| × | |
|---|---|
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] While similar to a lowercase X (x), the form is properly a four-fold rotationally symmetric saltire.[2]
The symbol is also used in botany, in botanical hybrid names.
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Transcription
We now, hopefully, know a little bit about the variables. And as we covered in the last video, a variable can be really any symbol. Although, we typically use letters because we're used to writing and typing letters. But it can be anything from an x to a y, a z, an a, a b. And so oftentimes, we'll start using Greek letters, like theta. But you can really use any simple to say, hey, this is going to vary. It can take on multiple values. But out of all of these, the one that's most typically used in algebra, or really in all of mathematics, is the variable x. Although, all of these are used to some degree. But given that x is used so heavily, it does introduce a slight problem. And that problem is it looks a lot like the multiplication symbol, or the one that we use in arithmetic. So in arithmetic, if I want to write 2 times 3, I literally write 2 times 3. But now that we're starting to use variables, and if I want to write 2 times x, well if I use this as the multiplication symbol, it would be 2 times x. And the times symbol and the x look awfully similar. And if I'm not really careful with my penmanship, it can get very confusing. Is this 2 x x? Is this 2 times times something? What exactly is going on here? And because this is confusing, this right over here is extremely confusing and it can be misinterpreted, we tend to not use this multiplication symbol when we are doing algebra. Instead of that, to represent multiplication, we have several options. Instead of writing 2 times x this way, we could write 2 dot x. And this dot-- I want to be very clear. This is not a decimal. This is written a little bit higher. And we write this so we don't get confusion between this and one of these variables right here. But this really can be interpreted as 2 times x. So for example, if someone says 2 dot x. What is 2 dot x when x is equal to 3? Well, this would be the same thing as 2 times 3 when x is equal to 3. Another way that you could write it is you could write 2, and then you could write the x in parentheses right next to it. This is also interpreted as 2 times x. Once again, so if in this situation x were 7, this would be 2 times 7, or 14. And then the most traditional way of doing it is to just write the x right after the 2. And sometimes this will be read as 2x. But this literally does mean 2 times x. And you might say, well, how come we didn't always do that? Well, it'd be literally confusing if we did it over here. If instead of writing 2 times 3, we just wrote 2 3, well, that looks like 23. This doesn't look like 2 times 3. And that's why we never did it. But here, since we're using a letter now, it's clear that this isn't part of that number. This isn't 20 something. This is 2 times this variable x. So all of these are really the same expression-- 2 times x, 2 times x, and 2 times x. And so with that out of the way, let's try a few worked examples, a few practice problems. And this will, hopefully, prepare you for the next exercise where you get a lot a chance to practice this. So if I were to say, what is 10 minus 3y? And what does this equal when y is equal to 2? Well, every time you see the y, you'd want that 2 there. So this is when y is equal to 2. Let's set y equal to 2. This is the same thing as 10 minus 3 times 2. You do the multiplication first. Multiplication takes precedence in order of operations. So 3 times 2 is 6. 10 minus 6 is equal to 4. Let's do another one. Let's say we had, I don't know, 7x minus 4. Let me do that in the same green color, 7x minus 4. And we want to evaluate that when x is equal to 3. Where will we see the x? We want to put a 3 there. So this is the same thing as 7 times 3. I'll actually use this notation because I can use that with numbers. 7 times 3 minus 4. And once again, multiplication takes precedence by order of operations over addition or subtraction. So we want to do the multiplying first. 7 times 3 is 21. 21 minus 4 is equal to 17. So hopefully that gives you a little bit of background. I really encourage you to try the next exercise. It will give you a lot of practice on being able to evaluate expressions like this.
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 |
|
| 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+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
- ^ 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
- ^ 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 page was last edited on 12 April 2024, at 20:24