×
Multiplication sign
In UnicodeU+00D7 × MULTIPLICATION SIGN (×)
Different from
Different fromU+0078 x LATIN SMALL LETTER X
Related
See alsoU+22C5 ⋅ DOT OPERATOR
U+00F7 ÷ DIVISION SIGN

The multiplication sign, also known as the times sign or the dimension sign, is the symbol ×, used in mathematics to denote the multiplication operation and its resulting product.[1] While similar to a lowercase X (x), the form is properly a four-fold rotationally symmetric saltire.[2]

History

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]

Two earlier uses of a ✕ notation have been identified, but do not stand critical examination.[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"[5]
  • 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".[6]

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⋅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[7]
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[8]
OpenOffice.org times
TeX
  • \times
  • \cdot
Unix-like (Linux, ChromeOS)

Unicode and HTML entities

  • 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

References

  1. 1 2 3 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.
  3. 1 2 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. ↑ Nykamp, Duane. "Cartesian product definition". Math Insight. Retrieved August 26, 2020.
  6. ↑ New Hart's rules: the handbook of style for writers and editors, Oxford University Press, 2005, p. 183, ISBN 978-0-19-861041-0
  7. ↑ "Mac Zeichenpalette" (in German). TypoWiki. Archived from the original on 2007-10-25. Retrieved 2009-10-09.
  8. ↑ "Unicode Character 'MULTIPLICATION SIGN' (U+00D7)". Fileformat.info. Retrieved 2017-01-13.
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