Multiplication sign

Short description: Mathematical symbol: ×

Multiplication sign

In Unicode| U+00D7 × MULTIPLICATION SIGN (HTML `&#215;` · `&times;`)
Related
See also| U+22C5 ⋅ DOT OPERATOR
U+00F7 ÷ DIVISION SIGN
Different from
Different from| U+0078 x LATIN SMALL LETTER X

The multiplication sign, also known as the times sign or the dimension sign, is the symbol ×, used in mathematics to denote the multiplication operation and its resulting product.[1] While similar to a lowercase X (x), the form is properly a four-fold rotationally symmetric saltire.[2]

## Contents

* 1 History
* 2 Uses
* 3 Similar notations
* 4 Typing the character
* 5 Unicode and HTML entities
* 6 See also
* 7 References
* 8 External links

## History

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

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

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). 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

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 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
* Via the Character Map utility: in the eighth row, or by searching
* The key combination using the numeric keypad[8]

OpenOffice.org | times
TeX |

* `\times`
* `\cdot`

Unix-like (Linux, ChromeOS)  |

*   *   * (UK extended layout)

## Unicode and HTML entities

* U+00D7 × MULTIPLICATION SIGN (HTML `&#215;` · `&times;`)

Other variants and related characters:

* U+002A * ASTERISK (HTML `&#42;`)
* U+2062 INVISIBLE TIMES (HTML `&#8290;`) (a zero-width space indicating multiplication)
* U+00B7 * MIDDLE DOT (HTML `&#183;` · `&middot;`) (the interpunct, may be easier to type than the dot operator)
* U+2297 ⊗ CIRCLED TIMES (HTML `&#8855;` · `&otimes;`)
* U+22C5 ⋅ DOT OPERATOR (HTML `&#8901;` · `&sdot;`)
* U+2715 ✕ MULTIPLICATION X (HTML `&#10005;`)
* U+2716 ✖ HEAVY MULTIPLICATION X (HTML `&#10006;`)
* U+2A09 ⨉ N-ARY TIMES OPERATOR (HTML `&#10761;`)
* U+2A2F ⨯ VECTOR OR CROSS PRODUCT (HTML `&#10799;`) (intended to explicitly denote the cross product of two vectors)
* U+2A30 ⨰ MULTIPLICATION SIGN WITH DOT ABOVE (HTML `&#10800;`)
* U+2A31 ⨱ MULTIPLICATION SIGN WITH UNDERBAR (HTML `&#10801;`)
* U+2A34 ⨴ MULTIPLICATION SIGN IN LEFT HALF CIRCLE (HTML `&#10804;`)
* U+2A35 ⨵ MULTIPLICATION SIGN IN RIGHT HALF CIRCLE (HTML `&#10805;`)
* U+2A36 ⨶ CIRCLED MULTIPLICATION SIGN WITH CIRCUMFLEX ACCENT (HTML `&#10806;`)
* U+2A37 ⨷ MULTIPLICATION SIGN IN DOUBLE CIRCLE (HTML `&#10807;`)
* U+2A3B ⨻ MULTIPLICATION SIGN IN TRIANGLE (HTML `&#10811;`)
* U+2AC1 ⫁ SUBSET WITH MULTIPLICATION SIGN BELOW (HTML `&#10945;`)
* U+2AC2 ⫂ SUPERSET WITH MULTIPLICATION SIGN BELOW (HTML `&#10946;`)

## See also

* Division sign
* List of mathematical symbols
* Plus and minus signs
* Reference mark
* X mark

## References

1. ↑ 1.0 1.1 1.2 Weisstein, Eric W.. "Multiplication" (in en). https://mathworld.wolfram.com/Multiplication.html.
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. ↑ 3.0 3.1 3.2 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. https://books.google.com/books?id=Ld5EAAAAcAAJ. "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". https://mathinsight.org/definition/cartesian_product.
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, https://archive.org/details/newhartsrules00rmri/page/183
7. ↑ "Mac Zeichenpalette". TypoWiki. http://www.typografie.info/typowiki/index.php?title=Mac_Zeichenpalette.
8. ↑ "Unicode Character 'MULTIPLICATION SIGN' (U+00D7)". https://www.fileformat.info/info/unicode/char/00D7/index.htm.

## External links

* "Letter Database". http://www.eki.ee/letter/chardata.cgi?ucode=00D7.
* "Unicode Character 'MULTIPLICATION SIGN' (U+00D7)". https://www.fileformat.info/info/unicode/char/00d7/index.htm.
* "Unicode Character 'VECTOR OR CROSS PRODUCT' (U+2A2F)". https://www.fileformat.info/info/unicode/char/2a2f/index.htm.

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