Multiplication 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]

×
Multiplication sign
In UnicodeU+00D7 × MULTIPLICATION SIGN (HTML &#215; · &times;)
Related
U+22C5 DOT OPERATOR
U+00F7 ÷ DIVISION SIGN
Different from
Different fromU+0078 x LATIN SMALL LETTER X (HTML &#120;)

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")
• 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 algebraic 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 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[7] Microsoft Windows In the Character Map utility in the eighth row, or by searching Alt+0215 using the numeric keypad[8] 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 &#215; · &times;)

Other variants and related characters:

• U+002A * ASTERISK (HTML &#42; · &ast;, &midast;)
• U+2062 INVISIBLE TIMES (HTML &#8290; · &InvisibleTimes;, &it;) (a zero-width space indicating multiplication)
• U+00B7 · MIDDLE DOT (HTML &#183; · &middot;, &CenterDot;, &centerdot;) (the interpunct, may be easier to type than the dot operator)
• U+2297 CIRCLED TIMES (HTML &#8855; · &CircleTimes;, &otimes;)
• U+22C5 DOT OPERATOR (HTML &#8901; · &sdot;)
• U+2715 MULTIPLICATION X (HTML &#10005;)
• U+2716 HEAVY MULTIPLICATION X (HTML &#10006;)
• U+2A09 N-ARY TIMES OPERATOR (HTML &#10761;)
• U+2A2F VECTOR OR CROSS PRODUCT (HTML &#10799; · &Cross;) (intended to explicitly denote the cross product of two vectors)
• U+2A30 MULTIPLICATION SIGN WITH DOT ABOVE (HTML &#10800; · &timesd;)
• U+2A31 MULTIPLICATION SIGN WITH UNDERBAR (HTML &#10801; · &timesbar;)
• U+2A34 MULTIPLICATION SIGN IN LEFT HALF CIRCLE (HTML &#10804; · &lotimes;)
• U+2A35 MULTIPLICATION SIGN IN RIGHT HALF CIRCLE (HTML &#10805; · &rotimes;)
• U+2A36 CIRCLED MULTIPLICATION SIGN WITH CIRCUMFLEX ACCENT (HTML &#10806; · &otimesas;)
• U+2A37 MULTIPLICATION SIGN IN DOUBLE CIRCLE (HTML &#10807; · &Otimes;)
• U+2A3B MULTIPLICATION SIGN IN TRIANGLE (HTML &#10811; · &tritime;)
• U+2AC1 SUBSET WITH MULTIPLICATION SIGN BELOW (HTML &#10945; · &submult;)
• U+2AC2 SUPERSET WITH MULTIPLICATION SIGN BELOW (HTML &#10946; · &supmult;)