Mathematical notation
As a mathematical notation is referred to in mathematics , logic and computer science the presentation of formulas and other mathematical objects by means of mathematical symbols . The mathematical notation corresponds to a language that is more formal than many natural languages and yet contains some ambiguities, which are characteristic of natural languages.
Components
The mathematical notation uses special symbols
 as for mathematical objects , such as functions or numbers ,
 Brackets for mapping purposes
 and for the construction of templates.
A distinction is made between the names for mathematical objects
 Constants (fixed values), i.e. general terms for frequently used objects such as and
 Variables (changeable values), for example names for objects that have yet to be found or about which one would like to say something in general.
Mathematical signs
Variable names
In mathematics , letters are usually used as characters when it comes to mutable objects . For the typesetting one is usually serif font used.
Examples of normal cases of the alphabet and text set used:
 Scalars : in italics :

Vectors : partly like scalars, partly with an arrow above or in semibold (DIN 1303): Previously also letters in Fraktur : or
 Complex quantities : such as real scalars, in engineering often with a horizontal line under the symbol (DIN 1304 and DIN 5483 ):
 Quantities : ordinary capital letters or, for quantities of numbers, with a double bar :

Matrices : Preferably capital letters, boldface, occasionally (DIN 1303): . In the past also capital letters in Gothic script :
Since the number of letters is not sufficient, they are often supplemented by indices (small, subscript numbers, letters or symbols):
More characters
Other characters, e.g. B. contain instructions, are assigned special mathematical symbols that only partially (originally) come from alphabets.
Examples:
character  meaning  Application example 
=  Equal sign  
<  Comparison sign "less than"  
+  Plus sign  
Sum symbol  
Real part of a complex number  
()  Brackets for changing the evaluation order  
Mathematical constant  
Power of the set of natural numbers 
Operator notation
In addition to specifying which characters are used for the individual operators (e.g. for addition), specifying the order of operators and their operands is important. Many variants are mixed in today's common mathematical notation:
Surname  description  Examples 
Prefix notation  Operator before operands  ( Sine ), ( cosine ), ( natural logarithm ), ( function of x) 
Postfix notation  Operator after operands  ( Faculty ), ( derivative ) 
Infix notation  Operator between operands  ( Equality ), ( Addition ), ( Subtraction ), ( Multiplication ), ( Division ), ( Concatenation ), ( Convolution ), ( Element of), ( Comparison ), ( Disjunction ), ( Conjunction ) 
Symbol over operands  ( Complex conjugation ), ( Fourier symbol ), (derivative)  
Parentheses of the operand  ( Gaussian bracket for rounding down ), ( Rounding up ), ( Amount ), ( Norm ), ( Scalar product ), ( Root )  
Operator application without symbols  ( Multiplication ), ( power )  
Other  ( Fraction ), ( binomial coefficient ) 
Infix notation
The most common in arithmetic is the infix notation, in which the operator is placed between the operands. With it, the order of calculation is determined by the value of the operations (" point calculation before line calculation "). By using brackets, you can define subexpressions that must be calculated first. Example:
Another example of an infix notation is the PeanoRussell notation used in logic :
Prefix notation
Expressions in infix notation can quickly become confusing. In the 1920s, the Polish logician and philosopher Jan Łukasiewicz therefore developed the Polish notation , a prefix notation that does not require brackets. The operators are designated with capital letters, e.g. B. for the material implication (sufficient condition) and for the disjunction (alternative). In Polish notation the aforementioned logical term is written like this:
Postfix notation
In Postfix notation, the operator is written after the arguments to be linked; it is therefore also called Reverse Polish Notation (UPN). Occasionally attributed to the Australian philosopher Charles Hamblin , it was very likely that Łukasiewicz also knew it. The UPN was never used in logic, but through the work of Hamblin it gained some importance in early computer science and in early compiler construction , because expressions in UPN are particularly easy to machine. For the same reason, they took over HewlettPackard for their scientific pocket calculators in the 1960s .
Other variants
Other notations without brackets are the conceptual notation by Gottlob Frege , the spelling of the first system of predicate logic , and the existential graphs by Charles S. Peirce . Both also differ greatly from the notations in use today because they are graphic, twodimensional notations.
Investigation of mathematical notation
Mathematical notation is the subject of investigation in the following areas, among others:
 semiotics
 History of mathematics
 Standardization ( DIN , ISO )
 Computer algebra
 Programming languages
 Artificial intelligence
 Formal term analysis and association theory
 Mathematics didactics
See also
literature
 Florian Cajori : A history of mathematical notations. 2 volumes, 1928, 1929. Dover Publications, New York 1993, ISBN 0486677664 .
Web links
 Jeff Miller: Earliest Uses of Various Mathematical Symbols.