# Mark

⋅ ×
Mathematical signs
arithmetic
Plus sign +
Minus sign - , ./.
Mark , ×
Divided sign : , ÷ , /
Plus minus sign ± ,
Comparison sign < , , = , , >
Root sign
Percent sign %
Analysis
Sum symbol Σ
Product mark Π
Difference sign , Nabla ,
Prime
Partial differential
Integral sign
Concatenation characters
Infinity symbol
geometry
Angle sign , , ,
Vertical , parallel ,
Triangle , square ,
Diameter sign
Set theory
Union , cut ,
Difference , complement ,
Element character
Subset , superset , , ,
Empty set
logic
Universal quantifier
Existential quantifier
Conjunction , disjunction ,
Negation sign ¬

As a mark , multiplication sign or product icons different are special characters called, mainly to represent the mathematical operator for multiplication can be used. In mathematics, the painting point ⋅, which is similar to the center point , is usually used, more rarely also a shape of the cross as a paint cross  × and especially in spreadsheet programs and programming languages the asterisk  *.

In vector calculation , the painting point (or a semi-bold variant) is also used for the scalar product and the painting cross for the cross product . In set theory the cross stands for the Cartesian product . The paint cross is also used when specifying dimensions .

## history

The oldest symbol seems to be the cross (×). It was first used by the English mathematician William Oughtred in his work Clavis Mathematicae , published in London in 1631 . Oughtred has probably been using the Malk Cross since 1618, if he wrote an anonymous appendix to the English translation of John Napier's Descriptio . The derivation of the symbol - from the letter ( X ) or from the St. Andrew's cross  - has not been clarified.

The German scientist Gottfried Wilhelm Leibniz rejected the Malkreuz because it could be confused with the letter X and preferred the point (⋅). Leibniz used the multiplication point in an exchange of letters from 1698, but probably introduced it into his mathematical notation as early as 1694 or even earlier.

Johann Rahn introduced the asterisk (∗) for multiplication. Together with the symbol for division (÷), this appears for the first time in his book Teutsche Algebra , published in 1659.

## use

### multiplication

In the basic arithmetic operation of multiplication, the painting point ⋅ and the cross × have the same meaning, but they are used differently.

The painting cross × is preferred

• if the multiplication has to be visually striking, for example on a poster:
5 × 2 free tickets!
• if the factors are not symbols or numbers, but words:
Trade tax = tax base × assessment rate
• if only the left factor is given, i.e. in the sense of "-mal" or "-fach":
Magnification: 6 ×
• in Anglo-American cultures as well as in France in school lessons and in everyday life.

The painting point ⋅, on the other hand, is preferred

• when calculations are reproduced with numbers:
${\ displaystyle a \, = \, 5 \ cdot 6 + 7}$
• in Germany in school lessons.

In some cases, such as the multiplication of unit symbols, the multiplication symbol can be omitted (example: Nm for Newton meter is a common, shortened notation for N ⋅ m) or indicated by a (narrow) space . In general, the painting point can be left out entirely if the design allows it and if there is no risk of confusion from the context.

U  = 2π r  - without white space
U  = 2 π  r  - narrow space

The asterisk ( ASCII  42) is usually used in programming languages . *

### Dot product and cross product

In vector calculus , ⋅ is used for the scalar product and × for the cross product . The two symbols are used here for completely different links. For example, in physics are the work

${\ displaystyle W = {\ vec {s}} \ cdot {\ vec {F}}}$

and the torque

${\ displaystyle {\ vec {M}} = {\ vec {s}} \ times {\ vec {F}}}$.

The Unicode character ⋅ (U + 22C5, HTML entity:) is &sdot;provided for the dot product . Graphically, it is laid out exactly like a bold&middot; or something in bold ; With normal font sizes, the two characters can hardly be distinguished visually. As a scientific work even with mechanical typewriters were written or ASCII submitted character set, was used here substitute the simple point ,

For the cross product there is currently - unlike the scalar product - no separate symbol; the cross × (Unicode U + 00D7, HTML-Entity:) is &times;to be used. In the past you had to use the asterisk * or, if necessary, the letter x.

### Cartesian product

In set theory the Cartesian product of two sets becomes and by ${\ displaystyle A}$${\ displaystyle B}$

${\ displaystyle A \ times B}$

written down. It is the set of all ordered pairs in which the first component is an element off and the second component is an element off . Building on this, the painting cross is also used for the direct product of groups or other algebraic structures. Typographically, the same applies to the Cartesian product as to the cross product. ${\ displaystyle A}$${\ displaystyle B}$

### Dimensions

When specifying dimensions such as width × length, the cross is also used. Exemplary uses are:

• an area of ​​3 m × 5 m
• the property is 30 × 40 m, the building on it (LBH) 10 × 8 × 4 m

The specification of dimensions does not represent a multiplication of the respective lengths . As in the second example, the unit of measurement can only be specified once.

## typography

### Formula set

All marks are centered and matched to the size and position of the digits in a font as well as other arithmetic symbols. Replacing the malmarks with other, similar characters (such as a small letter  x or the simple asterisk  *), which can be found on most computer keyboards in a different way than the typographically correct characters, leads to a typographically less demanding typeface, especially since the asterisk in many Fonts is superscripted because it is often used as a footnote mark .

### DIN 5008

The DIN 5008 describes word processing in the office and administration area.

In the current version from 2011, the point set “in the middle or on the baseline” is intended as the painting point. The “normal” end-of-sentence point  .is also permitted here as a representation of the painting point.

The lowercase letter is xintended for the paint cross . The reason for this is that mass-produced goods such as delivery notes and invoices can be created with the least possible expenditure of time and money, and that the simple push of a button xshould be sufficient. However, if in individual cases a document is to be designed in a particularly appealing way and additional effort is required, the standard does not prevent this.

By the way, all arithmetic symbols must always be enclosed in spaces, unless + or - occur as a sign of a number.

### DIN 55301

The DIN 55301 describes the design of statistical tables.

Here the symbol ✕ for “table compartment locked because statement does not make sense” is used as a value-substituting symbol (in contrast to value-supplementing symbols, also quality indicators ). The letter x (minuscule) is often incorrectly used instead in tables of official statistics .

## Representation in computer systems

### Coding

The Unicode character encoding standard contains multiple multipliers and similar characters. They are in the following positions:

Coding in Unicode, HTML and LaTeX
character Unicode Surname HTML Latex
· U + 00B7 MIDDLE DOT Focus & # x00B7; & # 183; · middot; \ cdot
× U + 00D7 MULTIPLICATION SIGN Paint cross & # x00D7; & # 215; x \ times
U + 2715 MULTIPLICATION X Cross (jewelry) & # x2715; & # 10005; - -
U + 2716 HEAVY MULTIPLICATION X Bold cross ( decorative symbol ) & # x2716; & # 10006; - -
U + 22C5 DOT OPERATOR Paint point, point operator & # x22C5; & # 8901; & sdot; \ cdot
U + 2219 BULLET OPERATOR Bold variant of the painting point for the dot product & # x2219; & # 8729; \ bullet
U + 2217 ASTERISK OPERATOR Hodge star operator & # x2217; & # 8727; & lowast; \ ast
U + 2062 INVISIBLE TIMES Invisible multiplication sign & # x2062; & # 8290; & InvisibleTimes; automatically

With the U + 2715/2716, however, it should be noted that they come from the Dingbats section and are only decorative versions of the normal cross U + 00D7; they have no special mathematical meaning and are not to be used in normal text.

The “invisible multiplication sign” is intended to first semantically separate the individual factors when coding in the scientific formula set. A narrow space of the desired width can then be used for the display (see above).

In ASCII - font from 1963, none of the characters is why many older computer systems they could not represent. The ASCII extensions ISO 6937 from 1983 and ISO 8859-1 (Latin 1) from 1986 already contain the Malkreuz.

### keyboard

The following is entered on keyboards with assignment T2 in accordance with the revised German standard DIN 2137 : 2012-06:

• The paint cross × with the key combination Alt Gr+ ^( caret key top left).
• The center point · as a substitute for the painting point that does not exist on the assignment with the key sequence Gruppenumschaltungfollowed by .(point).

In addition, common computer keyboards do not contain any of the syntactically and typographically correct malmarks listed above . Instead, both the asterisk key in the main keyboard block and the mark key in the numeric keypad produce the simple asterisk (*; Unicode U + 002A), which usually leads to an unsatisfactory typeface. Almost all computer programs with arithmetic functions and also in almost all programming languages , however, recognize the asterisk as the arithmetic symbol for multiplication, while the correct multiplication symbols are often not.

On Linux computers with a German keyboard layout , the center point · (U + 00B7) can be generated with the key combination Alt Gr +  ,; pressing the at the same time results in the multiplication symbol × (U + 00D7). When using the Swiss keyboard layout (German and French language variants), the center point is on the third assignment ( ) of the key and the markup is on the fourth assignment (  +  ) of the key . UmschalttasteAlt Gr.Alt Gr⇧ Umschalttaste,

Under Windows, the characters can be inserted using the character table or the key combinations Alt +  0183for the center point and Alt +  0215for the cross.

On Apple Macintosh computers, the characters can be found in the character palette using the appropriate designation, e.g. MULTIPLICATION SIGN .

With the Neo keyboard layout , the painting point can be entered using Mod3 +  × (Ziffernblock). The paint cross × lies on Mod4 +  × (Ziffernblock).

### Similar and related characters

Due to the lack of multiplication marks on the keyboard , characters are often replaced with the letter x (upper or lower case) or the simple asterisk (*). When using a sans serif font (such as Arial), the shape of the small x comes quite close to the mark, but of course it is still on the baseline and not in the middle.

Characters that can be confused
character Unicode Surname
position designation
X U + 0058 Latin capital letter X Latin capital letter X
x U + 0078 Latin small letter x Latin Small Letter X
* U + 002A Asterisk asterisk
U + 2022 Bullet Bold dot ( bullet )
U + 2024 One dot leader Guidance point
U + 2027 Hyphenation point Hyphenation point
· U + 0387 Greek ano teleia High point

## literature

• Florian Cajori : A History of Mathematical Notations. Dover Publications, New York NY 1993, ISBN 0-486-67766-4 (reprint of the original two volume work by Open Court Publishing Co. 1928/1929).