Plus and minus signs

This article is about the mathematical symbols. For other uses of "plus", see Plus (disambiguation). For other uses of "minus", see Minus (disambiguation).

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:^{[citation needed]}

or

In Europe in the early 15th century the letters "P" and "M" were generally used.^[1] The symbols (P with stroke for piu, i.e., plus, and M with stroke 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.^[2] The + is a simplification of the Latin "et" (comparable to the ampersand &).^{[citation needed]} 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.^{[citation needed]} In his 1489 treatise Johannes Widmann referred to the symbols − and + as minus and mer (Modern German mehr; "more"): "das − ist, das ist minus, und das + ist das mer".^[3]

A book published by Henricus Grammateus in 1518 makes another early use of + and − for addition and subtraction.^[4]

Robert Recorde, the designer of the equals sign, introduced plus and minus to Britain in 1557 in The Whetstone of Witte:^{[citation needed]} "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 emphasise 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 equivalent to, addition. Moreover, the symbolism has been extended to very different operations. Plus can mean:

• exclusive or (usually written ⊕): 1 + 1 = 0, 1 + 0 = 1
• logical disjunction (usually written ∨): 1 + 1 = 1, 1 + 0 = 1

Minus sign

The minus sign has three main uses in mathematics:^[5]

1. The subtraction operator: A binary operator to indicate the operation of subtraction, as in 5 − 3 = 2. Subtraction is the inverse of addition.
2. Directly in front of a number and when it is not a subtraction operator it means a negative number. For instance −5 is minus 5.
3. A unary operator that acts as an instruction to replace the operand by its opposite. For example, if x is 3, then −x is −3, but if x is −3, then −x is 3. Similarly, −(−2) is equal to 2.

All three uses can be referred to as "minus" in everyday speech. In modern US usage, −5 (for example) is normally pronounced "negative five" rather than "minus five". "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.^[6] In most other parts of the English-speaking world, "minus five" is more common. Textbooks in America encourage −x to be read as "the opposite of x" or even "the additive inverse of x" to avoid giving the impression that −x is necessarily negative.^[7]

In some contexts, different glyphs are used for these meanings; for instance in the computer language APL a raised minus sign is used in negative numbers (as in 2 − 5 gives ⁻3), but such usage is rare.

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 multiplication or division which binds more strongly than addition or subtraction. While strictly speaking, the unary minus is not subtraction, it is given the same place as subtraction. However in some programming languages and Excel in particular, unary operators bind strongest, so in these −5^2 is 25 but 0−5^2 is −25.^[8]

Use in elementary education

Some elementary teachers use raised plus and minus signs before numbers to show they are positive or negative numbers.^[9] 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,

or even as

⁺3 − ⁻5 becomes ⁺3 + ⁺5 which is ⁺8

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.

Positive and negative are sometimes abbreviated as +ve and −ve.

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⁻.

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" = "ab", although this usage is questioned by some for violating commutativity, a property addition is expected to have.

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) and 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. For example, x++ means "increment the value of x by one" and x-- means "decrement the value of x by one". By extension, "++" is sometimes used in computing terminology to signify an improvement, as in the name of the language C++.

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, the minus sign (rather than an en dash) is used for a single covalent bond between two atoms, in 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.^[10] The Unicode character used for the tone letter (U+02D7) is different from the mathematical minus sign.

Character codes

Plus, minus, and hyphen-minus.

Read	Character	Unicode	ASCII	URL	HTML (others)
Plus	+	U+002B	+	%2B
Minus	−	U+2212		%E2%88%92	− or &x2212; or −
Hyphen-minus	-	U+002D	-	%2D

The Unicode minus sign is designed to be the same length and height as the plus and equals signs. In most fonts these are the same width as digits in order to facilitate the alignment of numbers in tables.

The hyphen-minus sign (-) is the ASCII 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.

There is a commercial minus sign (⁒), which looks somewhat like an obelus, at U+2052 (HTML &x2052;).

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. This practice was adopted into Israeli schools (this practice goes back to at least the 1940s)^[11] and is still commonplace today in elementary schools (including secular schools) but in fewer secondary schools.^[12] It is also used occasionally in books by religious authors, but most books for adults use the international symbol "+". The usual explanation for this practice is that it avoids the writing of a symbol "+" that looks like a Christian cross.^[12] Unicode has this symbol at position U+FB29 "Hebrew letter alternative plus sign" (﬩).^[13]

See also

• Graft-chimaera for the meaning of + in botanical names
• List of international call prefixes that + can represent the numbers required to dial out of a country as seen in a phone number
• Plus-minus sign
• Table of mathematical symbols
• En dash, a dash that looks similar to the subtraction symbol but is used for a different purpose

References

1. ^ Stallings, Lynn (May 2000). "A brief history of algebraic notation". School Science and Mathematics. http://findarticles.com/p/articles/mi_qa3667/is_200005/ai_n8885415/. Retrieved 13 April 2009. 
2. ^ Alan Sangster, Greg Stoner & Patricia McCarthy: "The market for Luca Pacioli’s Summa Arithmetica" (Accounting, Business & Financial History Conference, Cardiff, September 2007) p.5
3. ^ "plus". Oxford English Dictionary. Oxford University Press. 2nd ed. 1989.
4. ^ Earliest Uses of Various Mathematical Symbols
5. ^ Henri Picciotto. The Algebra Lab. Creative Publications. p. 9. ISBN 9780884889649. http://books.google.com/books?id=nzukMBV6ReoC&pg=PA9. 
6. ^ Schwartzman, Steven (1994). The words of mathematics. The Mathematical Association of America. p. 136. 
7. ^ Wheeler, Ruric E. (2001). Modern Mathematics (11 ed.). pp. 171. 
8. ^ "Microsoft Office Excel Calculation operators and precedence". http://office.microsoft.com/en-us/excel/HP100788861033.aspx. Retrieved 2009-07-29. 
9. ^ Grant P. Wiggins; Jay McTighe (2005). Understanding by design. ACSD Publications. p. 210. ISBN 1416600353. 
10. ^ Hartell, Rhonda L., ed. (1993), The Alphabets of Africa. Dakar: UNESCO and SIL.
11. ^ The Holocaust in Three Generations (Page 107)
12. ^ ^{a b} 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)
13. ^ Unicode U+FB29 reference page This form of the plus sign is also used on the control buttons at individual seats onboard the El Al Israel Airlines aircraft.

External links

• The History of Mathematical Symbols – Plus and Minus
• Earliest uses of various mathematical symbols