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 , ${\ displaystyle +, \ sin, 2, x}$

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 ${\ displaystyle e, \ pi, 0,}$
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 :

{\ displaystyle a = 7.}

Vectors : partly like scalars, partly with an arrow above or in semi-bold (DIN 1303): Previously also letters in Fraktur : or ${\ displaystyle {\ vec {F}} \ equiv {\ boldsymbol {F}} = m \ cdot {\ boldsymbol {a}}.}$
${\ displaystyle {\ mathfrak {F}}}$ ${\ displaystyle {\ mathfrak {x}} = \ left (0 \ 2 \ 1 \ right).}$

Complex quantities : such as real scalars, in engineering often with a horizontal line under the symbol (DIN 1304 and DIN 5483 ):

{\ displaystyle {\ underline {z}}.}

Quantities : ordinary capital letters or, for quantities of numbers, with a double bar : ${\ displaystyle \ mathbb {N}, A \ cap B.}$

Matrices : Preferably capital letters, boldface, occasionally (DIN 1303): . In the past also capital letters in Gothic script : ${\ displaystyle \ det (M) = 4}$
${\ displaystyle {\ mathfrak {E}}: = {\ begin {pmatrix} 1 & \ cdots & 0 \\\ vdots & \ ddots & \ vdots \\ 0 & \ cdots & 1 \ end {pmatrix}}.}$

Since the number of letters is not sufficient, they are often supplemented by indices (small, subscript numbers, letters or symbols): ${\ displaystyle \ mathbb {N} _ {0}, {\ vec {F}} _ {\ text {G}}, E _ {\ perp}.}$

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	${\ displaystyle 3 ^ {2} = 9}$
<	Comparison sign "less than"	${\ displaystyle a <7}$
+	Plus sign	${\ displaystyle a + 4 = 7}$
${\ displaystyle \ sum}$	Sum symbol	${\ displaystyle a = \ sum _ {i = 1} ^ {n} a_ {i}}$
${\ displaystyle \ operatorname {Re}}$	Real part of a complex number	${\ displaystyle \ operatorname {Re} z}$
()	Brackets for changing the evaluation order	${\ displaystyle 3a + 7 \ neq 3 (a + 7)}$
${\ displaystyle \ pi}$	Mathematical constant	${\ displaystyle \ pi \ approx 3 {,} 14159}$
${\ displaystyle \ aleph _ {0}}$	Power of the set of natural numbers	${\ displaystyle \ aleph _ {0}: = \| \ mathbb {N} \|}$

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: ${\ displaystyle +}$

Surname	description	Examples
Prefix notation	Operator before operands	${\ displaystyle \ sin}$ ( Sine ), ( cosine ), ( natural logarithm ), ( function of x) ${\ displaystyle \ cos}$ ${\ displaystyle \ ln}$ ${\ displaystyle f (x)}$
Postfix notation	Operator after operands	${\ displaystyle!}$ ( Faculty ), ( derivative ) ${\ displaystyle '}$
Infix notation	Operator between operands	${\ displaystyle =}$ ( Equality ), ( Addition ), ( Subtraction ), ( Multiplication ), ( Division ), ( Concatenation ), ( Convolution ), ( Element of), ( Comparison ), ( Disjunction ), ( Conjunction ) ${\ displaystyle +}$ ${\ displaystyle -}$ ${\ displaystyle \ cdot}$ ${\ displaystyle:}$ ${\ displaystyle \ circ}$ ${\ displaystyle *}$ ${\ displaystyle \ in}$ ${\ displaystyle <,>}$ ${\ displaystyle \ lor}$ ${\ displaystyle \ land}$
	Symbol over operands	${\ displaystyle {\ bar {z}}}$ ( Complex conjugation ), ( Fourier symbol ), (derivative) ${\ displaystyle {\ widehat {f}}}$ ${\ displaystyle {\ dot {x}}}$
	Parentheses of the operand	${\ displaystyle \ lfloor \ cdot \ rfloor}$ ( Gaussian bracket for rounding down ), ( Rounding up ), ( Amount ), ( Norm ), ( Scalar product ), ( Root ) ${\ displaystyle \ lceil \ cdot \ rceil}$ ${\ displaystyle \| \ cdot \|}$ ${\ displaystyle \ \| \ cdot \ \|}$ ${\ displaystyle \ langle \ cdot, \ cdot \ rangle}$ ${\ displaystyle {\ sqrt {\ \}}}$
	Operator application without symbols	${\ displaystyle from}$ ( Multiplication ), ( power ) ${\ displaystyle a ^ {b}}$
	Other	${\ displaystyle {\ frac {a} {b}}}$ ( Fraction ), ( binomial coefficient ) ${\ displaystyle n \ choose k}$

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 sub-expressions that must be calculated first. Example:

{\ displaystyle (3 + 4 + 5) \ times 6 \ times 7 + 8}

Another example of an infix notation is the Peano-Russell notation used in logic :

{\ displaystyle (p \ rightarrow q) \ vee (q \ rightarrow p)}

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: ${\ displaystyle C}$ ${\ displaystyle A}$

{\ displaystyle ACpqCqp}

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 Hewlett-Packard 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, two-dimensional notations.

Investigation of mathematical notation

Mathematical notation is the subject of investigation in the following areas, among others:

literature

Florian Cajori : A history of mathematical notations. 2 volumes, 1928, 1929. Dover Publications, New York 1993, ISBN 0-486-67766-4 .

Web links

Jeff Miller: Earliest Uses of Various Mathematical Symbols.