Multiplication sign

Mathematical symbol

"×" and "Times sign" redirect here. Not to be confused with the letter X. For the Ed Sheeran album, see x (Ed Sheeran album). For other uses, see Sign of the times.

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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] While similar to a lowercase X (x), the form is properly a four-fold rotationally symmetric saltire.[2]

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

## History[edit]

The earliest known use of the × symbol to represent 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]

Two earlier uses of a ✕ notation have been identified, but do not stand critical examination.[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"[5]
* 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")
* Screen 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"

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, 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".[6]

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.

## Typing the character[edit]

HTML, SGML, XML | `&times;` or `&#215;`

|

macOS | In the Character Palette by searching for MULTIPLICATION SIGN[7]
Microsoft Windows |

* Via the Emoji and Symbol input panel, invoked with the `⊞ Win`+`.` key combination (Windows 10 version 1803 and later)
* Via the Touch Keyboard component of the Taskbar (Windows 10 and later)
* Some non-English keyboard layouts have it as an explicit keytop, like in Arabic keyboard.
* Using US International keyboard layout, use `Alt`+`=`
* Via the Character Map utility: in the eighth row, or by searching
* The `Alt`+`0``2``1``5` key combination using the numeric keypad[8]

OpenOffice.org | times
TeX |

* `\times`
* `\cdot`

Unix-like (Linux, ChromeOS)  |

* `Ctrl`+`⇧ Shift`+`U``D``7`
* `Compose``X``X`
* `AltGr`+`⇧ Shift`+`,` (UK extended layout)

## Unicode and HTML entities[edit]

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

Other variants and related characters:

* U+002A * ASTERISK (&ast;, &midast;)
* U+2062 INVISIBLE TIMES (&InvisibleTimes;, &it;) (a zero-width space indicating multiplication)
* 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
* X mark

## 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. ^ Nykamp, Duane. "Cartesian product definition". Math Insight. Retrieved August 26, 2020.
6. ^ New Hart's rules: the handbook of style for writers and editors, Oxford University Press, 2005, p. 183, ISBN 978-0-19-861041-0
7. ^ "Mac Zeichenpalette" (in German). TypoWiki. Archived from the original on 2007-10-25. Retrieved 2009-10-09.
8. ^ "Unicode Character 'MULTIPLICATION SIGN' (U+00D7)". Fileformat.info. Retrieved 2017-01-13.

## External links[edit]

* "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
* ⁂ 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

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

* © copyright symbol
* © copyleft
* ℗ 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
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* Whitespace

* Chinese punctuation
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