Plus and minus signs


+  −
Plus and minus signs
In UnicodeU+002B + PLUS SIGN (HTML &#43; · &plus;)
See alsoU+00B1 ± PLUS-MINUS SIGN
Different from
Different fromU+002D - HYPHEN-MINUS
(many)  - Dash


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]


Nicole Oresme's manuscripts from the 14th century show what may be one of the earliest uses of + as a sign for plus.[2]

In Europe in the early 15th century the letters "P" and "M" were generally used.[3][4] The symbols (P with overline, , for più (more), i.e., plus, and M with overline, , 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.[5] The + is a simplification of the Latin et (comparable to the evolution of 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 signEdit



The symbol is also used in chemistry and physics, see below.

Minus signEdit

The minus sign, , has three main uses in mathematics:[13]

  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. 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 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 just as 'semicircle' names a geometric figure, the difference being 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 either way. Sometimes, it does make a difference꞉ the programming language APL uses a raised minus sign ¯, (Unicode U+00AF) as a prefix rather than a function so that the interpreter of APL has less work when taking ¯5 as the number −5 rather than inverting the constant 5 by means of the minus sign considered as denoting a function (item 2 above). As described in the next section, some educators consider it important that elementary students realize that negative numbers are genuine entities that can be given names and so use a raised minus in the name of a negative number. Similarly, in 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).

All three uses can be referred to as "minus" in everyday speech, though the binary operator is sometimes read as "take away". In most English-speaking countries, −5 (for example) is normally referred to as "minus five", but in modern US usage it is instead usually called "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 (since x itself may already be negative)[15]

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 and in Microsoft Excel in particular, unary operators bind strongest, so in those cases −5^2 is 25 but 0−5^2 is −25.[16]

Again, the symbol is also used in chemistry and physics; see below.

Use in elementary educationEdit

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

3 − 5 becomes 3 + 5 = 8,

or even as

+3 − 5 becomes +3 + +5 = +8.

Use as a qualifierEdit

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.[18]


In mathematics the one-sided limit xa+ means x approaches a from the right, and xa means x approaches a from the left. For example, 1/x→+  as x→0+ but 1/x→−  as 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 computingEdit

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 (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 usesEdit

In chemistry, superscripted plus and minus signs are used to indicate an ion with a positive or negative charge of 1 (for example, NH+
). If the charge is greater than 1, a number indicating the charge is written before the sign (SO2−
). The minus sign is also used (rather than an en dash) for a single covalent bond between two atoms, as in the skeletal formula. For example, Sulfuric Acid: H

In physics, the use of plus and minus signs for different electrical charges was introduced by Georg Christoph Lichtenberg.

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, , 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 codesEdit

Plus, minus, and hyphen-minus.
Read Character Unicode ASCII in URL HTML notations
Plus + U+002B &#43; %2B &plus;
Minus U+2212 %E2%88%92 &minus; &#x2212; &#8722;
Hyphen-minus - U+002D &#45; %2D
Small Hyphen-minus U+FE63 %EF%B9%A3 &#xfe63; &#65123;
Full-width Plus U+FF0B %EF%BC%8B &#xff0b; &#65291;
Full-width Hyphen-minus U+FF0D %EF%BC%8D &#xff0d; &#65293;

The hyphen-minus sign, -, is the original ASCII version of the minus sign, which doubles as a hyphen. It is usually shorter in length than the plus sign and often at a different height to the plus-sign's cross bar. 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. Also several other software programs usable for calculations don't accept the U+2212 minus. For example pasting =3−2 into Excel or 3−2= into the Windows calculator won't work. The true minus is not available on most keyboard layouts.

There is a commercial minus sign, , which is used in Germany and Poland. The symbol ÷ is used to denote subtraction in Norway.

Alternative plus signEdit

A Jewish tradition that dates from at least the 19th century is to write plus using the symbol .[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 U+FB29 HEBREW LETTER ALTERNATIVE PLUS SIGN.[22]

See alsoEdit

References and footnotesEdit

  1. ^ 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.
  2. ^ The birth of symbols – Zdena Lustigova, Faculty of Mathematics and Physics Charles University, Prague Archived 2013-07-08 at
  3. ^ Ley, Willy (April 1965). "Symbolically Speaking". For Your Information. Galaxy Science Fiction. pp. 57–67.
  4. ^ Stallings, Lynn (May 2000). "A brief history of algebraic notation". School Science and Mathematics. Retrieved 13 April 2009.
  5. ^ 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.
  7. ^ 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. ^ Smith, D.E. (1951). History of Mathematics. 1. Courier Dover Publications. pp. 258, 330. ISBN 0486204308.
  10. ^ Earliest Uses of Various Mathematical Symbols
  11. ^ Cajori, Florian (2007), A History of Mathematical Notations, Cosimo, p. 164, ISBN 9781602066847.
  12. ^ Fraleigh, John B. (1989). A First Course in Abstract Algebra (4 ed.). United States: Addison-Wesley. p. 52. ISBN 0-201-52821-5.
  13. ^ Henri Picciotto (1990). The Algebra Lab. Creative Publications. p. 9. ISBN 978-0-88488-964-9.
  14. ^ Schwartzman, Steven (1994). The words of mathematics. The Mathematical Association of America. p. 136.
  15. ^ Wheeler, Ruric E. (2001). Modern Mathematics (11 ed.). p. 171.
  16. ^ "Microsoft Office Excel Calculation operators and precedence". Archived from the original on 2009-08-11. Retrieved 2009-07-29.
  18. ^ Castledine, George; Close, Ann (2009). Oxford Handbook of Adult Nursing. Oxford University Press. p. xvii. ISBN 9780191039676..
  20. ^ a b Kaufmann Kohler (1901–1906). "Cross". In Cyrus Adler; et al. (eds.). Jewish Encyclopedia.
  21. ^ 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)
  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.

External linksEdit