Mathematical notation is a system of symbolic representations of mathematical objects and ideas. Mathematical notations are used in mathematics, the physical sciences, engineering, and economics. Mathematical notations include relatively simple symbolic representations, such as the numbers 0, 1 and 2; function symbols such as sin; operator symbols such as "+"; conceptual symbols such as lim and dy/dx; equations and variables; and complex diagrammatic notations such as Penrose graphical notation and Coxeter–Dynkin diagrams.
Definition
A mathematical notation is a writing system used for recording concepts in mathematics.
The notation uses symbols or symbolic expressions that 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, and electronic media. Systematic adherence to mathematical concepts is a fundamental concept of mathematical notation.
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. 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. 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
"All x", "No x", "There is an 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.
History
Counting
It is believed that a mathematical notation to represent counting was first developed at least 50,000 years ago — early mathematical ideas such as finger counting have also been represented by collections of rocks, sticks, bone, clay, stone, wood carvings, and knotted ropes. The tally stick is a way of counting dating back to the Upper Paleolithic. 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. 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.
Geometry becomes analytic
The earliest 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.
Mathematically oriented markup languages such as TeX, LaTeX and, more recently, MathML are powerful enough to express a wide variety of mathematical notations. Theorem-proving software naturally comes with its own notations for mathematics; the seeks to provide an open commons for such notations; and the provides a basis for interoperability between other notations.
Non-Latin-based mathematical notation
is based mostly on the Arabic alphabet and is used widely in the Arab world, especially in pre-tertiary education. Some mathematical notations are mostly diagrammatic, and so are almost entirely script independent. Examples are Penrose graphical notation and Coxeter–Dynkin diagrams. Braille-based mathematical notations used by blind people include Nemeth Braille and GS8 Braille.
