Multiplication sign
 | This article needs additional citations for verification. (January 2017) |
| × | |
|---|---|
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 and the heavy goods vehicle industry, to calculate the amount of powered wheels.
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
[edit]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]
An earlier use of the symbol from the 1500s appears in The Ground of Arts, where it is used in the context of a mental arithmetic method for computing simple, single-digit multiplications. Rob Eastaway theorizes 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]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"[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 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".[7]
A monadic × symbol is used by the APL programming language to denote the sign function.
Similar notations
[edit]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#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
[edit]| HTML, SGML, XML | × or ×
|
| macOS | In the Character Palette by searching for MULTIPLICATION SIGN[8] |
| Microsoft Windows |
|
| OpenOffice.org | times |
| TeX |
|
| Unix-like (Linux, ChromeOS) |
|
Unicode and HTML entities
[edit]- U+00D7 × MULTIPLICATION SIGN (×)
Other variants and related characters:
- U+002A * ASTERISK (*, *)
- 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 (⊗, ⊗)
- 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
[edit]References
[edit]- ^ 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
- ^ "The Big X - Numberphile". YouTube. Retrieved 2024-07-07.
- ^ 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
[edit]- "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.