Mathematical notation is used in mathematics, and throughout the physical sciences, engineering, and economics. The complexity of such notation ranges from relatively simple symbolic representations, such as numbers 1 and 2; function symbols sin and +, to conceptual symbols, such as lim and dy/dx; to equations and variables. For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ... Physical science is the branch of science including chemistry and physics, usually contrasted with the social sciences and sometimes including and sometimes contrasted with natural or biological science. ... Engineering is the discipline and profession of applying scientific knowledge and utilizing natural laws and physical resources in order to design and implement materials, structures, machines, devices, systems, and processes that realize a desired objective and meet specified criteria. ... Face-to-face trading interactions on the New York Stock Exchange trading floor. ... This article is about the number one. ... This article does not cite any references or sources. ... This article is about functions in mathematics. ... In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ... 3 + 2 = 5 with apples, a popular choice in textbooks[1] This article is about addition in mathematics. ... Wikibooks Calculus has a page on the topic of Limits In mathematics, the concept of a limit is used to describe the behavior of a function as its argument either gets close to some point, or as it becomes arbitrarily large; or the behavior of a sequences elements as... This article is about derivatives and differentiation in mathematical calculus. ... An equation is a mathematical statement, in symbols, that two things are the same (or equivalent). ... In computer science and mathematics, a variable (pronounced ) (sometimes called an object or identifier in computer science) is a symbolic representation used to denote a quantity or expression. ...

Definition

A mathematical notation is a writing system (in fact, a formal language) used for recording concepts in mathematics. Writing systems of the world today. ... In mathematics, logic, and computer science, a formal language is a language that is defined by precise mathematical or machine processable formulas. ... For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ...

• The notation uses symbols or symbolic expressions which are intended to have a precise semantic meaning.
• In the history of mathematics, 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.

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, Cogency, Mathematical logic, Model theory, and Major themes in mathematics.) An expression is a combination of numbers, operators, grouping symbols (such as brackets and parentheses) and/or free variables and bound variables arranged in a meaningful way which can be evaluated. ... For a timeline of events in mathematics, see timeline of mathematics. ... The central question involved in discussing mathematics as a language can be stated as follows : What do we mean when we talk about the language of mathematics ? To what extent does mathematics meet generally accepted criteria of being a language ? A secondary question is : If it is valid to consider... In linguistics, the topic (or theme) is the part of the proposition that is being talked about (predicated). ... In logic, an argument is a set of statements, consisting of a number of premises, a number of inferences, and a conclusion, which is said to have the following property: if the premises are true, then the conclusion must be true or highly likely to be true. ... An argument is cogent if and only if the truth of the arguments premises would render the truth of the conclusion probable (i. ... Mathematical logic is a major area of mathematics, which grew out of symbolic logic. ... In mathematics, model theory is the study of the representation of mathematical concepts in terms of set theory, or the study of the structures that underlie mathematical systems. ... For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ...

Expressions

A mathematical expression 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 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 compilers. For more on expression evaluation, see the computer science topics: eager evaluation, lazy evaluation, and evaluation operator. An expression is a combination of numbers, operators, grouping symbols (such as brackets and parentheses) and/or free variables and bound variables arranged in a meaningful way which can be evaluated. ... In arithmetic and algebra, when a number or expression is both preceded and followed by a binary operation, a rule is required for which operation should be applied first. ... Look up computer language & a Brief History of it in Wiktionary, the free dictionary. ... A diagram of the operation of a typical multi-language, multi-target compiler. ... Computer science, or computing science, is the study of the theoretical foundations of information and computation and their implementation and application in computer systems. ... Eager evaluation is the evaluation model in most traditional programming languages. ... In computer programming, lazy evaluation is a technique that attempts to delay computation of expressions until the results of the computation are known to be needed. ... This article is being considered for deletion in accordance with Wikipedias deletion policy. ...

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 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 of that object has a heuristic 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. This article is about the word proposition as it is used in logic, philosophy, and linguistics. ... For other senses of this word, see sequence (disambiguation). ... An abstract model (or conceptual model) is a theoretical construct that represents something, with a set of variables and a set of logical and quantitative relationships between them. ... Semantics (Ancient σημαντικός semantikos significant, from semainein to signify, mean, from sema sign, token), is the study of meaning in communication. ... For other uses, see Heuristic (disambiguation). ... Deductive reasoning is the process of reaching a conclusion that is guaranteed to follow, if the evidence provided is true and the reasoning used to reach the conclusion is correct. ... An intensional definition gives the meaning of a term by giving all the properties required of something that falls under that definition; 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. This mathematical notation might include annotation such as The following table lists many specialized symbols commonly used in mathematics. ...

• "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"

History

Main article: History of mathematical notation

Mathematic notation comprises the symbols used in expressing mathematical expressions, equations, and formulas. ...

Counting

It is believed that a mathematical notation was first developed at least 50,000 years ago in order to assist with counting.^{[citation needed]} Early mathematical ideas for counting were represented by collections of rocks, sticks, bone, clay, stone, wood carvings, and knotted ropes. The tally stick is a timeless way of counting. Perhaps the oldest known mathematical texts are those of ancient Sumer. The Census Quipu of the Andes and the Ishango Bone from Africa both used the tally mark method of accounting for numerical concepts. Tally sticks are an ancient mnemonic device (memory aid) to record and document numbers or quantities even messages. ... Sumer (or Å umer; Sumerian: KI-EN-GIR [1]) was the earliest known civilization of the ancient Near East, located in lower Mesopotamia (modern Iraq), from the time of the earliest records in the mid 4th millennium BC until the rise of Babylonia in the late 3rd millennium BC. The term... Khipu, or quipa, or quipu were recording devices used by the Tahuantinsuyu (Inca empire) and its predecessor societies in the Andean region. ... The Ishango bone is a tally stick, made of bone, which contains sequences of prime numbers, and some series of multiples. ... Tally marks are a variation of the unary numeral system. ...

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 and Greek Egyptians, and then as an integer by the Mayans, Indians and Arabs. (See The history of zero for more information.) 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. ... Greek numerals are a system of representing numbers using letters of the Greek alphabet. ... Mayan numerals. ... For other uses, see Arabic numerals (disambiguation). ... Zero redirects here. ...

Geometry becomes analytic

The mathematical viewpoints in geometry did not lend themselves well to counting. The natural numbers, their relationship to fractions, 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 geometry by René Descartes 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's Principia Mathematica, for example. For other uses, see Geometry (disambiguation). ... In mathematics, a natural number can mean either an element of the set {1, 2, 3, ...} (i. ... For other uses, see Fraction (disambiguation). ... In mathematics, the word continuum sometimes denotes the real line. ... Analytic geometry, also called coordinate geometry and earlier referred to as Cartesian geometry or analytical geometry, is the study of geometry using the principles of algebra. ... Descartes redirects here. ... Sir Isaac Newton FRS (4 January 1643 – 31 March 1727) [ OS: 25 December 1642 – 20 March 1727][1] was an English physicist, mathematician, astronomer, natural philosopher, and alchemist. ... Newtons own copy of his Principia, with handwritten corrections for the second edition. ...

Counting is mechanized

After the rise of Boolean algebra and the development of positional notation, it became possible to mechanize simple circuits for counting, first by mechanical means, such as gears and rods, using rotation and translation to represent changes of state, 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. Boolean algebra is the finitary algebra of two values. ... A positional notation or place-value notation system is a numeral system in which each position is related to the next by a constant multiplier, a common ratio, called the base or radix of that numeral system. ... This article is about rotation as a movement of a physical body. ... Look up translate in Wiktionary, the free dictionary. ... In information processing, a state is the complete set of properties (for example, its energy level, etc. ...

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, 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). Many fields of mathematics bear the imprint of their creators for notation: the differential operator is due to Leibniz^[1], the cardinal infinities to Georg Cantor (in addition to the lemniscate (∞) of John Wallis), the congruence symbol (≡) to Gauss, and so forth. Leonhard Euler aged 49 (oil painting by Emanuel Handmann, 1756) Leonhard Euler (April 15, 1707 - September 18, 1783) (pronounced oiler) was a Swiss mathematician and physicist. ... In mathematics, the imaginary unit (or sometimes the Latin or the Greek iota, see below) allows the real number system to be extended to the complex number system . ... The title given to this article is incorrect due to technical limitations. ... William Jones is a common name, especially in Wales, and there have been several well-known individuals of this name, including: // Academics and authors William Jones (historian) (1860–1932) Sir William Jones (mathematician) (~1675–1749), father of Sir William Jones (philologist) Sir William Jones (philologist) (1746–1794) son of Sir... William Oughtred William Oughtred (March 5, 1575 – June 30, 1660) was an English mathematician. ... Gottfried Leibniz Gottfried Wilhelm von Leibniz (July 1, 1646 in Leipzig - November 14, 1716 in Hannover) was a German philosopher, scientist, mathematician, diplomat, librarian, and lawyer of Sorb descent. ... Aleph-0, the smallest infinite cardinal In mathematics, cardinal numbers, or cardinals for short, are a generalized kind of number used to denote the size of a set, known as its cardinality. ... Georg Ferdinand Ludwig Philipp Cantor (March 3, 1845[1] – January 6, 1918) was a German mathematician. ... A lemniscate In mathematics, a lemniscate is a type of curve described by a Cartesian equation of the form: Graphing this equation produces a curve similar to . ... John Wallis John Wallis (November 22, 1616 - October 28, 1703) was an English mathematician who is given partial credit for the development of modern calculus. ... Johann Carl Friedrich Gauss (pronounced ,  ; in German usually Gauß, Latin: ) (30 April 1777 – 23 February 1855) was a German mathematician and scientist who contributed significantly to many fields, including number theory, statistics, analysis, differential geometry, geodesy, electrostatics, astronomy, and optics. ...

 Yes correct. 

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 languages, 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. Internet shorthand notation is a notation widely used on Internet sites, where typing complicated mathematical symbols is rather cumbersome. ... A programming language is an artificial language that can be used to control the behavior of a machine, particularly a computer. ...

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. Graphic design is the applied art of arranging image and text to communicate a message. ... Vision can refer to: Visual perception is one of the senses. ... Hearing is one, the auditory, of the traditional five senses, and refers to the ability to detect sound. ... Tactition is the sense of pressure perception. ...

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. In mathematics, a Calabi-Yau manifold is a compact K hler manifold with a vanishing first Chern class. ... On a sphere, the sum of the angles of a triangle is not equal to 180° (see spherical trigonometry). ...

Examples of abstract visualization which properly belong to the mathematical imagination can be found, for example in computer graphics. The need for such models abounds, for example, when the measures for the subject of study are actually random variables and not really ordinary mathematical functions. Information visualization is a complex research area. ... This article is about the scientific discipline of computer graphics. ... In probability theory, a random variable is a quantity whose values are random and to which a probability distribution is assigned. ... In mathematics, a function is a relation, such that each element of a set (the domain) is associated with a unique element of another (possibly the same) set (the codomain, not to be confused with the range). ...

Non-Latin-based mathematical notation

Modern Arabic mathematical notation is based mostly on the Arabic alphabet and is used widely in the Arab world, especially in pre-university levels of education. Modern Arabic mathematical notation is a mathematical notation that is based on the Arabic script. ... The Arabic alphabet is the script used for writing languages such as Arabic, Persian, Urdu, and others. ... Arab States redirects here. ...

See also

• Abuse of notation
• Notation in probability
• Begriffsschrift
• Typographical conventions in mathematical formulae
• Rendering mathematical formulas in Wikipedia
• History of mathematical notation
• Table of mathematical symbols
• Scientific notation
• ISO_31-11

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 (while being unlikely to introduce errors or cause confusion). ... In probability theory and statistics, some special forms of mathematical notation are of interest : Random variables (for example, the height of students) are written in upper case. ... Begriffsschrift is the title of a short book on logic by Gottlob Frege, published in 1879, and is also the name of the formal system set out in that book. ... Roman letters used in mathematics Greek letters used in mathematics See also: Mathematical alphanumeric symbols Table of mathematical symbols Categories: Mathematics stubs ... Mathematic notation comprises the symbols used in expressing mathematical expressions, equations, and formulas. ... The following table lists many specialized symbols commonly used in mathematics. ... Scientific notation, also known as standard form, is a notation for writing numbers that is often used by scientists and mathematicians to make it easier to write large and small numbers. ... ISO 31-11 is the part of international standard ISO 31 that defines mathematical signs and symbols for use in physical sciences and technology. ...

Notes

• Florian Cajori, A History of Mathematical Notations (1929), 2 volumes. ISBN 0-486-67766-4

1. ^ Gottfried Wilhelm Leibnitz (1646 - 1716)

Florian Cajori at Colorado College Florian Cajori was born February 28, 1859 in St Aignan (near Thusis), Graubünden, Switzerland. ...

External links

• Mathematics as a Language at cut-the-knot
• Earliest Uses of Various Mathematical Symbols
• Mathematical ASCII Notation how to type math notation in any text editor.
• Stephen Wolfram: Mathematical Notation: Past and Future. October 2000. Transcript of a keynote address presented at MathML and Math on the Web: MathML International Conference.