This article needs additional citations for verification. (January 2017) |
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.
| × | |
|---|---|
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 symbol is also used in botany, in botanical hybrid names.
The form is properly a four-fold rotationally symmetric saltire. 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. This appendix has been attributed to William Oughtred, 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.
Other works have been identified in which crossed diagonals appear in diagrams involving multiplied numbers, such as Robert Recorde's The Ground of Arts and Oswald Schreckenfuchs's 1551 edition of Almagest, but these are not symbolizations.
Uses
In mathematics, the symbol × has a number of uses, including
- Multiplication of two numbers, where it is read as "times" or "multiplied by"
- Cross product of two vectors, where it is usually read as "cross"
- Cartesian product of two sets, where it is usually read as "cross"
- 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".
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.
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
| HTML, SGML, XML | × or × |
| macOS | In the Character Palette by searching for MULTIPLICATION SIGN |
| Microsoft Windows |
|
| OpenOffice.org | times |
| TeX |
|
| Unix-like (Linux, ChromeOS) |
|
Unicode and HTML entities
- U+00D7 × MULTIPLICATION SIGN (×)
Other variants and related characters:
- U+002A * ASTERISK (*, *)
- U+2217 ∗ ASTERISK OPERATOR (∗)
- U+2062 INVISIBLE TIMES (⁢, ⁢) (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 )
- U+00B7 · MIDDLE DOT (·, ·, ·) (the interpunct, may be easier to type than the dot operator)
- U+2297 ⊗ CIRCLED TIMES (⊗, ⊗)
- 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 (⨯) (intended to explicitly denote the cross product of two vectors)
- U+2A30 ⨰ MULTIPLICATION SIGN WITH DOT ABOVE (⨰)
- U+2A31 ⨱ MULTIPLICATION SIGN WITH UNDERBAR (⨱)
- U+2A34 ⨴ MULTIPLICATION SIGN IN LEFT HALF CIRCLE (⨴)
- U+2A35 ⨵ MULTIPLICATION SIGN IN RIGHT HALF CIRCLE (⨵)
- U+2A36 ⨶ CIRCLED MULTIPLICATION SIGN WITH CIRCUMFLEX ACCENT (⨶)
- U+2A37 ⨷ MULTIPLICATION SIGN IN DOUBLE CIRCLE (⨷)
- U+2A3B ⨻ MULTIPLICATION SIGN IN TRIANGLE (⨻)
- U+2AC1 ⫁ SUBSET WITH MULTIPLICATION SIGN BELOW (⫁)
- U+2AC2 ⫂ SUPERSET WITH MULTIPLICATION SIGN BELOW (⫂)
See also
- Division sign
- List of mathematical symbols
- Plus and minus signs
- Reference mark
- X mark
References
- 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.
- 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.
This article needs additional citations for verification Please help improve this article by adding citations to reliable sources Unsourced material may be challenged and removed Find sources Multiplication sign news newspapers books scholar JSTOR January 2017 Learn how and when to remove this message 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 Multiplication signIn UnicodeU 00D7 MULTIPLICATION SIGN amp times Different fromDifferent fromU 0078 x LATIN SMALL LETTER XRelatedSee alsoU 22C5 DOT OPERATOR U 00F7 DIVISION SIGN The symbol is also used in botany in botanical hybrid names The form is properly a four fold rotationally symmetric saltire The multiplication sign is similar to a lowercase X x which is not a four fold rotationally symmetric saltire HistoryThe 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 This appendix has been attributed to William Oughtred 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 Other works have been identified in which crossed diagonals appear in diagrams involving multiplied numbers such as Robert Recorde s The Ground of Arts and Oswald Schreckenfuchs s 1551 edition of Almagest but these are not symbolizations UsesIn mathematics the symbol has a number of uses including Multiplication of two numbers where it is read as times or multiplied by Cross product of two vectors where it is usually read as cross Cartesian product of two sets where it is usually read as cross 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 A monadic symbol is used by the APL programming language to denote the sign function Similar notationsThe 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 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 characterHTML SGML XML amp times or amp 215 macOS In the Character Palette by searching for MULTIPLICATION SIGNMicrosoft 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 keypadOpenOffice org timesTeX times cdotUnix like Linux ChromeOS Ctrl Shift UD7 ComposeXX AltGr Shift UK extended layout Unicode and HTML entitiesU 00D7 MULTIPLICATION SIGN amp times Other variants and related characters U 002A ASTERISK amp ast amp midast U 2217 ASTERISK OPERATOR amp lowast U 2062 INVISIBLE TIMES amp InvisibleTimes amp it 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 lt U 2062 gt x U 00B7 MIDDLE DOT amp middot amp CenterDot amp centerdot the interpunct may be easier to type than the dot operator U 2297 CIRCLED TIMES amp CircleTimes amp otimes U 22C5 DOT OPERATOR amp sdot U 2715 MULTIPLICATION X U 2716 HEAVY MULTIPLICATION X U 2A09 N ARY TIMES OPERATOR U 2A2F VECTOR OR CROSS PRODUCT amp Cross intended to explicitly denote the cross product of two vectors U 2A30 MULTIPLICATION SIGN WITH DOT ABOVE amp timesd U 2A31 MULTIPLICATION SIGN WITH UNDERBAR amp timesbar U 2A34 MULTIPLICATION SIGN IN LEFT HALF CIRCLE amp lotimes U 2A35 MULTIPLICATION SIGN IN RIGHT HALF CIRCLE amp rotimes U 2A36 CIRCLED MULTIPLICATION SIGN WITH CIRCUMFLEX ACCENT amp otimesas U 2A37 MULTIPLICATION SIGN IN DOUBLE CIRCLE amp Otimes U 2A3B MULTIPLICATION SIGN IN TRIANGLE amp tritime U 2AC1 SUBSET WITH MULTIPLICATION SIGN BELOW amp submult U 2AC2 SUPERSET WITH MULTIPLICATION SIGN BELOW amp supmult See alsoDivision sign List of mathematical symbols Plus and minus signs Reference mark X markReferencesWeisstein 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 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 nota in vel vel plerumque absque nota 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
