Multiplication sign

The multiplication sign, also known as the times sign or the dimension sign, is the symbol ×. While similar to a lowercase X (x), the form is properly a rotationally symmetric saltire.[1]

"×" redirects here. It is not to be confused with the letter X.
×
Multiplication sign
In UnicodeU+00D7 × MULTIPLICATION SIGN (HTML × · ×)
Related
See alsoU+2715 ✕ MULTIPLICATION X
U+22C5 ⋅ DOT OPERATOR
U+00F7 ÷ DIVISION SIGN
Different from
Different fromU+0078 x LATIN SMALL LETTER X (HTML x)

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.[2] This appendix has been attributed to William Oughtred,[2] 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".[3] Two earlier uses of a ✕ notation have been identified but do not stand critical examination.[2]

Uses

In mathematics, the symbol × has a number of uses, including

  • Multiplication of two scalar 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" (for example: "10 feet by 12 feet")
  • 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".[4]

A monadic × symbol is used by the APL programming language to denote the sign function.

Similar notations
Main article: Multiplication: Notation and terminology

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 algebraic notation to resolve ambiguity, for instance "b times 2" may be written 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 standard numeric keypad, 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[5]
Microsoft Windows
OpenOffice.org times
TeX \times
Unix-like (Linux, Chrome OS)
  • Ctrl+⇧ Shift+UD7
  • ComposeXX
  • AltGr+⇧ Shift+, (UK extended layout)

Unicode and HTML entities
  • U+00D7 × MULTIPLICATION SIGN (HTML × · ×)

Other variants and related characters:

  • U+002A * ASTERISK (HTML * · *, *)
  • U+2062 INVISIBLE TIMES (HTML ⁢ · ⁢, ⁢) (a zero-width space indicating multiplication)
  • U+00B7 · MIDDLE DOT (HTML · · ·, ·, ·) (the interpunct, may be easier to type than the dot operator)
  • U+2297 ⊗ CIRCLED TIMES (HTML ⊗ · ⊗, ⊗)
  • U+22C5 ⋅ DOT OPERATOR (HTML ⋅ · ⋅)
  • U+2715 ✕ MULTIPLICATION X (HTML ✕)
  • U+2716 ✖ HEAVY MULTIPLICATION X (HTML ✖)
  • U+2A09 ⨉ N-ARY TIMES OPERATOR (HTML ⨉)
  • U+2A2F ⨯ VECTOR OR CROSS PRODUCT (HTML ⨯ · ⨯) (intended to explicitly denote the cross product of two vectors)
  • U+2A30 ⨰ MULTIPLICATION SIGN WITH DOT ABOVE (HTML ⨰ · ⨰)
  • U+2A31 ⨱ MULTIPLICATION SIGN WITH UNDERBAR (HTML ⨱ · ⨱)
  • U+2A34 ⨴ MULTIPLICATION SIGN IN LEFT HALF CIRCLE (HTML ⨴ · ⨴)
  • U+2A35 ⨵ MULTIPLICATION SIGN IN RIGHT HALF CIRCLE (HTML ⨵ · ⨵)
  • U+2A36 ⨶ CIRCLED MULTIPLICATION SIGN WITH CIRCUMFLEX ACCENT (HTML ⨶ · ⨶)
  • U+2A37 ⨷ MULTIPLICATION SIGN IN DOUBLE CIRCLE (HTML ⨷ · ⨷)
  • U+2A3B ⨻ MULTIPLICATION SIGN IN TRIANGLE (HTML ⨻ · ⨻)
  • U+2AC1 ⫁ SUBSET WITH MULTIPLICATION SIGN BELOW (HTML ⫁ · ⫁)
  • U+2AC2 ⫂ SUPERSET WITH MULTIPLICATION SIGN BELOW (HTML ⫂ · ⫂)
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See also

References
  1. 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.
  2. Florian Cajori (1929). A History of Mathematical Notations. Dover Books on Mathematics. Dover. p. 251f. ISBN 9780486677668.
  3. 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
  4. New Hart's rules: the handbook of style for writers and editors, Oxford University Press, 2005, p. 183, ISBN 978-0-19-861041-0
  5. "Mac Zeichenpalette" (in German). TypoWiki. Archived from the original on 2007-10-25. Retrieved 2009-10-09.
  6. "Unicode Character 'MULTIPLICATION SIGN' (U+00D7)". Fileformat.info. Retrieved 2017-01-13.
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