Plus and minus signs

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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.[1] Their use has been extended to many other meanings, more or less analogous. Script error: No such module "Lang". and Script error: No such module "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 Page Template:Mono/styles.css has no content.U+FB29 ﬩ Script error: No such module "Unicode data". for plus and Page Template:Mono/styles.css has no content.U+2052 ⁒ Script error: No such module "Unicode data". for minus.

History

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:[2]

<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.[3]

In early 15th century Europe, the letters "P" and "M" were generally used.[4][5] The symbols (P with overline, Template:Char, for Script error: No such module "Lang". (more), i.e., plus, and M with overline, Template:Char, for Script error: No such module "Lang". (less), i.e., minus) appeared for the first time in Luca Pacioli's mathematics compendium, Script error: No such module "Lang"., first printed and published in Venice in 1494.[6]

The Template:Char sign is a simplification of the Template:Langx (comparable to the evolution of the ampersand Template:Char).[7] The Template:Char may be derived from a macron Template:Char 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
From Johannes Widmann's book on "handy and pretty arithmetic for all merchants"[9][10]

In his 1489 treatise, Johannes Widmann referred to the symbols Template:Char and Template:Char as minus and mer (Modern German Script error: No such module "Lang".; "more"): Script error: No such module "Lang"..[9]Template:R/superscript[10]Template:R/superscript[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 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

Lua error in package.lua at line 80: module 'Module:Hatnote list' not found. 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.[15]

The symbol is also used in chemistry and physics. For more, see § Other uses.

Minus sign

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"−" (minus sign) redirects here. For the Ed Sheeran album, see − (album)

The minus sign (Template:Char) has three main uses in mathematics:[16]

  1. The subtraction operator: a binary operator to indicate the operation of subtraction, as in 5 − 3 = 2. Subtraction is the inverse of addition.[1]
  2. 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.
  3. A 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 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.[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" 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).[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,[22] and on batteries and cell terminals are often marked with Template:Char and Template:Char.

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, 1x→+∞ as x → 0+ but 1x→−∞ as x → 0−.

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, ℤ+ is the set of all positive integers and ℤ− is the set of all negative integers. In these cases, a subscript 0 may also be added to clarify that 0 is included.

Blood

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.

Music

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 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 Page Template:Mono/styles.css has no content.U+00AF ¯ Script error: No such module "Unicode data".) 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 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 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. Hyphen-minus usually indicates a range ([A-Z] - any capital from 'A' to 'Z'), although it can stand for itself ([ABCDE-] 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 (−∞) whereas 1 / 0 is positive infinity (∞).

+ is also used to denote added lines in diff output in the Template:Pslink or the Template:Pslink.

Other uses

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., 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 International Direct Dialing.[23] 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.[24] The Unicode character used for the tone letter (Page Template:Mono/styles.css has no content.U+02D7 ˗ Script error: No such module "Unicode data".) is different from the mathematical minus sign.

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 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

Template:Infobox symbol Variants of the symbols have unique codepoints in Unicode:

Alternative plus sign

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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.[26][27] This practice was adopted into Israeli schools and was still commonplace as of 1986 in elementary schools (including secular schools) but in fewer secondary schools.[needs update?]Template:DMCA[27] 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 Page Template:Mono/styles.css has no content.U+FB29 ﬩ Script error: No such module "Unicode data"..[28]

Alternative minus signs

File:Skjermbilete 2012-11-03 kl. 02.48.36.png
"÷" being used as a minus sign (not as a division sign) in an excerpt from an official Norwegian trading statement form called «Næringsoppgave 1» for the taxation year 2010

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.[29]

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.[30] The name hyphen-minus derives from the original ASCII standard,[31] where it was called hyphen–(minus).[32] The character is referred to as a hyphen, a minus sign, or a dash according to the context where it is being used.

See also

Notes

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  1. ^ at least the early Texas Instruments models, including the TI-81 and TI-82

References

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  1. ^ a b Page Module:Citation/CS1/styles.css has no content.Weisstein, Eric W. "Subtraction". mathworld.wolfram.com. Archived from the original on 2020-09-14. Retrieved 2020-08-26.
  2. ^ Page Module:Citation/CS1/styles.css has no content.Karpinski, Louis C. (1917). "Algebraical Developments Among the Egyptians and Babylonians". The American Mathematical Monthly. 24 (6): 257–265. doi:10.2307/2973180. JSTOR 2973180. MR 1518824.
  3. ^ The birth of symbols – Zdena Lustigova, Faculty of Mathematics and Physics Charles University, Prague Script error: No such module "webarchive".
  4. ^ Page Module:Citation/CS1/styles.css has no content.Ley, Willy (April 1965). "Symbolically Speaking". For Your Information. Galaxy Science Fiction. pp. 57–67.
  5. ^ Page Module:Citation/CS1/styles.css has no content.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. Retrieved 13 April 2009.
  6. ^ Page Module:Citation/CS1/styles.css has no content.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. S2CID 107010686. Archived (PDF) from the original on 2018-01-26. Retrieved 2012-04-29.
  7. ^ Page Module:Citation/CS1/styles.css has no content.Cajori, Florian (1928). "Origin and meanings of the signs + and -". A History of Mathematical Notations, Vol. 1. The Open Court Company, Publishers.
  8. ^ Page Module:Citation/CS1/styles.css has no content.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
  9. ^ a b Page Module:Citation/CS1/styles.css has no content.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.
  10. ^ a b Page Module:Citation/CS1/styles.css has no content.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.
  11. ^ Script error: No such module "template wrapper". (Subscription or participating institution membership required.)
  12. ^ Page Module:Citation/CS1/styles.css has no content.Smith, D.E. (1951). History of Mathematics. Vol. 1. Courier Dover Publications. pp. 258, 330. ISBN 0486204308. {{cite book}}: ISBN / Date incompatibility (help)
  13. ^ Page Module:Citation/CS1/styles.css has no content."Earliest Uses of Symbols of Operation". Archived from the original on 2022-04-29. Retrieved 2022-05-03.
  14. ^ Page Module:Citation/CS1/styles.css has no content.Cajori, Florian (2007), A History of Mathematical Notations, Cosimo, p. 164, ISBN 9781602066847.
  15. ^ Page Module:Citation/CS1/styles.css has no content.Fraleigh, John B. (1989). A First Course in Abstract Algebra (4 ed.). United States: Addison-Wesley. p. 52. ISBN 0-201-52821-5.
  16. ^ Page Module:Citation/CS1/styles.css has no content.Henri Picciotto (1990). The Algebra Lab. Creative Publications. p. 9. ISBN 978-0-88488-964-9.
  17. ^ Page Module:Citation/CS1/styles.css has no content."Subtraction". www.mathsisfun.com. Archived from the original on 2020-08-12. Retrieved 2020-08-26.
  18. ^ Page Module:Citation/CS1/styles.css has no content.Schwartzman, Steven (1994). The words of mathematics. The Mathematical Association of America. p. 136. ISBN 9780883855119.
  19. ^ Page Module:Citation/CS1/styles.css has no content.Wheeler, Ruric E. (2001). Modern Mathematics (11 ed.). p. 171.
  20. ^ Page Module:Citation/CS1/styles.css has no content."Microsoft Office Excel Calculation operators and precedence". Archived from the original on 2009-08-11. Retrieved 2009-07-29.
  21. ^ Page Module:Citation/CS1/styles.css has no content.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.
  22. ^ Page Module:Citation/CS1/styles.css has no content.Castledine, George; Close, Ann (2009). Oxford Handbook of Adult Nursing. Oxford University Press. p. xvii. ISBN 9780191039676..
  23. ^ Page Module:Citation/CS1/styles.css has no content."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.
  24. ^ Hartell, Rhonda L., ed. (1993), The Alphabets of Africa. Dakar: UNESCO and SIL.
  25. ^ Page Module:Citation/CS1/styles.css has no content.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.
  26. ^ Page Module:Citation/CS1/styles.css has no content.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.
  27. ^ a b Page Module:Citation/CS1/styles.css has no content.Osten-Sacken, Peter von der (1986). Christian-Jewish dialogue: theological foundations. Philadelphia: Fortress Press. p. 96. ISBN 0-8006-0771-6. Retrieved 11 December 2025. 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.
  28. ^ Unicode U+FB29 reference page Script error: No such module "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.
  29. ^ Page Module:Citation/CS1/styles.css has no content."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.
  30. ^ Page Module:Citation/CS1/styles.css has no content.Korpela, Jukka K. (2006). Unicode explained. O'Reilly. p. 382. ISBN 978-0-596-10121-3.
  31. ^ Page Module:Citation/CS1/styles.css has no content."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'.
  32. ^ Page Module:Citation/CS1/styles.css has no content."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.

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