“Plus and minus signs”的意思、由来-开放百科全书

The plus and minus signs (+ and −) are mathematical symbols used to represent the notions of positive and negative as well as the operations of addition and subtraction. Their use has been extended to many other meanings, more or less analogous. Plus and minus are Latin terms meaning "more" and "less", respectively.

History

Though the signs now seem as familiar as the alphabet or the Hindu-Arabic numerals, they are not of great antiquity. The Egyptian hieroglyphic sign for addition, for example, resembled 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:^[1]

In Europe in the early 15th century the letters "P" and "M" were generally used.^[3]^[4]

The symbols (P with line p̄ for più, i.e., plus, and M with line m̄ for meno, 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.^[5] The + is a simplification of the Latin "et" (comparable to the ampersand &).^[6] The − may be derived from a tilde written over m when used to indicate subtraction; or it may come from a shorthand version of the letter m itself.^[7] 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 + ist das mer".^[8] They weren't used for addition and subtraction here, but to indicate surplus and deficit; their first use in their modern sense appears in a book by Henricus Grammateus in 1518.^[9]^[10]

Robert Recorde, the designer of the equals sign, introduced plus and minus to Britain in 1557 in The Whetstone of Witte:^[11] "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 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 when contrasting with the negative (+5 versus −5).

The plus sign can also indicate many other operations, depending on the mathematical system under consideration. Many algebraic structures have some operation which is called, or is equivalent to, addition. It is conventional to use the plus sign to only denote commutative operations.^[12] Moreover, the symbolism has been extended to very different operations; plus can also mean:

Minus sign

All three uses can be referred to as "minus" in everyday speech. In most English-speaking countries, −5 (for example) is normally pronounced "minus five", but in modern US usage it is instead usually pronounced "negative five"; here, "minus" may be used by speakers born before 1950, and is still popular in some contexts, but "negative" is usually taught as the only correct reading.^[14] 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.^[15]

In some contexts, different glyphs are used for these meanings; for instance in the computer language APL and the expression language used by Texas Instruments graphing calculators (definitely at least the early models including the TI-81 and TI-82) a raised minus sign is used in negative numbers (as in 2 − 5 shows ⁻3), but such usage is uncommon.

In mathematics and most programming languages, the rules for the order of operations mean that −5² is equal to −25: Powers bind more strongly than the unary minus, which binds more strongly than multiplication or division. However, in some programming languages and Microsoft Excel in particular, unary operators bind strongest, so in those cases −5^2 is 25 but 0−5^2 is −25.^[16]

Use in elementary education

Some elementary teachers use raised plus and minus signs before numbers to show they are positive or negative numbers.^[17] For example, subtracting −5 from 3 might be read as "positive three take away negative 5" and be shown as

Use as a qualifier

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. Sometimes this is extended to two plus or minus signs; for example A++ is two grades higher than A.

In mathematics the one-sided limit x→a⁺ means x approaches a from the right, and x→a⁻ means x approaches a from the left. For example, when calculating what x⁻¹ is when x approaches 0, because x⁻¹→+∞ when x→0⁺ but x⁻¹→−∞ when x→0⁻.

Blood types are often qualified with a plus or minus to indicate the presence or absence of the Rh factor; for instance, A+ means A-type blood with the Rh factor present, while B− means B-type blood with the Rh factor absent.

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". Also used as superscript.

Uses in computing

As well as the normal mathematical usage plus and minus 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 (Unicode U+00AF) 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".

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 (∞).

Other uses

In chemistry, superscripted plus and minus signs are used to indicate an ion with a positive or negative charge of 1 (for example, NH₄⁺). If the charge is greater than 1, a number indicating the charge is written before the sign (SO₄²⁻). The minus sign is also used (rather than an en dash) for a single covalent bond between two atoms, as in the skeletal formula.

Subscripted plus and minus signs are used as diacritics in the International Phonetic Alphabet 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.^[19] The Unicode character used for the tone letter (U+02D7) is different from the mathematical minus sign.

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. A double plus (++) is sometimes used to denote double check. Combinations of the plus and minus signs are used to evaluate a move (+/−, +/=, =/+, −/+).

Character codes

The hyphen-minus sign (-) is the ASCII alternative/version of the minus sign, and doubles as a hyphen. It is usually shorter in length than the plus sign and sometimes at a different height. It can be used as a substitute for the true minus sign when the character set is limited to ASCII. Most programming languages and other computer readable languages do this, since ASCII is generally available as a subset of most character encodings, while U+2212 is a Unicode feature only.

There is a commercial minus sign (⁒), which looks somewhat like an obelus, at U+2052 (HTML ⁒).

Alternative plus sign

A Jewish tradition that dates from at least the 19th century is to write plus using a symbol like an inverted T.^[20] This practice was adopted into Israeli schools and is still commonplace today in elementary schools (including secular schools) but in fewer secondary schools.^[21] 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 a Christian cross.^[20]^[21] Unicode has this symbol at position {{Unichar|FB29|HEBREW LETTER ALTERNATIVE PLUS SIGN}}.^[22]

See also

References and footnotes

1. ^{{cite journal | last = Karpinski | first = Louis C. | doi = 10.2307/2973180 | issue = 6 | journal = The American Mathematical Monthly | mr = 1518824 | pages = 257–265 | title = Algebraical Developments Among the Egyptians and Babylonians | volume = 24 | year = 1917}}
2. ^The birth of symbols – Zdena Lustigova, Faculty of Mathematics and Physics Charles University, Prague {{webarchive|url=https://archive.is/20130708153352/http://educ.ubc.ca/courses/etec540/Sep02/ResearchAssignment/LustigovaZ/ra-LustigovaZ.htm |date=2013-07-08 }}
3. ^{{Cite magazine |last=Ley |first=Willy |date=April 1965 |title=Symbolically Speaking |department=For Your Information |url=https://archive.org/stream/Galaxy_v23n04_1965-04#page/n57/mode/2up |magazine=Galaxy Science Fiction |pages=57-67 |type=}}
4. ^{{cite journal|last=Stallings|first=Lynn|date=May 2000|title=A brief history of algebraic notation|journal=School Science and Mathematics|url=http://findarticles.com/p/articles/mi_qa3667/is_200005/ai_n8885415/|accessdate=13 April 2009}}
5. ^{{cite journal |first=Alan |last=Sangster |first2=Greg |last2=Stoner |first3=Patricia |last3=McCarthy |title=The market for Luca Pacioli’s Summa Arithmetica |journal=Accounting Historians Journal |volume=35 |issue=1 |year=2008 |pages=111–134 [p. 115] |url=http://eprints.mdx.ac.uk/3201/1/final_final_proof_Market_paper_050308.pdf }}
6. ^{{cite book|last=Cajori|first=Florian|title=A History of Mathematical Notations, Vol. 1|year=1928|publisher=The Open Court Company, Publishers|chapter=Origin and meanings of the signs + and -}}
7. ^{{cite book|title=Intermediate Algebra|edition=4th|year=2000|first1=D. Franklin|last1=Wright|first2=Bill D.|last2=New|publisher=Thomson Learning|page=1|quote=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}}
8. ^{{OED|plus}}
9. ^{{cite book|last=Smith|author-link=David Eugene Smith|first=D.E.|title=History of Mathematics|isbn=0486204308 |publisher=Courier Dover Publications|year=1951|volume=1|pages=258, 330}}
10. ^Earliest Uses of Various Mathematical Symbols
11. ^{{citation|title=A History of Mathematical Notations|first=Florian|last=Cajori|authorlink=Florian Cajori|publisher=Cosimo|year=2007|isbn=9781602066847|page=164|url=https://books.google.com/books?id=rhEh8jPGQOcC&pg=PA164}}.
12. ^{{cite book | last = Fraleigh | first = John B. | title = A First Course in Abstract Algebra | publisher = Addison-Wesley | series = | volume = | edition = 4 | year = 1989 | location = United States | pages = 52 | language = | url = | doi = | id = | isbn = 0-201-52821-5 | mr = | zbl = | jfm = }}
13. ^{{Cite book|title= The Algebra Lab | url= https://books.google.com/books?id=nzukMBV6ReoC&pg=PA9 | author=Henri Picciotto | publisher=Creative Publications | page=9 | isbn=978-0-88488-964-9}}
14. ^{{Cite book|title=The words of mathematics |first=Steven |last=Schwartzman |year=1994 |publisher=The Mathematical Association of America |page=136}}
15. ^{{Cite book|title=Modern Mathematics |first=Ruric E. |last=Wheeler |year=2001 |edition=11 |pages=171}}
16. ^{{cite web|url=http://office.microsoft.com/en-us/excel/HP100788861033.aspx |title=Microsoft Office Excel Calculation operators and precedence |accessdate=2009-07-29 |deadurl=yes |archiveurl=https://web.archive.org/web/20090811090433/http://office.microsoft.com/en-us/excel/HP100788861033.aspx |archivedate=2009-08-11 |df= }}
17. ^{{cite book|title=Understanding by design|author1=Grant P. Wiggins|author2=Jay McTighe|page=210|year=2005|publisher=ACSD Publications|isbn=1-4166-0035-3}}
18. ^{{cite book|title=Oxford Handbook of Adult Nursing|first1=George|last1=Castledine|first2=Ann|last2=Close|publisher=Oxford University Press|year=2009|isbn=9780191039676|page=xvii|url=https://books.google.com/books?id=R6icAwAAQBAJ&pg=PR17}}.
19. ^Hartell, Rhonda L., ed. (1993), The Alphabets of Africa. Dakar: UNESCO and SIL.
20. ^¹{{cite book|author=Kaufmann Kohler|editor=Cyrus Adler|display-editors=etal|title=Jewish Encyclopedia|chapter=Cross|year=1901–1906|url=http://www.jewishencyclopedia.com/articles/4776-cross}}
21. ^¹[https://books.google.com/books?id=m8sWAAAAIAAJ&dq=Jewish+plus+sign&q=%22plus+sign+used+in+mathematics%22&pgis=1 Christian-Jewish Dialogue: Theological Foundations By Peter von der Osten-Sacken (1986 – Fortress Press)] {{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)
22. ^Unicode U+FB29 reference page This form of the plus sign is also used on the control buttons at individual seats on board the El Al Israel Airlines aircraft.