Multiplication sign

×
×
Multiplication sign
InUnicode	U+00D7×MULTIPLICATION SIGN(×)
Different from
Different from	U+0078xLATIN SMALL LETTER X
Related
See also	U+22C5⋅DOT OPERATOR; U+00F7÷DIVISION SIGN

Themultiplication sign(×), also known as thetimes signor thedimension sign,is amathematical symbolused to denote the operation ofmultiplication,which results in aproduct.^[1]While similar to a lowercaseX(x), the form is properly a four-foldrotationally symmetric saltire.^[2]

The symbol is also used inbotany,in botanicalhybrid namesand theheavy goods vehicleindustry, to calculate the amount of powered wheels.

History[edit]

The earliest known use of the×symbol to represent multiplication appears in an anonymous appendix to the 1618 edition ofJohn Napier'sMirifici Logarithmorum Canonis Descriptio.^[3]This appendix has been attributed toWilliam 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]

An earlier use of the symbol from the 1500s appears inThe Ground of Arts,where it is used in the context of a mental arithmetic method for computing simple, single-digit multiplications.Rob Eastawaytheorizes that this may have been the original influence for John Napier's more general usage.^[5]

Two even earlier uses of a✕notation have been identified, but do not stand critical examination.^[3]

Uses[edit]

Inmathematics,thesymbol× has a number of uses, including

Multiplicationof twonumbers,where it is read as "times" or "multiplied by"^[1]
Cross productof twovectors,where it is usually read as "cross"
Cartesian productof two sets, where it is usually read as "cross"^[6]
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 resolutionin pixels, such as 1920 pixels across × 1080 pixels down. Read as "by".
Dimensions of amatrix,where it is usually read as "by"
Astatistical interactionbetween two explanatoryvariables,where it is usually read as "by"

Inbiology,the multiplication sign is used in a botanicalhybrid name,for instanceCeanothus papillosus×impressus(a hybrid betweenC. papillosusandC. impressus) orCrocosmia × crocosmiiflora(a hybrid between two other species ofCrocosmia). 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 byhistoriansfor an event between twodates.When employed between two dates – for example 1225 and 1232 – the expression "1225×1232" means "no earlier than 1225 and no later than 1232".^[7]

Amonadic×symbol is used by theAPL programming languageto denote thesign function.

Similar notations[edit]

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

Inalgebraicnotation, 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 asGermany,the primary symbol for multiplication is the "dot operator"⋅(as in $a\cdotb$ ). This symbol is also used in compoundunits of measurement,e.g., N⋅m (seeInternational System of Units#Lexicographic conventions). In algebra, it is a notation to resolve ambiguity (for instance, "b times 2" may be written as $b \cdot2$ ,to avoid being confused with a value called $b 2$ ). This notation is used wherever multiplication should be written explicitly, such as in " $ab = a \cdot2$ for $b = 2$ ";this usage is also seen in English-language texts. In some languages, the use offull stopas a multiplication symbol, such as $a . b$ ,is common when the symbol fordecimal pointiscomma.

Historically, computer languagesyntaxwas restricted to theASCIIcharacter set, and theasterisk*became the de facto symbol for the multiplication operator. This selection is reflected in thenumeric keypadon 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	`×`or`×`
macOS	In the Character Palette by searching forMULTIPLICATION SIGN^[8]
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 inArabic keyboard. UsingUS International keyboard layout,use`Alt`+`=` Via theCharacter Maputility: in the eighth row, or by searching The`Alt`+`0215`key combination using thenumeric keypad^[9]
OpenOffice.org	times
TeX	`\times` `\cdot`
Unix-like(Linux,ChromeOS)	`Ctrl`+`⇧ Shift`+`UD7` `ComposeXX` `AltGr`+`⇧ Shift`+`,`(UK extended layout)

Unicode and HTML entities[edit]

U+00D7×MULTIPLICATION SIGN(×)

Other variants and related characters:

U+002A*ASTERISK(&ast;, &midast;)
U+2062INVISIBLE TIMES(⁢, ⁢) (a zero-width space indicating multiplication)
U+00B7·MIDDLE DOT(·, ·, ·) (theinterpunct,may be easier to type than the dot operator)
U+2297⊗CIRCLED TIMES(&CircleTimes;, &otimes;)
U+22C5⋅DOT OPERATOR(⋅)
U+2715✕MULTIPLICATION X
U+2716✖HEAVY MULTIPLICATION X
U+2A09⨉N-ARY TIMES OPERATOR
U+2A2F⨯VECTOR OR CROSS PRODUCT(&Cross;) (intended toexplicitlydenote thecross productof 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;)

References[edit]

^^a ^b ^cWeisstein, Eric W."Multiplication".mathworld.wolfram.Retrieved2020-08-26.
^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.
^^a ^b ^cCajori, Florian(1928).A History of Mathematical Notations, Volume I: Notations in Elementary Mathematics.Open Court. pp.251–252.
^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
^"The Big X - Numberphile".YouTube.Retrieved2024-07-07.
^Nykamp, Duane."Cartesian product definition".Math Insight.RetrievedAugust 26,2020.
^New Hart's rules: the handbook of style for writers and editors,Oxford University Press, 2005, p.183,ISBN 978-0-19-861041-0
^"Mac Zeichenpalette"(in German). TypoWiki. Archived fromthe originalon 2007-10-25.Retrieved2009-10-09.
^"Unicode Character 'MULTIPLICATION SIGN' (U+00D7)".Fileformat.info.Retrieved2017-01-13.

External links[edit]

"Letter Database".Eki.ee.Retrieved2017-01-13.
"Unicode Character 'MULTIPLICATION SIGN' (U+00D7)".Fileformat.info.Retrieved2017-01-13.
"Unicode Character 'VECTOR OR CROSS PRODUCT' (U+2A2F)".Fileformat.info.Retrieved2017-01-13.

[:1-1] Weisstein, Eric W."Multiplication".mathworld.wolfram.Retrieved2020-08-26.

[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.

[Cajori-3] 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] "The Big X - Numberphile".YouTube.Retrieved2024-07-07.

[6] Nykamp, Duane."Cartesian product definition".Math Insight.RetrievedAugust 26,2020.

[7] New Hart's rules: the handbook of style for writers and editors,Oxford University Press, 2005, p.183,ISBN 978-0-19-861041-0

[8] "Mac Zeichenpalette"(in German). TypoWiki. Archived fromthe originalon 2007-10-25.Retrieved2009-10-09.

[9] "Unicode Character 'MULTIPLICATION SIGN' (U+00D7)".Fileformat.info.Retrieved2017-01-13.

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