Multiplication sign

Mathematical symbol

"×" redirects here and is not to be confused with the letter X.

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Multiplication sign

In Unicode| U+00D7 × MULTIPLICATION SIGN (&times;)
Different from
Different from| U+0078 x LATIN SMALL LETTER X
Related
See also| U+22C5 ⋅ DOT OPERATOR
U+00F7 ÷ DIVISION SIGN

The multiplication sign (×), also known as the times sign or the dimension sign, is a mathematical symbol used to denote the operation of multiplication, which results in a product.[1]

The symbol is also used in botany, in botanical hybrid names.

The form is properly a four-fold rotationally symmetric saltire.[2] The multiplication sign × is similar to a lowercase X (x).

## History

[edit]

The earliest known use of the × symbol to indicate multiplication appears in an anonymous appendix to the 1618 edition of John Napier's Mirifici Logarithmorum Canonis Descriptio.[3] This appendix has been attributed to William Oughtred,[3] who used the same symbol in his 1631 algebra text, Clavis Mathematicae, stating:

> Multiplication of species [i.e. unknowns] connects both proposed magnitudes with the symbol 'in' or ×: or ordinarily without the symbol if the magnitudes be denoted with one letter.[4]

Other works have been identified in which crossed diagonals appear in diagrams involving multiplied numbers, such as Robert Recorde's The Ground of Arts[5][6] and Oswald Schreckenfuchs's 1551 edition of Almagest, but these are not symbolizations.[3]

## Uses

[edit]

In mathematics, the symbol × has a number of uses, including

* Multiplication of two numbers, where it is read as "times" or "multiplied by"[1]
* Cross product of two vectors, where it is usually read as "cross"
* Cartesian product of two sets, where it is usually read as "cross"[7]
* Geometric dimension of an object, such as noting that a room is 10 feet × 12 feet in area, where it is usually read as "by" (e.g., "10 feet by 12 feet")
* Display resolution in pixels, such as 1920 pixels across × 1080 pixels down. Read as "by".
* Dimensions of a matrix, where it is usually read as "by"
* A statistical interaction between two explanatory variables, where it is usually read as "by"
* the optical magnification is sometimes referred to as "power" (for example "10× power")

In biology, the multiplication sign is used in a botanical hybrid name, for instance Ceanothus papillosus × impressus (a hybrid between C. papillosus and C. impressus) or Crocosmia × crocosmiiflora (a hybrid between two other species of Crocosmia). However, the communication of these hybrid names with a Latin letter "x" is common, especially when the actual "×" symbol is not readily available.

The multiplication sign is also used by historians for an event between two dates. When employed between two dates – for example 1225 and 1232 – the expression "1225×1232" means "no earlier than 1225 and no later than 1232".[8]

A monadic × symbol is used by the APL programming language to denote the sign function.

## Similar notations

[edit]

Main article: Multiplication: Notation

The lower-case Latin letter x is sometimes used in place of the multiplication sign. This is considered incorrect in mathematical writing.

In algebraic notation, widely used in mathematics, a multiplication symbol is usually omitted wherever it would not cause confusion: "a multiplied by b" can be written as ab or a b.[1]

Other symbols can also be used to denote multiplication, often to reduce confusion between the multiplication sign × and the common variable x. In some countries, such as Germany, the primary symbol for multiplication is the "dot operator" ⋅ (as in a⋅b). This symbol is also used in compound units of measurement, e.g., N⋅m (see International System of Units § Lexicographic conventions). In algebra, it is a notation to resolve ambiguity (for instance, "b times 2" may be written as b⋅2, to avoid being confused with a value called b2). This notation is used wherever multiplication should be written explicitly, such as in "ab = a⋅2 for b = 2"; this usage is also seen in English-language texts. In some languages, the use of full stop as a multiplication symbol, such as a.b, is common when the symbol for decimal point is comma.

Historically, computer language syntax was restricted to the ASCII character set, and the asterisk * became the de facto symbol for the multiplication operator. This selection is reflected in the numeric keypad on English-language keyboards, where the arithmetic operations of addition, subtraction, multiplication and division are represented by the keys `+`, `-`, `*` and `/`, respectively.

## Unicode and HTML entities

[edit]

* U+00D7 × MULTIPLICATION SIGN (&times;)

Other variants and related characters:

* U+002A * ASTERISK (&ast;, &midast;)
* U+2217 ∗ ASTERISK OPERATOR (&lowast;)
* U+2062 INVISIBLE TIMES (&InvisibleTimes;, &it;) (a zero-width space indicating multiplication; The invisible times codepoint is used in mathematical type-setting to indicate the multiplication of two terms without a visible multiplication operator, e.g. when type-setting 2x (the multiplication of the number 2 and the variable x), the invisible times codepoint can be inserted in-between: 2 <U+2062> x )
* U+00B7 · MIDDLE DOT (&middot;, &CenterDot;, &centerdot;) (the interpunct, may be easier to type than the dot operator)
* U+2297 ⊗ CIRCLED TIMES (&CircleTimes;, &otimes;)
* U+22C5 ⋅ DOT OPERATOR (&sdot;)
* U+2715 ✕ MULTIPLICATION X
* U+2716 ✖ HEAVY MULTIPLICATION X
* U+2A09 ⨉ N-ARY TIMES OPERATOR
* U+2A2F ⨯ VECTOR OR CROSS PRODUCT (&Cross;) (intended to explicitly denote the cross product of two vectors)
* U+2A30 ⨰ MULTIPLICATION SIGN WITH DOT ABOVE (&timesd;)
* U+2A31 ⨱ MULTIPLICATION SIGN WITH UNDERBAR (&timesbar;)
* U+2A34 ⨴ MULTIPLICATION SIGN IN LEFT HALF CIRCLE (&lotimes;)
* U+2A35 ⨵ MULTIPLICATION SIGN IN RIGHT HALF CIRCLE (&rotimes;)
* U+2A36 ⨶ CIRCLED MULTIPLICATION SIGN WITH CIRCUMFLEX ACCENT (&otimesas;)
* U+2A37 ⨷ MULTIPLICATION SIGN IN DOUBLE CIRCLE (&Otimes;)
* U+2A3B ⨻ MULTIPLICATION SIGN IN TRIANGLE (&tritime;)
* U+2AC1 ⫁ SUBSET WITH MULTIPLICATION SIGN BELOW (&submult;)
* U+2AC2 ⫂ SUPERSET WITH MULTIPLICATION SIGN BELOW (&supmult;)

## See also

[edit]

* Division sign
* List of mathematical symbols
* Plus and minus signs
* Reference mark
* Unicode input – Input characters using their Unicode code points (a general guide to entering "off-keyboard" symbols).
* X mark – Symbol denoting 'no' or 'incorrect'

## References

[edit]

1. ^ a b c Weisstein, Eric W. "Multiplication". mathworld.wolfram.com. Retrieved 2020-08-26.
2. ^ Stallings, L. (2000). "A Brief History of Algebraic Notation". School Science and Mathematics. 100 (5): 230–235\. doi:10.1111/j.1949-8594.2000.tb17262.x. ISSN 0036-6803.
3. ^ a b c Cajori, Florian (1928). A History of Mathematical Notations, Volume I: Notations in Elementary Mathematics. Open Court. pp. 251–252.
4. ^ William Oughtred (1667). Clavis Mathematicae. p. 10\. "Multiplicatio speciosa connectit utramque magintudinem propositam cum notâ in vel ×: vel plerumque absque notâ, si magnitudines denotentur unica litera"
5. ^ Recorde, Robert (1618). The Ground of Arts. London: John Beale.
6. ^ The diagonals do not appear in the original 1543 edition, leaving their priority to Oughtred uncertain.
7. ^ Nykamp, Duane. "Cartesian product definition". Math Insight. Retrieved August 26, 2020.
8. ^ New Hart's rules: the handbook of style for writers and editors, Oxford University Press, 2005, p. 183, ISBN 978-0-19-861041-0

## External links

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* "Letter Database". Eki.ee. Retrieved 2017-01-13.
* "Unicode Character 'MULTIPLICATION SIGN' (U+00D7)". Fileformat.info. Retrieved 2017-01-13.
* "Unicode Character 'VECTOR OR CROSS PRODUCT' (U+2A2F)". Fileformat.info. Retrieved 2017-01-13.

* v
* t
* e

Common punctuation and other typographical symbols

*      space
*   ,   comma
*   :   colon
*   ;   semicolon
*   ‐   hyphen
*   ’   '   apostrophe
*   ′   ″   ‴   prime
*   .   full stop

*   &  ampersand
*   @   at sign
*   ^   caret
*   /   slash
*   \   backslash
*   …   ellipsis
*   *   asterisk
*   ※   Reference mark
*   ⁂   asterism
*     •  •  •     dinkus

*   \-   hyphen-minus
*   ‒   –   —   dash
*   ⹀   ⸗   double hyphen

*   ?   question mark
*   !   exclamation mark
*   ‽   interrobang
*   ¡   ¿   inverted ! and ?
*   ⸮   irony punctuation

*   #   number sign
*   №   numero sign
*   º   ª   ordinal indicator
*   %   percent sign
*   ‰   per mille
*   ‱   basis point
*   °   degree symbol
*   ⌀   diameter sign

*   \+   −   plus and minus signs
*   ×   multiplication sign
*   ÷   division sign
*   ~   tilde
*   ±   plus–minus sign
*   ∓   minus-plus sign
*   √   radical symbol

*   _   underscore
*   ⁀  tie
*   |   ¦   ‖   vertical bar
*   •   bullet
*   ·   interpunct

*   ©   copyright symbol
*   ℗   sound recording copyright
*   ®   registered trademark
*   SM  service mark symbol
*   TM  trademark symbol

*   ‘ ’   “ ”   ' '   " "   quotation mark
*   ‹ ›   « »   guillemet
*   ( )   [ ]   { }   ⟨ ⟩   bracket
*   ”   ditto mark

*   †   ‡   dagger
*   ❧   fleuron (hedera, aldus)
*   ☞   manicule
*   ◊   ⌑   lozenge
*   ¶   ⸿   pilcrow (paragraph mark)
*   §   section mark

* Version of this table as a sortable list

* Currency symbols
* Diacritics (accents)
* Logic symbols
* Math symbols
* Whitespace

* Chinese punctuation
* Hebrew punctuation
* Japanese punctuation
* Korean punctuation
* Vietnamese punctuation

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