can be contrasted with 'categorical' models which propose qualitatively different states.-In physics:...

quantities actually took millennia to take form, and even longer to allow for the development of notation. It was not until the invention of analytic geometry
Analytic geometry
Analytic geometry, or analytical geometry has two different meanings in mathematics. The modern and advanced meaning refers to the geometry of analytic varieties...

by 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...

that geometry became more subject to a numerical notation. Some symbolic shortcuts for mathematical concepts came to be used in the publication of geometric proofs. Moreover, the power and authority of geometry's theorem and proof structure greatly influenced non-geometric treatises, Isaac Newton
Isaac Newton
Sir Isaac Newton PRS was an English physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian, who has been "considered by many to be the greatest and most influential scientist who ever lived."...

's Principia Mathematica
Philosophiae Naturalis Principia Mathematica
Philosophiæ Naturalis Principia Mathematica, Latin for "Mathematical Principles of Natural Philosophy", often referred to as simply the Principia, is a work in three books by Sir Isaac Newton, first published 5 July 1687. Newton also published two further editions, in 1713 and 1726...

, for example.

### Counting is mechanized

After the rise of Boolean algebra and the development of positional notation
Positional notation
Positional notation or place-value notation is a method of representing or encoding numbers. Positional notation is distinguished from other notations for its use of the same symbol for the different orders of magnitude...

, it became possible to mechanize simple circuits for counting, first by mechanical means, such as gears and rods, using rotation
Rotation
A rotation is a circular movement of an object around a center of rotation. A three-dimensional object rotates always around an imaginary line called a rotation axis. If the axis is within the body, and passes through its center of mass the body is said to rotate upon itself, or spin. A rotation...

and translation
Translation
Translation is the communication of the meaning of a source-language text by means of an equivalent target-language text. Whereas interpreting undoubtedly antedates writing, translation began only after the appearance of written literature; there exist partial translations of the Sumerian Epic of...

to represent changes of state
State (computer science)
In computer science and automata theory, a state is a unique configuration of information in a program or machine. It is a concept that occasionally extends into some forms of systems programming such as lexers and parsers....

, then by electrical means, using changes in voltage and current to represent the analogs of quantity. Today, computers use standard circuits to both store and change quantities, which represent not only numbers but pictures, sound, motion, and control.

### Modern notation

The 18th and 19th centuries saw the creation and standardization of mathematical notation as used today. Euler was responsible for many of the notations in use today: the use of a, b, c for constants and x, y, z for unknowns, e for the base of the natural logarithm, sigma (Σ) for summation, i for the imaginary unit
Imaginary unit
In mathematics, the imaginary unit allows the real number system ℝ to be extended to the complex number system ℂ, which in turn provides at least one root for every polynomial . The imaginary unit is denoted by , , or the Greek...

, and the functional notation f(x). He also popularized the use of π for Archimedes constant (due to William Jones' proposal for the use of π in this way based on the earlier notation of 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...

). Many fields of mathematics bear the imprint of their creators for notation: the differential operator is due to Leibniz, the cardinal
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...

infinities to 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,...

(in addition to the lemniscate (∞) of John Wallis), the congruence symbol (≡) to 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...

, and so forth.

### Computerized notation

The rise of expression evaluators such as calculators and slide rules were only part of what was required to mathematicize civilization. Today, keyboard-based notations are used for the e-mail of mathematical expressions, the Internet shorthand notation. The wide use of programming language
Programming language
A programming language is an artificial language designed to communicate instructions to a machine, particularly a computer. Programming languages can be used to create programs that control the behavior of a machine and/or to express algorithms precisely....

s, which teach their users the need for rigor in the statement of a mathematical expression (or else the compiler will not accept the formula) are all contributing toward a more mathematical viewpoint across all walks of life. Mathematically oriented markup languages such as
TeX
TeX is a typesetting system designed and mostly written by Donald Knuth and released in 1978. Within the typesetting system, its name is formatted as ....

,
LaTeX
LaTeX is a document markup language and document preparation system for the TeX typesetting program. Within the typesetting system, its name is styled as . The term LaTeX refers only to the language in which documents are written, not to the editor used to write those documents. In order to...

and, more recently, MathML
MathML
Mathematical Markup Language is an application of XML for describing mathematical notations and capturing both its structure and content. It aims at integrating mathematical formulae into World Wide Web pages and other documents...

are powerful enough that they qualify as mathematical notations in their own right.

For some people, computerized visualizations have been a boon to comprehending mathematics that mere symbolic notation could not provide. They can benefit from the wide availability of devices, which offer more graphical, visual, aural, and tactile feedback.

### Ideographic notation

In the history of writing, ideographic symbols arose first, as more-or-less direct renderings of some concrete item. This has come full circle with the rise of computer visualization systems, which can be applied to abstract visualizations as well, such as for rendering some projections of a Calabi-Yau manifold
Manifold
In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....

.

Examples of abstract visualization
Information visualization
Information visualization is the interdisciplinary study of "the visual representation of large-scale collections of non-numerical information, such as files and lines of code in software systems, library and bibliographic databases, networks of relations on the internet, and so forth".- Overview...

which properly belong to the mathematical imagination can be found, for example in computer graphics
Computer graphics
Computer graphics are graphics created using computers and, more generally, the representation and manipulation of image data by a computer with help from specialized software and hardware....

. The need for such models abounds, for example, when the measures for the subject of study are actually random variable
Random variable
In probability and statistics, a random variable or stochastic variable is, roughly speaking, a variable whose value results from a measurement on some type of random process. Formally, it is a function from a probability space, typically to the real numbers, which is measurable functionmeasurable...

s and not really ordinary mathematical functions.

## Non-Latin-based mathematical notation

Modern Arabic mathematical notation
Modern Arabic mathematical notation
The designation modern Arabic mathematical notation is used for a mathematical notation based on the Arabic script that is widely used in the Arab world, especially at pre-university levels of education. Its form is mostly derived from Western notation, but has some notable features that set it...

is based mostly on the Arabic alphabet
Arabic alphabet
The Arabic alphabet or Arabic abjad is the Arabic script as it is codified for writing the Arabic language. It is written from right to left, in a cursive style, and includes 28 letters. Because letters usually stand for consonants, it is classified as an abjad.-Consonants:The Arabic alphabet has...

and is used widely in the Arab world
Arab world
The Arab world refers to Arabic-speaking states, territories and populations in North Africa, Western Asia and elsewhere.The standard definition of the Arab world comprises the 22 states and territories of the Arab League stretching from the Atlantic Ocean in the west to the Arabian Sea in the...

, especially in pre-university levels of education.

Some mathematical notations are mostly diagrammatic, and so are almost entirely script independent. Examples are Penrose graphical notation
Penrose graphical notation
In mathematics and physics, Penrose graphical notation or tensor diagram notation is a visual depiction of multilinear functions or tensors proposed by Roger Penrose. A diagram in the notation consists of several shapes linked together by lines, much like tinker toys...

and Coxeter-Dynkin diagram
Coxeter-Dynkin diagram
In geometry, a Coxeter–Dynkin diagram is a graph with numerically labeled edges representing the spatial relations between a collection of mirrors...

s.

Braille-based mathematical notations used by blind people include Nemeth Braille
Nemeth Braille
The Nemeth Braille Code for Mathematics is a Braille code for encoding mathematical and scientific notation linearly using standard six-dot Braille cells for tactile reading by the visually impaired. The code was developed by Abraham Nemeth....

and GS8 Braille
GS8 Braille
The GS8 or Gardner-Salinas Braille code is a method of encoding mathematical and scientific notation linearly using eight-dot Braille cells for tactile reading by the visually impaired....

.

• Abuse of notation
Abuse of notation
In mathematics, abuse of notation occurs when an author uses a mathematical notation in a way that is not formally correct but that seems likely to simplify the exposition or suggest the correct intuition . Abuse of notation should be contrasted with misuse of notation, which should be avoided...

• Begriffsschrift
Begriffsschrift
Begriffsschrift is a book on logic by Gottlob Frege, published in 1879, and the formal system set out in that book...

• Bourbaki dangerous bend symbol
Bourbaki dangerous bend symbol
The dangerous bend or caution symbol ☡ was created by the Nicolas Bourbaki group of mathematicians and appears in the margins of mathematics books written by the group...

• History of mathematical notation
History of mathematical notation
Mathematical notation comprises the symbols used to write mathematical equations and formulas. It includes Hindu-Arabic numerals, letters from the Roman, Greek, Hebrew, and German alphabets, and a host of symbols invented by mathematicians over the past several centuries.The development of...

• ISO 31-11
ISO 31-11
ISO 31-11 was the part of international standard ISO 31 that defines mathematical signs and symbols for use in physical sciences and technology...

• ISO/IEC 80000
ISO/IEC 80000
International standard ISO 80000 or IEC 80000—depending on which of the two international standards bodies International Organization for Standardization and International Electrotechnical Commission is in charge of each respective part—is a style guide for the use of physical quantities and units...

-2
• Mathematical Alphanumeric Symbols
Mathematical alphanumeric symbols
Mathematical Alphanumeric Symbols is a Unicode block of Latin and Greek letters and decimal digits that enable mathematicians to denote different notions with different letter styles .Unicode now includes many such symbols Mathematical Alphanumeric Symbols is a Unicode block of Latin and Greek...

• Notation in probability
Notation in probability
Probability theory and statistics has some commonly used conventions of its own, in addition to standard mathematical notation and mathematical symbols.-Probability theory:* Random variables are usually written in upper case roman letters: X, Y, etc....

• Scientific notation
Scientific notation
Scientific notation is a way of writing numbers that are too large or too small to be conveniently written in standard decimal notation. Scientific notation has a number of useful properties and is commonly used in calculators and by scientists, mathematicians, doctors, and engineers.In scientific...

• Table of mathematical symbols
Table of mathematical symbols
This is a listing of common symbols found within all branches of mathematics. Each symbol is listed in both HTML, which depends on appropriate fonts being installed, and in , as an image.-Symbols:-Variations:...

• Typographical conventions in mathematical formulae
Typographical conventions in mathematical formulae
Typographical conventions in mathematical formulae provide uniformity across mathematical texts and help the readers of those texts to grasp new concepts quickly....

• Modern Arabic mathematical notation
Modern Arabic mathematical notation
The designation modern Arabic mathematical notation is used for a mathematical notation based on the Arabic script that is widely used in the Arab world, especially at pre-university levels of education. Its form is mostly derived from Western notation, but has some notable features that set it...

• Earliest Uses of Various Mathematical Symbols
• Mathematical ASCII Notation how to type math notation in any text editor.
• Mathematics as a Language at cut-the-knot
Cut-the-knot
Cut-the-knot is a free, advertisement-funded educational website maintained by Alexander Bogomolny and devoted to popular exposition of many topics in mathematics. The site has won more than 20 awards from scientific and educational publications, including a Scientific American Web Award in 2003,...

• Stephen Wolfram
Stephen Wolfram
Stephen Wolfram is a British scientist and the chief designer of the Mathematica software application and the Wolfram Alpha computational knowledge engine.- Biography :...

: Mathematical Notation: Past and Future. October 2000. Transcript of a keynote address presented at MathML
MathML
Mathematical Markup Language is an application of XML for describing mathematical notations and capturing both its structure and content. It aims at integrating mathematical formulae into World Wide Web pages and other documents...

and Math on the Web: MathML International Conference.