Template:Short description Template:For Template:Redirect-distinguish Template:Infobox symbol
The plus sign (Template:Char) and the minus sign (Template:Char) are mathematical symbols used to denote positive and negative functions, respectively. In addition, the symbol Template:Char represents the operation of addition, which results in a sum, while the symbol Template:Char represents subtraction, resulting in a difference.<ref name="Weisstein">{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> Their use has been extended to many other meanings, more or less analogous. {{#invoke:Lang|lang}} and {{#invoke:Lang|lang}} are Latin terms meaning 'more' and 'less', respectively.
The forms Template:Char and Template:Char are used in many countries around the world. Other designs include Template:Unichar for plus and Template:Unichar for minus.
HistoryEdit
Though the signs now seem as familiar as the alphabet or the Arabic numerals, they are not of great antiquity. The Egyptian hieroglyphic sign for addition, for example, resembles a pair of legs walking in the direction in which the text was written (Egyptian could be written either from right to left or left to right), with the reverse sign indicating subtraction:<ref> Template:Cite journal</ref>
| <hiero>D54</hiero> or <hiero>D55</hiero> |
Nicole Oresme's manuscripts from the 14th century show what may be one of the earliest uses of Template:Char as a sign for plus.<ref>The birth of symbols – Zdena Lustigova, Faculty of Mathematics and Physics Charles University, Prague Template:Webarchive</ref>
In early 15th century Europe, the letters "P" and "M" were generally used.<ref name="ley196504">Template:Cite magazine</ref><ref>Template:Cite journal</ref> The symbols (P with overline, Template:Char, for {{#invoke:Lang|lang}} (more), i.e., plus, and M with overline, Template:Char, for {{#invoke:Lang|lang}} (less), i.e., minus) appeared for the first time in Luca Pacioli's mathematics compendium, {{#invoke:Lang|lang}}, first printed and published in Venice in 1494.<ref>Template:Cite journal</ref>
The Template:Char sign is a simplification of the Template:Langx (comparable to the evolution of the ampersand Template:Char).<ref>Template:Cite book</ref> The Template:Char may be derived from a macron Template:Char written over Template:Angbr when used to indicate subtraction; or it may come from a shorthand version of the letter Template:Angbr itself.<ref>Template:Cite book</ref>
In his 1489 treatise, Johannes Widmann referred to the symbols Template:Char and Template:Char as minus and mer (Modern German {{#invoke:Lang|lang}}; "more"): {{#invoke:Lang|lang}}.Template:RTemplate:R<ref name="OED">Template:OED</ref> They were not used for addition and subtraction in the treatise, but were used to indicate surplus and deficit; usage in the modern sense is attested in a 1518 book by Henricus Grammateus.<ref>Template:Cite book</ref><ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>
Robert Recorde, the designer of the equals sign, introduced plus and minus to Britain in 1557 in The Whetstone of Witte:<ref>Template:Citation.</ref> "There be other 2 signes in often use of which the first is made thus + and betokeneth more: the other is thus made − and betokeneth lesse."
Plus signEdit
Template:Redirect The plus sign (Template:Char) is a binary operator that indicates addition, as in 2 + 3 = 5. It can also serve as a unary operator that leaves its operand unchanged (+x means the same as x). This notation may be used when it is desired to emphasize the positiveness of a number, especially in contrast with the negative numbers (+5 versus −5).
The plus sign can also indicate many other operations, depending on the mathematical system under consideration. Many algebraic structures, such as vector spaces and matrix rings, have some operation which is called, or is equivalent to, addition. It is though conventional to use the plus sign to only denote commutative operations.<ref name="Fraleigh">Template:Cite book</ref>
The symbol is also used in chemistry and physics. For more, see Template:Section link.
Minus signEdit
Template:Redirect {{#invoke:Hatnote|hatnote}} The minus sign (Template:Char) has three main uses in mathematics:<ref>Template:Cite book</ref>
- The subtraction operator: a binary operator to indicate the operation of subtraction, as in 5 − 3 = 2. Subtraction is the inverse of addition.<ref name="Weisstein" />
- The function whose value for any real or complex argument is the additive inverse of that argument. For example, if x = 3, then −x = −3, but if x = −3, then −x = +3. Similarly, −(−x) = x.
- Template:AnchorA prefix of a numeric constant. When it is placed immediately before an unsigned number, the combination names a negative number, the additive inverse of the positive number that the numeral would otherwise name. In this usage, '−5' names a number the same way 'semicircle' names a geometric figure, with the caveat that 'semi' does not have a separate use as a function name.
In many contexts, it does not matter whether the second or the third of these usages is intended: −5 is the same number. When it is important to distinguish them, a raised minus sign (Template:Char) is sometimes used for negative constants, as in elementary education, the programming language APL, and some early graphing calculators.Template:Efn
All three uses can be referred to as "minus" in everyday speech, though the binary operator is sometimes read as "take away".<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> In American English nowadays, −5 (for example) is generally referred to as "negative five" though speakers born before 1950 often refer to it as "minus five". (Temperatures tend to follow the older usage; −5° is generally called "minus five degrees".)<ref>Template:Cite book</ref> Further, a few textbooks in the United States encourage −x to be read as "the opposite of x" or "the additive inverse of x"—to avoid giving the impression that −x is necessarily negative (since x itself may already be negative).<ref>Template:Cite book</ref>
In mathematics and most programming languages, the rules for the order of operations mean that −52 is equal to −25: Exponentiation binds more strongly than the unary minus, which binds more strongly than multiplication or division. However, in some programming languages (Microsoft Excel in particular), unary operators bind strongest, so in those cases <syntaxhighlight lang="text" class="" style="" inline="1">−5^2</syntaxhighlight> is 25, but <syntaxhighlight lang="text" class="" style="" inline="1">0−5^2</syntaxhighlight> is −25.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>
Similar to the plus sign, the minus sign is also used in chemistry and physics. (For more, see Template:Section link below.)
Use in elementary education Template:AnchorEdit
Some elementary teachers use raised minus signs before numbers to disambiguate them from the operation of subtraction.<ref>Template:Cite journal</ref> The same convention is also used in some computer languages. For example, subtracting −5 from 3 might be read as "positive three take away negative 5", and be shown as
- 3 − −5 becomes 3 + 5 = 8,
which can be read as:
- +3 −1(−5)
or even as
- +3 − −5 becomes +3 + +5 = +8.
Use as a qualifierEdit
When placed after a number, a plus sign can indicate an open range of numbers. For example, "18+" is commonly used as shorthand for "ages 18 and up" although "eighteen plus", for example, is now common usage.
In US grading systems, the plus sign indicates a grade one level higher and the minus sign a grade lower. For example, Template:Char ("B minus") is one grade lower than Template:Char. In some occasions, this is extended to two plus or minus signs (e.g., Template:Char being two grades higher than Template:Char).Template:Citation needed
A common trend in branding, particularly with streaming video services, has been the use of the plus sign at the end of brand names, e.g. Google+, Disney+, Paramount+, and Apple TV+. Since the word "plus" can mean an advantage, or an additional amount of something, such "+" signs imply that a product offers extra features or benefits.
Positive and negative are sometimes abbreviated as Template:Char and Template:Char,<ref>Template:Cite book.</ref> and on batteries and cell terminals are often marked with Template:Char and Template:Char.
MathematicsEdit
In mathematics the one-sided limit Template:Math means Template:Math approaches Template:Math from the right (i.e., right-sided limit), and Template:Math means Template:Math approaches Template:Math from the left (i.e., left-sided limit). For example, Template:Math as Template:Math but Template:Math as Template:Math.
When placed after special sets of numbers, plus and minus signs are used to indicate that only positive numbers and negative numbers are included, respectively. For example, <math>\mathbb{Z}^+</math> is the set of all positive integers and <math>\mathbb{Z}^-</math> is the set of all negative integers. In these cases, a subscript 0 may also be added to clarify that 0 is included.
BloodEdit
Blood types are often qualified with a plus or minus to indicate the presence or absence of the Rh factor. For example, A+ means type A blood with the Rh factor present, while B− means type B blood with the Rh factor absent.
MusicEdit
In music, augmented chords are symbolized with a plus sign, although this practice is not universal (as there are other methods for spelling those chords). For example, "C+" is read "C augmented chord". Sometimes the plus is written as a superscript.
Uses in computingEdit
As well as the normal mathematical usage, plus and minus signs may be used for a number of other purposes in computing.
Plus and minus signs are often used in tree view on a computer screen—to show if a folder is collapsed or not.
In some programming languages, concatenation of strings is written <syntaxhighlight lang="text" class="" style="" inline="1">"a" + "b"</syntaxhighlight>, and results in <syntaxhighlight lang="text" class="" style="" inline="1">"ab"</syntaxhighlight>.
In most programming languages, subtraction and negation are indicated with the ASCII hyphen-minus character, <syntaxhighlight lang="text" class="" style="" inline="1">-</syntaxhighlight>. In APL a raised minus sign (here written using Template:Unichar) is used to denote a negative number, as in <syntaxhighlight lang="text" class="" style="" inline="1">¯3</syntaxhighlight>. While in J a negative number is denoted by an underscore, as in <syntaxhighlight lang="text" class="" style="" inline="1">_5</syntaxhighlight>.
In C and some other computer programming languages, two plus signs indicate the increment operator and two minus signs a decrement; the position of the operator before or after the variable indicates whether the new or old value is read from it. For example, if x equals 6, then <syntaxhighlight lang="text" class="" style="" inline="1">y = x++</syntaxhighlight> increments x to 7 but sets y to 6, whereas <syntaxhighlight lang="text" class="" style="" inline="1">y = ++x</syntaxhighlight> would set both x and y to 7. By extension, <syntaxhighlight lang="text" class="" style="" inline="1">++</syntaxhighlight> is sometimes used in computing terminology to signify an improvement, as in the name of the language C++.
In regular expressions, <syntaxhighlight lang="text" class="" style="" inline="1">+</syntaxhighlight> is often used to indicate "1 or more" in a pattern to be matched. For example, <syntaxhighlight lang="text" class="" style="" inline="1">x+</syntaxhighlight> means "one or more of the letter x". This is the Kleene plus notation. Hyphen-minus usually indicates a range (<syntaxhighlight lang="text" class="" style="" inline="1">[A-Z]</syntaxhighlight> - any capital from 'A' to 'Z'), although it can stand for itself (<syntaxhighlight lang="text" class="" style="" inline="1">[ABCDE-]</syntaxhighlight> any capital from 'A' to 'E' or '-').
There is no concept of negative zero in mathematics, but in computing −0 may have a separate representation from zero. In the IEEE floating-point standard, 1 / −0 is negative infinity (<math>-\infty</math>) whereas 1 / 0 is positive infinity (<math>\infty</math>).
<syntaxhighlight lang="text" class="" style="" inline="1">+</syntaxhighlight> is also used to denote added lines in <syntaxhighlight lang="text" class="" style="" inline="1">diff</syntaxhighlight> output in the Template:Pslink or the Template:Pslink.
Other usesEdit
In physics, the use of plus and minus signs for different electrical charges was introduced by Georg Christoph Lichtenberg.
In chemistry, superscripted plus and minus signs are used to indicate an ion with a positive or negative charge of 1 (e.g., NHTemplate:Subsup). If the charge is greater than 1, a number indicating the charge is written before the sign (as in SOTemplate:Subsup).
A plus sign prefixed to a telephone number is used to indicate the form used for International Direct Dialing.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> Its precise usage varies by technology and national standards. In the International Phonetic Alphabet, subscripted plus and minus signs are used as diacritics to indicate advanced or retracted articulations of speech sounds.
The minus sign is also used as tone letter in the orthographies of Dan, Krumen, Karaboro, Mwan, Wan, Yaouré, Wè, Nyabwa, and Godié.<ref>Hartell, Rhonda L., ed. (1993), The Alphabets of Africa. Dakar: UNESCO and SIL.</ref> The Unicode character used for the tone letter (Template:Unichar) is different from the mathematical minus sign.
The plus sign sometimes represents Template:IPAslink in the orthography of Huichol.<ref>Template:Cite thesis</ref>
In the algebraic notation used to record games of chess, the plus sign Template:Char is used to denote a move that puts the opponent into check, while a double plus Template:Char is sometimes used to denote double check. Combinations of the plus and minus signs are used to evaluate a move (+/−, +/=, =/+, −/+).
In linguistics, a superscript plus Template:Char sometimes replaces the asterisk, which denotes unattested linguistic reconstruction.
In botanical names, a plus sign denotes graft-chimaera.
In Catholicism, the plus sign before a last name denotes a Bishop, and a double plus is used to denote an Archbishop.
Unicode Edit
Template:Infobox symbol Variants of the symbols have unique codepoints in Unicode:
- Template:Unichar
- Template:Unichar
- Template:Unichar
- Template:Unichar
- Template:Unichar
- Template:Unichar
- Template:Unichar
- Template:Unichar
- Template:Unichar
- Template:Unichar
- Template:Unichar
- Template:Unichar (a contiguity operator indicating addition)
- Template:Unichar
- Template:Unichar
- Template:Unichar
- Template:Unichar
- Template:Unichar
- Template:Unichar
- Template:Unichar
- Template:Unichar
- Template:Unichar
- Template:Unichar
- Template:Unichar
- Template:Unichar
- Template:Unichar
- Template:Unichar
- Template:Unichar
- Template:Unichar
- Template:Unichar
- Template:Unichar
- Template:Unichar
- Template:Unichar
- Template:Unichar
- Template:Unichar
- Template:Unichar
- Template:Unichar
- Template:Unichar
- Template:Unichar
- Template:Unichar
- Template:Unichar
Alternative minus signsEdit
There is a commercial minus sign, Template:Char, which is (or was) used in Germany and Scandinavia. The symbol Template:Char, still used in many Anglophone countries as a division sign, is (or was) used to denote subtraction in Scandinavia.<ref>Template:Cite book</ref>
The hyphen-minus symbol (Template:Char) is the form of hyphen most commonly used in digital documents. On most keyboards, it is the only character that resembles a minus sign or a dash so it is also used for these.<ref>Template:Cite book</ref> The name hyphen-minus derives from the original ASCII standard,<ref>Template:Cite book</ref> where it was called hyphen–(minus).<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> The character is referred to as a hyphen, a minus sign, or a dash according to the context where it is being used.
Alternative plus signEdit
A Jewish tradition that dates from at least the 19th century is to write plus using the symbol Template:Char, to avoid the writing of a symbol Template:Char that could look like a Christian cross.<ref name=JE>Template:Cite book</ref><ref name="University of California">Christian-Jewish Dialogue: Theological Foundations By Peter von der Osten-Sacken (1986 – Fortress Press) Template:Webarchive Template:ISBN "In Israel the plus sign used in mathematics is represented by a horizontal stroke with a vertical hook instead of the sign otherwise used all over the world, because the latter is reminiscent of a cross." (Page 96)</ref> This practice was adopted into Israeli schools and is still commonplace today in elementary schools (including secular schools) but in fewer secondary schools.<ref name="University of California"/> It is also used occasionally in books by religious authors, but most books for adults use the international symbol Template:Char. Unicode has this symbol at position Template:Unichar.<ref>Unicode U+FB29 reference page Template:Webarchive This form of the plus sign is also used on the control buttons at individual seats on board the El Al Israel Airlines aircraft.</ref>
See alsoEdit
- En dash, a dash that looks similar to the subtraction symbol but is used for different purposes
- Glossary of mathematical symbols
- ⊕ (disambiguation)