Plus and minus signs
+ − | |
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Plus and minus signs | |
In |
The plus sign (+) and the minus sign (−) are
The forms ⟨+⟩ and ⟨−⟩ are used in many countries around the word. Other designs include ⟨
History
Though the signs now seem as familiar as the
|
Nicole Oresme's manuscripts from 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, for più (more), i.e., plus, and M with overline, m̄, for meno (less), i.e., minus) appeared for the first time in Luca Pacioli's mathematics compendium, Summa de arithmetica, geometria, proportioni et proportionalità, first printed and published in Venice in 1494.[6]
The + sign is a simplification of the
In his 1489 treatise, Johannes Widmann referred to the symbols − and + as minus and mer (Modern German mehr; "more"): "[...] was − ist das ist minus [...] und das + das ist mer das zu addirst"[9][10][11] They weren't 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.[12][13]
Robert Recorde, the designer of the equals sign, introduced plus and minus to Britain in 1557 in The 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
The plus sign (+) is a
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.[15]
The symbol is also used in chemistry and physics. For more, see § Other uses.
Minus sign
The minus sign (−) has three main uses in mathematics:[16]
- The binary operator to indicate the operation of subtraction, as in 5 − 3 = 2. Subtraction is the inverse of addition.[1]
- 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.
- A prefix of 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: −5 is the same number. When it is important to distinguish them, a raised minus sign (¯) is sometimes used for negative constants, as in elementary education, the programming language APL, 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 −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).[19]
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 −5^2
is 25, but 0−5^2
is −25.[20]
Similar to the plus sign, the minus sign is also used in chemistry and physics. For more, see § Other uses below.
Use in elementary education
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 − −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 qualifier
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 grading systems (such as examination marks), 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 than B. In some occasions, this is extended to two plus or minus signs (e.g., A++ being two grades higher than A).
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 +ve and −ve.[22]
Mathematics
In mathematics the one-sided limit x → a+ means x approaches a from the right (i.e., right-sided limit), and x → a− means x approaches a from the left (i.e., left-sided limit). For example, 1/x → + as x → 0+ but 1/x → − as x → 0−.
Blood
Music
In music,
Uses in computing
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 "a" + "b"
, and results in "ab"
.
In most programming languages, subtraction and negation are indicated with the ASCII hyphen-minus character, -
. In APL a raised minus sign (here written using Unicode U+00AF MACRON) is used to denote a negative number, as in ¯3
. While in J a negative number is denoted by an underscore, as in _5
.
In
y = x++
increments x to 7 but sets y to 6, whereas y = ++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 language C++In regular 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 the Kleene plus notation.
There is no concept of negative zero in mathematics, but in computing
+
is also used to denote added lines in diff
output in the context format or the unified format.
Other uses
In physics, the use of plus and minus signs for different
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 for
The minus sign is also used as tone letter in the orthographies of
The plus sign sometimes represents /ɨ/ in the orthography of Huichol.[25]
In the algebraic notation used to record games of chess, the plus sign + is used to denote a move that puts the opponent into check, while a double plus ++ 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 + 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.
Character codes
- + − | |
---|---|
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
There is a commercial minus sign, ⁒, which is used in Germany and Scandinavia. The symbol ÷ is used to denote subtraction in Scandinavia.[26]
The
Alternative plus sign
A
See also
- En dash, a dash that looks similar to the subtraction symbol but is used for different purposes
- Glossary of mathematical symbols
- ⊕ (disambiguation)
Notes
- ^ at least the early Texas Instruments models, including the TI-81 and TI-82
References
- ^ a b Weisstein, Eric W. "Subtraction". mathworld.wolfram.com. Archived from the original on 2020-09-14. Retrieved 2020-08-26.
- ^
Karpinski, Louis C. (1917). "Algebraical Developments Among the Egyptians and Babylonians". MR 1518824.
- ^ The birth of symbols – Zdena Lustigova, Faculty of Mathematics and Physics Charles University, Prague Archived 2013-07-08 at archive.today
- ^ Ley, Willy (April 1965). "Symbolically Speaking". For Your Information. Galaxy Science Fiction. pp. 57–67.
- . Retrieved 13 April 2009.
- (PDF) from the original on 2018-01-26. Retrieved 2012-04-29.
- ^ 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
- ^ a b Widmann, Johannes (1489). "Behe[n]de vnd hubsche Rechenung auff allen kauffmanschafft". Leipzig : Konrad Kachelofen. p. 176. Archived from the original on 2022-05-03. Retrieved 2022-05-03.
- ^ a b Widmann, Johannes (1508). "Behend vnd hüpsch Rechnung vff allen Kauffmanschafften". Kolophon: Gedruck zů Pfhortzheim von Thoman Anßhelm. p. 122. Archived from the original on 2022-05-03. Retrieved 2022-05-03.
- ^ "plus". Oxford English Dictionary (Online ed.). Oxford University Press. (Subscription or participating institution membership required.)
- ISBN 0486204308.
- ^ "Earliest Uses of Symbols of Operation". Archived from the original on 2022-04-29. Retrieved 2022-05-03.
- ISBN 9781602066847.
- ISBN 0-201-52821-5.
- ISBN 978-0-88488-964-9.
- ^ "Subtraction". www.mathsisfun.com. Archived from the original on 2020-08-12. Retrieved 2020-08-26.
- ISBN 9780883855119.
- ^ Wheeler, Ruric E. (2001). Modern Mathematics (11 ed.). p. 171.
- ^ "Microsoft Office Excel Calculation operators and precedence". Archived from the original on 2009-08-11. Retrieved 2009-07-29.
- .
- ISBN 9780191039676..
- ^ "Recommendation E.123: Notation for national and international telephone numbers, e-mail addresses and Web addresses". International Telecommunication Union. 2001. Archived from the original on 2021-05-05. Retrieved 2021-03-18.
- ^ 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. Retrieved 2021-05-29.
- ^ "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. Retrieved 2022-04-11.
- ISBN 978-0-596-10121-3.
- ^ "3.1 General scripts" (PDF). Unicode Version 1.0 · Character Blocks. p. 30. Archived (PDF) from the original on 21 November 2021. Retrieved 10 December 2021.
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. Retrieved 10 December 2021.
- ^ a b Kaufmann Kohler (1901–1906). "Cross". In Cyrus Adler; et al. (eds.). Jewish Encyclopedia. Archived from the original on 2017-01-06. Retrieved 2017-02-12.
- ^ ISBN 0-8006-0771-6"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 page Archived 2009-01-26 at the Wayback Machine This 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
- The dictionary definition of plus sign at Wiktionary
- The dictionary definition of minus sign at Wiktionary