Plus and minus signs
+ − | |
---|---|
Plus and minus signs | |
InUnicode | U+002B+PLUS SIGN(+) U+2212−MINUS SIGN(−) |
Different from | |
Different from | U+002D-HYPHEN-MINUS U+2010‐HYPHEN (many)-Dash |
Related | |
See also | U+00B1±PLUS-MINUS SIGN U+2213∓MINUS-OR-PLUS SIGN U+2052⁒COMMERCIAL MINUS SIGN |
Theplus sign(+) and theminus sign(−) aremathematical symbolsused to denotepositiveandnegativefunctions, respectively. In addition,+represents the operation ofaddition,which results in asum,while−representssubtraction,resulting in adifference.[1]Their use has been extended to many other meanings, more or less analogous.PlusandminusareLatinterms meaning "more" and "less", respectively.
The forms⟨+⟩and⟨−⟩are used in many countries around the word. Other designs include⟨﬩⟩for plus and⟨⁒⟩for minus.
History
[edit]Though the signs now seem as familiar as theAlpha betor theHindu–Arabic numerals,they are not of great antiquity. TheEgyptian hieroglyphicsign for addition, for example, resembled a pair of legs walking in the direction in which the text was written (Egyptiancould be written either from right to left or left to right), with the reverse sign indicating subtraction:[2]
|
Nicole Oresme'smanuscriptsfrom the 14th century show what may be one of the earliest uses of+as a sign for plus.[3]
In early 15th century Europe, the letters "P" and "M" were generally used.[4][5] The symbols (P with overline,p̄,forpiù(more), i.e., plus, and M with overline,m̄,formeno(less), i.e., minus) appeared for the first time inLuca Pacioli's mathematicscompendium,Summa de arithmetica, geometria, proportioni et proportionalità,first printed and published inVenicein 1494.[6]
The+sign is a simplification of theLatin:et(comparable to the evolution of theampersand&).[7]The−may be derived from amacron◌̄written over⟨m⟩when used to indicate subtraction; or it may come from a shorthand version of the letter⟨m⟩itself.[8]
![A page from Johannes Widmann's book](https://upload.wikimedia.org/wikipedia/commons/thumb/3/3b/Johannes_Widmann-Mercantile_Arithmetic_1489.jpg/220px-Johannes_Widmann-Mercantile_Arithmetic_1489.jpg)
In his 1489 treatise,Johannes Widmannreferred to the symbols−and+asminusandmer(Modern Germanmehr;"more" ):"[...] was − ist das ist minus [...] und das + das ist mer das zu addirst".[9][10][11]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 byHenricus Grammateus.[12][13]
Robert Recorde,the designer of theequals sign,introduced plus and minus to Britain in 1557 inThe Whetstone of Witte:[14]"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 sign
[edit]The plus sign (+) is abinary operatorthat indicatesaddition,as in2 + 3 = 5.It can also serve as aunary operatorthat leaves itsoperandunchanged(+xmeans the same asx). This notation may be used when it is desired to emphasize the positiveness of a number, especially in contrast with thenegative numbers(+5versus−5).
The plus sign can also indicate many other operations, depending on the mathematical system under consideration. Manyalgebraic structures,such asvector spacesandmatrix rings,have some operation which is called, or is equivalent to, addition. It is though conventional to use the plus sign to only denotecommutative operations.[15]
The symbol is also used inchemistryandphysics.For more, see§ Other uses.
Minus sign
[edit]The minus sign (−) has three main uses in mathematics:[16]
- Thesubtractionoperator: abinary operatorto indicate the operation of subtraction, as in5 − 3 = 2.Subtraction is the inverse of addition.[1]
- Thefunctionwhosevaluefor anyrealorcomplexargumentis theadditive inverseof that argument. For example, ifx= 3,then−x= −3,but ifx= −3,then−x= +3.Similarly,−(−x) =x.
- Aprefixof a numeric constant. When it is placed immediately before an unsigned numeral, 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:−5is the same number. When it is important to distinguish them, a raised minus sign (¯) is sometimes used for negative constants, as inelementary education,the programming languageAPL,and some early graphing calculators.[a]
All three uses can be referred to as "minus" in everyday speech, though the binary operator is sometimes read as "take away".[17]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".)[18]Further, a few textbooks in the United States encourage−xto be read as "the opposite ofx"or" the additive inverse ofx"—to avoid giving the impression that−xis necessarily negative (sincexitself may already be negative).[19]
In mathematics and most programming languages, the rules for theorder of operationsmean that−52is equal to−25:Exponentiationbinds more strongly than the unary minus, which binds more strongly than multiplication or division. However, in some programming languages (Microsoft Excelin particular), unary operators bind strongest, so in those cases−5^2
is 25, but0−5^2
is −25.[20]
Similar to the plus sign, the minus sign is also used inchemistryandphysics.For more, see§ Other usesbelow.
Use in elementary education
[edit]Some elementary teachers use raised minus signs before numbers to disambiguate them from the operation of subtraction.[21]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 −−5becomes3 + 5 = 8,
which can be read as:
- +3 −1 (−5)
or even as
- +3 −−5becomes+3 ++5 =+8.
Use as a qualifier
[edit]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".
In US grading systems, the plus sign indicates a grade one level higher and the minus sign a grade lower. For example,B−( "B minus" ) is one grade lower thanB.In some occasions, this is extended to two plus or minus signs (e.g.,A++being two grades higher thanA).
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+andApple 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+veand−ve.[22]
Mathematics
[edit]In mathematics theone-sided limitx→a+meansxapproachesafrom the right (i.e., right-sided limit), andx→a−meansxapproachesafrom the left (i.e., left-sided limit). For example,1/x→ +asx→ 0+but1/x→ −asx→ 0−.
Blood
[edit]Blood typesare often qualified with a plus or minus to indicate the presence or absence of theRh factor.For example, A+ meanstype A bloodwith the Rh factor present, while B− means type B blood with the Rh factor absent.
Music
[edit]In music,augmented chordsare 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 asuperscript.
Uses in computing
[edit]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 intree viewon a computer screen—to show if a folder is collapsed or not.
In some programming languages,concatenationofstringsis written"a" + "b"
,and results in"ab"
.
In most programming languages, subtraction and negation are indicated with the ASCIIhyphen-minuscharacter,-
.InAPLa raised minus sign (here written using Unicode U+00AF MACRON) is used to denote a negative number, as in¯3
.While inJa negative number is denoted by anunderscore,as in_5
.
InCand some other computer programming languages, two plus signs indicate theincrement operatorand 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, theny = x++
increments x to 7 but sets y to 6, whereasy = ++x
would set both x and y to 7. By extension,++
is sometimes used in computing terminology to signify an improvement, as in the name of the languageC++.
Inregular expressions,+
is often used to indicate "1 or more" in a pattern to be matched. For example,x+
means "one or more of the letter x". This is theKleene plusnotation.
There is no concept of negative zero in mathematics, but in computing−0may have a separate representation from zero. In theIEEE floating-point standard,1 / −0 isnegative infinity() whereas 1 / 0 ispositive infinity().
+
is also used to denote added lines indiff
output in thecontext formator theunified format.
Other uses
[edit]In physics, the use of plus and minus signs for differentelectrical chargeswas introduced byGeorg 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., NH+
4). If the charge is greater than 1, a number indicating the charge is written before the sign (as in SO2−
4).
A plus sign prefixed to a telephone number is used to indicate the form used forInternational Direct Dialing.[23]Its precise usage varies by technology and national standards. In theInternational Phonetic Alphabet,subscriptedplus and minus signs are used as diacritics to indicateadvanced or retracted articulationsof speech sounds.
The minus sign is also used as tone letter in the orthographies ofDan,Krumen,Karaboro,Mwan,Wan,Yaouré,Wè,NyabwaandGodié.[24]The Unicode character used for the tone letter (U+02D7) is different from the mathematical minus sign.
The plus sign sometimes represents/ɨ/in the orthography ofHuichol.[25]
In thealgebraic notationused to record games ofchess,the plus sign+is used to denote a move that puts the opponent intocheck,while a double plus++is sometimes used to denotedouble check.Combinations of the plus and minus signs are used to evaluate a move (+/−, +/=, =/+, −/+).
In linguistics, a superscript plus+sometimes replaces theasterisk,which denotes unattestedlinguistic reconstruction.
Inbotanical names,a plus sign denotesgraft-chimaera.
In Catholicism, the plus sign before a last name denotes aBishop,and a double plus is used to denote an Archbishop.
Character codes
[edit]- + − | |
---|---|
hyphen-minus, plus, minus signs compared |
- U+002B+PLUS SIGN
- U+2212−MINUS SIGN
- U+002D-HYPHEN-MINUS
- U+FE63﹣SMALL HYPHEN-MINUS
- U+FE62﹢SMALL PLUS SIGN
- U+FF0B+FULLWIDTH PLUS SIGN
- U+FF0D-FULLWIDTH HYPHEN-MINUS
Alternative minus signs
[edit]![](https://upload.wikimedia.org/wikipedia/commons/thumb/2/2e/Skjermbilete_2012-11-03_kl._02.48.36.png/220px-Skjermbilete_2012-11-03_kl._02.48.36.png)
There is acommercial minus sign,⁒,which is used in Germany and Scandinavia. The symbol÷is used to denote subtractioninScandinavia.[26]
Thehyphen-minussymbol (-) is the form ofhyphenmost commonly used in digitaldocuments.On most keyboards, it is the only character that resembles aminus signor adashso it is also used for these.[27]The namehyphen-minusderives from the originalASCIIstandard,[28]where it was calledhyphen–(minus).[29]The character is referred to as ahyphen,aminus sign,or adashaccording to the context where it is being used.
Alternative plus sign
[edit]AJewishtradition that dates from at least the 19th century is to writeplususing the symbol﬩.[30]This practice was adopted intoIsraelischools and is still commonplace today inelementary schools(includingsecularschools) but in fewersecondary schools.[31]It is also used occasionally in books by religious authors, but most books for adults use the international symbol+.The reason for this practice is that it avoids the writing of a symbol+that looks like aChristian cross.[30][31]Unicodehas this symbol at positionU+FB29﬩HEBREW LETTER ALTERNATIVE PLUS SIGN.[32]
See also
[edit]- En dash,a dash that looks similar to the subtraction symbol but is used for different purposes
- Glossary of mathematical symbols
- ⊕ (disambiguation)
Notes
[edit]- ^at least the earlyTexas Instrumentsmodels, including theTI-81andTI-82
References
[edit]- ^abWeisstein, Eric W."Subtraction".mathworld.wolfram.Archivedfrom the original on 2020-09-14.Retrieved2020-08-26.
- ^ Karpinski, Louis C. (1917). "Algebraical Developments Among the Egyptians and Babylonians".The American Mathematical Monthly.24(6): 257–265.doi:10.2307/2973180.JSTOR2973180.MR1518824.
- ^The birth of symbols – Zdena Lustigova, Faculty of Mathematics and Physics Charles University, PragueArchived2013-07-08 atarchive.today
- ^Ley, Willy (April 1965)."Symbolically Speaking".For Your Information.Galaxy Science Fiction.pp. 57–67.
- ^Stallings, Lynn (May 2000)."A brief history of algebraic notation".School Science and Mathematics.100(5): 230–235.doi:10.1111/j.1949-8594.2000.tb17262.x.Retrieved13 April2009.
- ^Sangster, Alan; Stoner, Greg; McCarthy, Patricia (2008)."The market for Luca Pacioli's Summa Arithmetica"(PDF).Accounting Historians Journal.35(1): 111–134 [p. 115].doi:10.2308/0148-4184.35.1.111.S2CID107010686.Archived(PDF)from the original on 2018-01-26.Retrieved2012-04-29.
- ^Cajori, Florian (1928). "Origin and meanings of the signs + and -".A History of Mathematical Notations, Vol. 1.The Open Court Company, Publishers.
- ^Wright, D. Franklin; New, Bill D. (2000).Intermediate Algebra(4th ed.). Thomson Learning. p. 1.
The minus sign or bar, —, is thought to be derived from the habit of early scribes of using a bar to represent the letter m
- ^abWidmann, Johannes (1489)."Behe[n]de vnd hubsche Rechenung auff allen kauffmanschafft".Leipzig: Konrad Kachelofen. p. 176.Archivedfrom the original on 2022-05-03.Retrieved2022-05-03.
- ^abWidmann, Johannes (1508)."Behend vnd hüpsch Rechnung vff allen Kauffmanschafften".Kolophon: Gedruck zů Pfhortzheim von Thoman Anßhelm. p. 122.Archivedfrom the original on 2022-05-03.Retrieved2022-05-03.
- ^"plus".Oxford English Dictionary(Online ed.).Oxford University Press.(Subscription orparticipating institution membershiprequired.)
- ^Smith, D.E.(1951).History of Mathematics.Vol. 1. Courier Dover Publications. pp. 258, 330.ISBN0486204308.
- ^"Earliest Uses of Symbols of Operation".Archivedfrom the original on 2022-04-29.Retrieved2022-05-03.
- ^Cajori, Florian(2007),A History of Mathematical Notations,Cosimo, p. 164,ISBN9781602066847.
- ^Fraleigh, John B. (1989).A First Course in Abstract Algebra(4 ed.). United States:Addison-Wesley.p. 52.ISBN0-201-52821-5.
- ^Henri Picciotto (1990).The Algebra Lab.Creative Publications. p. 9.ISBN978-0-88488-964-9.
- ^"Subtraction".mathsisfun.Archivedfrom the original on 2020-08-12.Retrieved2020-08-26.
- ^Schwartzman, Steven (1994).The words of mathematics.The Mathematical Association of America. p.136.ISBN9780883855119.
- ^Wheeler, Ruric E. (2001).Modern Mathematics(11 ed.). p. 171.
- ^"Microsoft Office Excel Calculation operators and precedence".Archived fromthe originalon 2009-08-11.Retrieved2009-07-29.
- ^Gaskill, H.S.; Lopez, Robert J. (May 1978). "Let's bring back subtraction".International Journal of Mathematical Education in Science and Technology.9(2): 221–229.doi:10.1080/0020739780090211.
- ^Castledine, George; Close, Ann (2009).Oxford Handbook of Adult Nursing.Oxford University Press. p. xvii.ISBN9780191039676..
- ^"Recommendation E.123: Notation for national and international telephone numbers, e-mail addresses and Web addresses".International Telecommunication Union.2001.Archivedfrom the original on 2021-05-05.Retrieved2021-03-18.
- ^Hartell, Rhonda L., ed. (1993),The Alphabets of Africa.Dakar: UNESCO and SIL.
- ^Biglow, Brad Morris (2001).Ethno-Nationalist Politics and Cultural Preservation: Education and Bordered Identities Among the Wixaritari (Huichol) of Tateikita, Jalisco, Mexico(PDF)(PhD). University of Florida. p. 284.Archived(PDF)from the original on 2021-06-02.Retrieved2021-05-29.
- ^"6. Writing Systems and Punctuation".The Unicode Standard: Version 10.0 – Core Specification(PDF).Unicode Consortium. June 2017. p. 280, Obelus.Archived(PDF)from the original on 2021-10-04.Retrieved2022-04-11.
- ^Korpela, Jukka K. (2006).Unicode explained.O'Reilly. p. 382.ISBN978-0-596-10121-3.
- ^"3.1 General scripts"(PDF).Unicode Version 1.0 · Character Blocks.p. 30.Archived(PDF)from the original on 21 November 2021.Retrieved10 December2021.
Loose vs. Precise Semantics.Some ASCII characters have multiple uses, either through ambiguity in the original standards or through accumulated reinterpretations of a limited codeset. For example, 27 hex is defined in ANSI X3.4 as apostrophe (closing single quotation mark; acute accent), and 2D hex as hyphen minus. In general, the Unicode standard provides the same interpretation for the equivalent code values, without adding to or subtracting from their semantics. The Unicode standard supplies unambiguous codes elsewhere for the most useful particular interpretations of these ASCII values; the corresponding unambiguous characters are cross-referenced in the character names list for this block. In a few cases, the Unicode standard indicates the generic interpretation of an ASCII code in the name of the corresponding Unicode character, for example U+0027 is APOSTROPHE-QUOTE'.
- ^"American National Standard X3.4-1977: American Standard Code for Information Interchange"(PDF).National Institute of Standards and Technology. p. 10 (4.2 Graphic characters).Archived(PDF)from the original on 9 October 2022.Retrieved10 December2021.
- ^abKaufmann Kohler (1901–1906)."Cross".InCyrus Adler;et al. (eds.).Jewish Encyclopedia.Archivedfrom the original on 2017-01-06.Retrieved2017-02-12.
- ^abChristian-Jewish Dialogue: Theological Foundations By Peter von der Osten-Sacken (1986 – Fortress Press)Archived2023-04-08 at theWayback MachineISBN0-8006-0771-6"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)
- ^Unicode U+FB29 reference pageArchived2009-01-26 at theWayback MachineThis form of the plus sign is also used on the control buttons at individual seats on board the El Al Israel Airlines aircraft.
External links
[edit]The dictionary definition ofplus signat Wiktionary
The dictionary definition ofminus signat Wiktionary