Multiplication sign

×
×
Multiplication sign
In Unicode	U+00D7 × MULTIPLICATION SIGN (×)
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) which is not a four-fold rotationally symmetric saltire.

History

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

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")
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".^[8]

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 $a\cdotb$ ). 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 $b \cdot2$ , 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 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	`×` or `×`
macOS	In the Character Palette by searching for MULTIPLICATION SIGN^[9]
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`+`0215` key combination using the numeric keypad^[10]
OpenOffice.org	times
TeX	`\times` `\cdot`
Unix-like (Linux, ChromeOS)	`Ctrl`+`⇧ Shift`+`UD7` `ComposeXX` `AltGr`+`⇧ Shift`+`,` (UK extended layout)

Unicode and HTML entities

U+00D7 × MULTIPLICATION SIGN (×)

Other variants and related characters:

U+002A * ASTERISK (&ast;, &midast;)
U+2062 INVISIBLE TIMES (⁢, ⁢) (a zero-width space indicating multiplication)
U+00B7 · MIDDLE DOT (·, ·, ·) (the interpunct, 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 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;)

References

^ ^a ^b ^c Weisstein, Eric W. "Multiplication". mathworld.wolfram.com. Retrieved 2020-08-26.
^ 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 ^c Cajori, 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
^ Recorde, Robert (1618). The Ground of Arts. London: John Beale.
^ The diagonals do not appear in the original 1543 edition, leaving their priority to Oughtred uncertain.
^ Nykamp, Duane. "Cartesian product definition". Math Insight. Retrieved August 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 from the original on 2007-10-25. Retrieved 2009-10-09.
^ "Unicode Character 'MULTIPLICATION SIGN' (U+00D7)". Fileformat.info. Retrieved 2017-01-13.

External links

"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.

[:1-1] 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.

[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] 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

[9] "Mac Zeichenpalette" (in German). TypoWiki. Archived from the original on 2007-10-25. Retrieved 2009-10-09.

[10] "Unicode Character 'MULTIPLICATION SIGN' (U+00D7)". Fileformat.info. Retrieved 2017-01-13.

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