View source for Multiplication sign

{{Short description|Mathematical symbol}}
{{Redirect-distinguish2|×|the letter [[X]]}}
{{Refimprove|date=January 2017}}
{{Infobox symbol
|mark=×
|unicode={{unichar|00D7|Multiplication SIGN|html=}}
|see also= {{unichar|22C5|DOT OPERATOR|nlink=Multiplication#Notation and terminology}}{{br}}{{unichar|F7|nlink=Division sign}}
|different from= {{unichar|0078|Latin small letter x|nlink=X}}
}}

The '''multiplication sign''' ({{char|×}}), also known as the '''times sign''' or the '''dimension sign''', is a [[Glossary of mathematical symbols|mathematical symbol]] used to denote the operation of [[multiplication]], which results in a [[Product (mathematics)|product]].<ref name=":1">{{Cite web|last=Weisstein|first=Eric W.|title=Multiplication|url=https://mathworld.wolfram.com/Multiplication.html|access-date=2020-08-26|website=mathworld.wolfram.com|language=en}}</ref> 

The symbol is also used in [[botany]], in  botanical [[hybrid name (botany)|hybrid name]]s.

The form is properly a four-fold [[Rotational symmetry|rotationally symmetric]] [[saltire]].<ref>{{cite journal |doi=10.1111/j.1949-8594.2000.tb17262.x |last=Stallings |first=L. |year=2000 |title=A Brief History of Algebraic Notation |journal=School Science and Mathematics |volume=100 |issue=5 |pages=230–235 |issn=0036-6803 }}</ref> The multiplication sign {{char|×}} is similar to a lowercase [[X]] ({{char|x}}).

== History ==
The earliest known use of the {{char|×}} symbol to indicate multiplication appears in an anonymous appendix to the 1618 edition of [[John Napier]]'s {{lang|la|[[Mirifici Logarithmorum Canonis Descriptio]]}}.<ref name=Cajori /> This appendix has been attributed to [[William Oughtred]],<ref name=Cajori /> who used the same symbol in his 1631 algebra text, {{lang|la|Clavis Mathematicae}}, stating:<blockquote>Multiplication of species [i.e. unknowns] connects both proposed magnitudes with the symbol 'in' or {{char|×}}: or ordinarily without the symbol if the magnitudes be denoted with one letter.<ref>{{cite book|url= https://books.google.com/books?id=Ld5EAAAAcAAJ | title = Clavis Mathematicae | author = William Oughtred | year= 1667 | page=10| quote = {{lang|la|Multiplicatio speciosa connectit utramque magintudinem propositam cum notâ in vel ×: vel plerumque absque notâ, si magnitudines denotentur unica litera}} }}</ref> </blockquote>

Other works  have been identified in which crossed diagonals appear in diagrams involving multiplied numbers, such as [[Robert Recorde]]'s ''[[The Ground of Arts]]''<ref>{{cite book|last=Recorde|first=Robert|title=The Ground of Arts|year=1618|publisher=John Beale|place=London}}</ref>{{refn|The diagonals do not appear in the original 1543 edition, leaving their priority to Oughtred uncertain.}} and [[Oswald Schreckenfuchs]]'s 1551 edition of ''[[Almagest]]'', but these are not symbolizations.<ref name="Cajori">{{cite book|last=Cajori|first=Florian|author-link=Florian Cajori |date= 1928 |  title=A History of Mathematical Notations, Volume I: Notations in Elementary Mathematics|title-link=A History of Mathematical Notations |publisher= Open Court | pages=[https://archive.org/details/historyofmathema031756mbp/page/n269 251–252] }}</ref>

== Uses ==
In [[mathematics]], the [[symbol]] × has a number of uses, including
* [[Multiplication]] of two [[numbers]], where it is read as "times" or "multiplied by"<ref name=":1" />
* [[Cross product]] of two [[Vector space|vectors]], where it is usually read as "cross"
* [[Cartesian product]] of two sets, where it is usually read as "cross"<ref>{{Cite web|last=Nykamp|first=Duane|title=Cartesian product definition|url=https://mathinsight.org/definition/cartesian_product|access-date=August 26, 2020|website=Math Insight}}</ref>
* 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 (mathematics)|matrix]], where it is usually read as "by"
* A [[Interaction (statistics)|statistical interaction]] between two explanatory [[Variable (mathematics)|variables]], where it is usually read as "by"
* the [[Magnification#Size_ratio_(optical_magnification)|optical magnification]] is sometimes referred to as "power" (for example "10× power")

In [[biology]], the multiplication sign is used in a botanical [[hybrid name (botany)|hybrid name]], for instance ''Ceanothus papillosus'' × ''impressus'' (a hybrid between ''[[Ceanothus papillosus|C. papillosus]]'' and ''[[Ceanothus impressus|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 [[historian]]s for an event between two [[Calendar date|dates]]. When employed between two dates{{snd}} for example 1225 and 1232{{snd}} the expression "1225×1232" means "no earlier than 1225 and no later than 1232".<ref>{{citation |title=New Hart's rules: the handbook of style for writers and editors |year=2005 |publisher=Oxford University Press |isbn=978-0-19-861041-0 |page=[https://archive.org/details/newhartsrules00rmri/page/183 183] |url-access=registration |url=https://archive.org/details/newhartsrules00rmri/page/183 }}</ref>

A [[Monad (functional programming)|monadic]] {{char|×}} symbol is used by the [[APL (programming language)|APL programming language]] to denote the [[sign function]].

== Similar notations ==
{{Main|Multiplication#Notation|l1=Multiplication: Notation}}

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

In [[algebra]]ic notation, widely used in mathematics, a multiplication symbol is usually omitted wherever it would not cause confusion: "{{mvar|a}} multiplied by {{mvar|b}}" can be written as {{math|''ab''}} or {{math|''a b''}}.<ref name=":1" />

Other symbols can also be used to denote multiplication, often to reduce confusion between the multiplication sign × and the common variable {{mvar|x}}. In some countries, such as [[Germany]], the primary symbol for multiplication is the "[[dot operator]]" {{char|⋅}} (as in {{math|''a''⋅''b''}}). This symbol is also used in compound [[units of measurement]], e.g., N⋅m (see ''{{slink|International System of Units#Lexicographic conventions}}''). In algebra, it is a notation to resolve ambiguity (for instance, "{{math|''b''}} times {{math|2}}" may be written as {{math|''b''⋅2}}, to avoid being confused with a value called {{math|''b''<sub>2</sub>}}). This notation is used wherever multiplication should be written explicitly, such as in "{{math|1= ''ab'' = ''a''⋅2}} for {{math|1=''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 {{math|''a''.''b''}}, is common when the symbol for [[decimal point]] is [[comma]].

Historically, computer language [[syntax (programming)|syntax]] was restricted to the [[ASCII]] character set, and the [[asterisk]] {{char|*}} 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 {{key press|+}}, {{key press|-}}, {{key press|*}} and {{key press|/}}, respectively.

== Typing the character ==
<!-- Note: This table is also maintained on German (:de) Version -->
{| class="wikitable"
|-
|[[HTML]], [[SGML]], [[XML]]
|<code>&amp;times;</code> or <code>&amp;#215;</code>
|-
|[[macOS]]
|In the Character Palette by searching for ''MULTIPLICATION SIGN''<ref>{{cite web |url=http://www.typografie.info/typowiki/index.php?title=Mac_Zeichenpalette |title=Mac Zeichenpalette |publisher=TypoWiki |access-date=2009-10-09 |url-status=dead |archive-url=https://web.archive.org/web/20071025090903/http://www.typografie.info/typowiki/index.php?title=Mac_Zeichenpalette |archive-date=2007-10-25 |lang=de}}</ref>
|-
|[[Microsoft Windows]]
|
* Via the Emoji and Symbol input panel, invoked with the {{key press|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 [[QWERTY#US-International|US International keyboard layout]], use {{Key press|Alt|2==}} 
* Via the [[Character Map (Windows)|Character Map]] utility: in the eighth row, or by searching
* The {{key press|Alt|0}}{{key press|2}}{{key press|1}}{{key press|5}} key combination using the [[numeric keypad]]<ref>{{cite web|url=https://www.fileformat.info/info/unicode/char/00D7/index.htm |title=Unicode Character 'MULTIPLICATION SIGN' (U+00D7) |website=Fileformat.info |access-date=2017-01-13}}</ref>
|-
|[[OpenOffice.org]]
|times
|-
|[[TeX]]
|
* <code>\times</code>
* <code>\cdot</code>
|-
|[[Unix-like]] ([[Linux]], [[ChromeOS]])
|
* {{keypress|Ctrl|Shift|U}}{{keypress|D|7|chain=}}
* {{keypress|[[Compose key|Compose]]|X|X|chain=}}
* {{keypress|AltGr|Shift|,}} ([[QWERTY#United Kingdom (Extended) Layout|UK extended layout]]) 
|}

== Unicode and HTML entities ==
* {{unichar|00d7|MULTIPLICATION SIGN|html=}}
Other variants and related characters:
* {{unichar|002A|Asterisk|html=}}
* {{unichar|2217|Asterisk operator|html=}}
* {{unichar|2062|INVISIBLE TIMES|html=}} (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 )
* {{unichar|00B7|Middle dot|html=}} (the [[interpunct]], may be easier to type than the dot operator)
* {{unichar|2297|CIRCLED TIMES|html=}}
* {{unichar|22C5|dot operator|html=}}
* {{unichar|2715|MULTIPLICATION X|html=}}
* {{unichar|2716|HEAVY MULTIPLICATION X|html=}}
* {{unichar|2a09|N-ARY TIMES OPERATOR|html=}}
* {{unichar|2a2f|VECTOR OR CROSS PRODUCT|html=}} (intended to ''explicitly'' denote the [[cross product]] of two vectors)
* {{unichar|2a30|MULTIPLICATION SIGN WITH DOT ABOVE|html=}}
* {{unichar|2a31|MULTIPLICATION SIGN WITH UNDERBAR|html=}}
* {{unichar|2a34|MULTIPLICATION SIGN IN LEFT HALF CIRCLE|html=}}
* {{unichar|2a35|MULTIPLICATION SIGN IN RIGHT HALF CIRCLE|html=}}
* {{unichar|2a36|CIRCLED MULTIPLICATION SIGN WITH CIRCUMFLEX ACCENT|html=}}
* {{unichar|2a37|MULTIPLICATION SIGN IN DOUBLE CIRCLE|html=}}
* {{unichar|2a3b|MULTIPLICATION SIGN IN TRIANGLE|html=}}
* {{unichar|2ac1|SUBSET WITH MULTIPLICATION SIGN BELOW|html=}}
* {{unichar|2ac2|SUPERSET WITH MULTIPLICATION SIGN BELOW|html=}}

== See also ==
* [[Division sign]]
* [[List of mathematical symbols]]
* [[Plus and minus signs]]
* [[Reference mark]]
* [[X mark]]

== References ==
{{reflist}}

==External links==
* {{cite web |url=http://www.eki.ee/letter/chardata.cgi?ucode=00D7 |title=Letter Database |website=Eki.ee |access-date=2017-01-13}}
* {{cite web |url=https://www.fileformat.info/info/unicode/char/00d7/index.htm |title=Unicode Character 'MULTIPLICATION SIGN' (U+00D7) |website=Fileformat.info |access-date=2017-01-13}}
* {{cite web |url=https://www.fileformat.info/info/unicode/char/2a2f/index.htm |title=Unicode Character 'VECTOR OR CROSS PRODUCT' (U+2A2F) |website=Fileformat.info |access-date=2017-01-13}}


{{navbox punctuation}}
[[Category:Mathematical symbols]]
[[Category:Multiplication]]