A
mathematical notation is a system of symbolic representations of mathematical objects and ideas. Mathematical notations are used in
mathematicsMathematics is the science and study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions....
and the physical sciences,
engineeringEngineering is the discipline, art and profession of acquiring and applying technical, scientific and mathematical knowledge to design and implement materials, structures, machines, devices, systems, and processes that safely realize a desired objective or inventions.The American Engineers' Council...
and
economicsEconomics is the social science that studies 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 numbers
1{| class="infobox" style="width: 20em;"|-! colspan="2" align="center" style="font: 10em times; background:#ccc;" | 1|-| colspan="2" | |-| Cardinal| 1
one|-| Ordinal| 1st
first|-| Numeral system| unary|-| Factorization| |-...
and
2class="infobox" style="width: 20em;"|-! colspan="2" align="center" style="font: 10em times; background:#ccc;" | 2|-| colspan="2" | |-| Ordinal number| 2nd
second|-| Numeral system| binary|-| Factorisation| prime|-| Gaussian integer factorisation...
,
functionIn mathematics, a function is a relation between a given set of elements and another set of elements , which associates each element in the domain with exactly one element in the codomain...
symbols
sinMaurice Sinet, known as Siné is a French cartoonist.As a young man he studied drawing and graphic arts, while earning a living as a cabaret singer. After his military service he started publishing his drawings and also worked as a photo-retoucher for porn magazines. His first published drawing...
and
+Addition is the mathematical process of combining quantities. 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. Therefore, 3 + 2 = 5...
; conceptual symbols, such as
lim,
dy/dxIn calculus 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 a quantity is changing at a given point; for example, the derivative of the position of a vehicle with respect to time is the instantaneous velocity...
,
equationAn equation is a mathematical statement, in symbols, that two things are exactly the same . Equations are written with an equal sign, as in...
s and
variablesA variable is a symbol that stands for a value that may vary; the term usually occurs in opposition to constant, which is a symbol for a non-varying value, i.e. completely fixed or fixed in the context of use...
; and complex diagramatic notations such as
Penrose graphical notationIn 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 diagramIn geometry, a Coxeter–Dynkin diagram is a graph with numerically labelled edges representing the spatial relations between a collection of mirrors...
s.
Definition
A mathematical notation is a
writing systemA writing system is a type of 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 one must usually understand something of the associated spoken language to...
(in fact, a
formal languageA formal language is a set of words, i.e. finite strings of letters, symbols, or tokens. The set from which these letters are taken is called the alphabet over which the language is defined...
) used for recording concepts in
mathematicsMathematics is the science and study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions....
.
- The notation uses symbol
A symbol is something such as an object, picture, written word, sound, or particular mark that represents something else by association, resemblance, or convention. For example, a red octagon may stand for "STOP". On maps, crossed sabres may indicate a battlefield...
s or symbolic expressionsIn mathematics, the word expression is a term for any well-formed combination of mathematical symbols. An algebraic expression is only a phrase, not a whole sentence, so it cannot contain an equality sign . For example,is an expression, while...
which are intended to have a precise semantic meaning.
- In the 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 languageThe central question involved in discussing mathematics as a language can be stated as follows:A secondary question is:-What is a language?:To answer the first question, we need some definitions of language:...
.
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. The
systematic adherence to mathematical concepts is a fundamental concept of mathematical notation. (See also some related concepts: Topic (linguistics), Logical argument,
CogencyAn argument is cogent if and only if the truth of the argument's premises would render the truth of the conclusion probable , and the argument's premises are, in fact, true...
,
Mathematical logicMathematical logic is a subfield of mathematics with close connections to 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...
,
Model theoryIn mathematics, model theory is the study of mathematical structures such as groups, fields, graphs, or even universes of set theory, using tools from mathematical logic. A structure that gives meaning to the sentences of a formal language is called a model for the language...
, and Major themes in mathematics.)
Expressions
A
mathematical expressionIn mathematics, the word expression is a term for any well-formed combination of mathematical symbols. An algebraic expression is only a phrase, not a whole sentence, so it cannot contain an equality sign . For example,is an expression, while...
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 operationsIn mathematics and computer programming, an expression or string of symbols is intended to represent a numerical value; a properly-formed expression may be evaluated in an unambiguous way. But in practice, an expression with multiple terms and operators may be written with parentheses, in...
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
compilerA compiler is a computer program that transforms source code written in a computer language into another computer language...
s. For more on expression evaluation, see the
computer scienceComputer science is the study of the theoretical foundations of information and computation, and of practical techniques for their implementation and application in computer systems. It is frequently described as the systematic study of algorithmic processes that create, describe and transform...
topics:
eager evaluationEager 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...
,
lazy evaluationIn computer programming, lazy evaluation is the technique of delaying a computation until the result is required.The benefits of lazy evaluation include: performance increases due to avoiding unnecessary calculations, avoiding error conditions in the evaluation of compound expressions, the ability...
, 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
sequenceIn 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...
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
semanticsSemantics is the study of meaning, usually in language. The word "semantics" itself denotes a range of ideas, from the popular to the highly technical. It is often used in ordinary language to denote a problem of understanding that comes down to word selection or connotation. This problem of...
of that object has a
heuristicHeuristic is an adjective for experience-based techniques that help in problem solving, learning and discovery. A heuristic method is particularly used to rapidly come to a solution that is hoped to be close to the best possible answer, or 'optimal solution'...
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 definitionIn 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-Common symbols:This is a listing of common symbols found within all branches of mathematics.-See also:* Greek letters used in mathematics* ISO 31-11* List of mathematical abbreviations* Mathematical alphanumeric symbols* Mathematical notation...
. 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 problematical if the author assumes the reader is already familiar with the notation in use.
Counting
It is believed that a mathematical notation to represent
countingCounting is the mathematical action of repeatedly adding one, usually to find out how many objects there are or to set aside a desired number of objects , or for well-ordered objects, to find the ordinal number of a...
was first developed at least 50,000 years ago — early mathematical ideas such as
finger countingFinger counting, or dactylonomy, is the art of counting along one's fingers. Though marginalized in modern societies by the Arabic numeral system, formerly different systems flourished in many cultures, including educated methods far more sophisticated than the one-by-one finger count taught...
have also been represented by collections of rocks, sticks, bone, clay, stone, wood carvings, and knotted ropes. The
tally stickA 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...
is a timeless way of counting. Perhaps the oldest known mathematical texts are those of ancient
SumerSumer was a civilization and historical region in southern Iraq . It is the earliest known civilization in the world and is known as the Cradle of Civilization...
. The Census Quipu of the Andes and the
Ishango BoneThe 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 or writing...
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
BabyloniansBabylonian 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 EgyptiansGreek 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
MayansThe Pre-Columbian Maya civilization used a vigesimal numeral system.The numerals are made up of three symbols; zero , one and five . For example, nineteen is written as four dots in a horizontal row above three horizontal lines stacked upon each other.- Numbers above 19 :
Numbers after 19...
,
IndiansMost of the positional base 10 numeral systems in the world have originated from India, which first developed the concept of positional numerology...
and
ArabsThe Arabic numerals are the ten digits . They are descended from Indian numerals and the Hindu-Arabic numeral system developed by Indian mathematicians, by which a sequence of digits such as "975" is read as a whole number...
. (See The history of zero for more information.)
Geometry becomes analytic
The mathematical viewpoints in
geometryGeometry 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 numberIn mathematics, there are two conventions for the set of natural numbers: it is either the set of positive integers {, , , ...} according to the traditional definition or the set of non-negative integers {, 1, 2, ...} according to...
s, their relationship to
fractionA fraction is a number that can represent part of a whole.The earliest fractions were reciprocals of integers, symbols representing one half, one third, one quarter, and so on...
s, and the identification of
continuous 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 geometryAnalytic geometry, also known as coordinate geometry, analytical geometry, or Cartesian geometry, is the study of geometry using a coordinate system and the principles of algebra and analysis...
by
René DescartesRené Descartes , , also known as Renatus Cartesius , was a French philosopher, mathematician, physicist, and writer who spent most of his adult life in the Dutch Republic...
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 NewtonSir Isaac Newton FRS was an English physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian who is perceived and considered by a substantial number of scholars and the general public as one of the most influential men in history...
's
Principia MathematicaThe Philosophiæ Naturalis Principia Mathematica, Latin for "Mathematical Principles of Natural Philosophy", often Principia or Principia Mathematica for short, is a work in three books by Isaac Newton, first published on 5 July 1687. Newton also published two further editions, the second in 1713,...
, for example.
Counting is mechanized
After the rise of Boolean algebra and the development of
positional notationA positional notation or place-value notation system is a numeral system in which each position is related to the next by a constant multiplier, called the base or radix of that numeral system. The value of each digit position is the value of its digit multiplied by a power of the base; the power...
, it became possible to mechanize simple circuits for counting, first by mechanical means, such as gears and rods, using
rotationA rotation is a movement of an object in a circular motion. A two-dimensional object rotates around a center of rotation. A three-dimensional object rotates around a line called an axis. If the axis of rotation is within the body, the body is said to rotate upon itself, or spin—which implies...
and
translationTranslation is the interpreting of the meaning of a text and the subsequent production of an equivalent text, likewise called a "translation," that communicates the same message in another language...
to represent changes of
stateIn 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 unitIn mathematics, physics, and engineering, the imaginary unit is denoted by i or the Latin j or the Greek iota...
, and the functional notation
f(
x). He also popularized the use of π for Archimedes constant (due to
William Jones-Academics and authors:*William Jones , Welsh mathematician who proposed the use of the symbol π*William Jones , English optics manufacturer and instrument maker....
' proposal for the use of π in this way based on the earlier notation of
William OughtredWilliam 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
cardinalIn 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 CantorGeorg Ferdinand Ludwig Phillip Cantor was a German mathematician, born in Russia. He is best known as the creator of set theory, which has become a fundamental theory in mathematics...
(in addition to the
lemniscateIn algebraic geometry, lemniscate refers to any of several figure-eight or ∞ shaped curves, of which the best known is the Lemniscate of Bernoulli.
The most general visual form for this curve is seen as a toric section of a torus.
...
(∞) of
John WallisJohn Wallis was an English mathematician who is given partial credit for the development of modern calculus. Between 1643 and 1689 he served as chief cryptographer for Parliament and, later, the royal court. He is also credited with introducing the symbol ∞ for infinity...
), the congruence symbol (≡) to
GaussJohann 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...
, 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 languageA programming language is an artificial language designed to express computations that can be performed by a machine, particularly a computer. Programming languages can be used to create programs that control the behavior of a machine, to express algorithms precisely, or as a mode of human...
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
MathMLMathematical 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 documents...
and
LaTeX is a document markup language and document preparation system for the TeX typesetting program. Within the typesetting system, its name is styled as ....
have become so powerful and widespread 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 manifoldIn mathematics, more specifically in differential geometry and topology, a manifold is a mathematical space that on a small enough scale resembles the Euclidean space of a certain dimension, called the dimension of the manifold....
.
Examples of
abstract visualizationInformation 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".- Overview :...
which properly belong to the mathematical imagination can be found, for example in
computer graphicsComputer graphics are graphics created using computers and, more generally, the representation and manipulation of pictorial data by a computer....
. The need for such models abounds, for example, when the measures for the subject of study are actually
random variableIn mathematics, random variables are used in the study of probability. They were developed to assist in the analysis of games of chance, stochastic events, and the results of scientific experiments by capturing only the mathematical properties necessary to answer probabilistic questions...
s and not really ordinary mathematical functions.
Non-Latin-based mathematical notation
Modern Arabic mathematical notationThe 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 alphabetThe Arabic alphabet is the script used for writing several languages of Asia and Africa, such as Arabic and Urdu. After the Latin alphabet, it is the second-most widely used alphabet around the world....
and is used widely in the
Arab worldThe Arab World refers to Arabic-speaking countries stretching from the Atlantic Ocean in the west to the Arabian Sea in the east, and from the Mediterranean Sea in the north to the Horn of Africa and the Indian Ocean in the southeast...
, especially in pre-university levels of education.
Some mathematical notations are mostly diagramatic, and so are almost entirely script independent. Examples are
Penrose graphical notationIn 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 diagramIn geometry, a Coxeter–Dynkin diagram is a graph with numerically labelled edges representing the spatial relations between a collection of mirrors...
s.
Braille-based mathematical notations used by blind people include
Nemeth BrailleThe 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 BrailleThe 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....
.
See also
- 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 is the title of a book on logic by Gottlob Frege, published in 1879, and is also the name of the formal system set out in that book....
- History of mathematical notation
Mathematical notation comprises the symbols used to write mathematical equations and formulas. It includes Arabic numerals, letters from the Roman, Greek, Hebrew, and German alphabets, and a host of symbols invented by mathematicians over the past several centuries.-Beginning of notation:Written...
- ISO 31-11
ISO 31-11 is the part of international standard ISO 31 that defines mathematical signs and symbols for use in physical sciences and technology.Its definitions include:-Mathematical logic:-Sets:-Miscellaneous signs and symbols:-Operations:...
- 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 written in upper case....
- Rendering mathematical formulas in Wikipedia
- Scientific notation
Scientific notation, also known as standard form or as exponential notation, is a way of writing numbers that accommodates values too large or small to be conveniently written in standard decimal notation...
- Table of mathematical symbols
-Common symbols:This is a listing of common symbols found within all branches of mathematics.-See also:* Greek letters used in mathematics* ISO 31-11* List of mathematical abbreviations* Mathematical alphanumeric symbols* Mathematical notation...
- 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
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...
External links
- 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 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 is a British physicist, software developer, mathematician, computer programmer, author and businessman, known for his work in theoretical particle physics, cosmology, cellular automata, complexity theory, computer algebra and the Wolfram Alpha computational knowledge engine.-...
: Mathematical Notation: Past and Future. October 2000. Transcript of a keynote address presented at MathMLMathematical 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 documents...
and Math on the Web: MathML International Conference.