Mar 122013

The symbols for the arithmetic operations of addition (plus; “+”) and subtraction (minus; “–”) are so common today we hardly ever think about the fact that they didn’t always exist.  In fact, someone first had to invent these symbols (or at least other ones that later evolved into the current form), and some time surely had to pass before the symbols were universally adopted.  When I started looking into the history of these signs, I discovered to my surprise that they did not have their origin in antiquity.  Much of what we know is based on an impressively comprehensive and still unsurpassed research (in 1928–1929) entitled History of Mathematical Notations by the Swiss-American historian of mathematics, Florian Cajori (1859–1930).

The ancient Greeks expressed addition mostly by juxtaposition, but sporadically used the slash symbol “/” for addition and a semi-elliptical curve for subtraction.  In the famous Egyptian Ahmes papyrus, a pair of legs walking forward marked addition, and walking away subtraction.  The Hindus, like the Greeks, usually had no mark for addition, except that “yu” was used in the Bakhshali manuscript Arithmetic (which probably dates to the third or fourth century).  Towards the end of the fifteenth century, the French mathematician Chuquet (in 1484) and the Italian Pacioli (in 1494) used “\boldmath{\bar{\bf p}}” or “p” (indicating plus) for addition and “\boldmath{\widetilde{\bf m}}” or “m” (indicating minus) for subtraction.

There is little doubt that our + sign has its roots in one of the forms of the word “et,” meaning “and” in Latin.  The first person who may have used the + sign as an abbreviation for et was the astronomer Nicole d’Oresme (author of The Book of the Sky and the World) at the middle of the fourteenth century.  A manuscript from 1417 also has the + symbol (although the downward stroke is not quite vertical) as a descendent of one of the forms of et.

The origin of the – sign is much less clear, and speculations range all the way from hieroglyphic or Alexandrian grammar ancestry, to a bar symbol used by merchants to separate the tare from the total weight of goods.

Figure 1. The first use of the + and – symbols in print in Johannes Widman’s Behëde und Lubsche Rechenung auff allen Kauffmanschafft, Augsburg edition of 1526.

The first use of the modern algebraic sign – appears in a German algebra manuscript from 1481 that was found in the Dresden Library.  In a Latin manuscript from the same period (also in the Dresden Library), both symbols + and – appear.  Johannes Widman is known to have examined and annotated both of those manuscripts.  In 1489, in Leipzig, he published the first printed book (Mercantile Arithmetic) in which the two signs + and – occurred (Figure 1).  The fact that Widman used the symbols as if they were generally known points to the possibility that they were derived from merchants’ practices.  An anonymous manuscript—probably written around the same time—also used the same symbols, and it provided input for two additional books published in 1518 and 1525.

In Italy, the symbols + and – were adopted by the astronomer Christopher Clavius (a German who lived in Rome), the mathematicians Gloriosi, and Cavalieri at the beginning of the seventeenth century.

The first appearance of + and – in English was in the 1551 book on algebra The Whetstone of Witte by the Oxford mathematician Robert Recorde, who also introduced the equal sign as the rather longer than today’s symbol “═.”  In describing the plus and minus signs Recorde wrote: “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.”

As a historical curiosity, I should note that even once adopted, not everybody used precisely the same symbol for +.  Widman himself introduced it as a Greek cross + (the sign we use today), with the horizontal stroke sometimes a bit longer than the vertical one.  Mathematicians such as Recorde, Harriot and Descartes used this form.  Others (e.g., Hume, Huygens, and Fermat) used the Latin cross “†,” sometimes placed horizontally, with the crossbar at one end or the other.  Finally, a few (e.g., De Hortega, Halley) used the more ornamental form “\maltese.”

The practices of denoting subtraction were somewhat less fanciful, but perhaps more confusing (to us at least), since instead of the simple –, German, Swiss, and Dutch books sometimes used the symbol “÷,” which we now use for division.  A few seventeenth century books (e.g., by Descartes and by Mersenne) used two dots “∙∙” or three dots “∙∙∙” for subtraction.

Overall, what is perhaps most impressive in this story is the fact that symbols which first appeared in print only about five hundred years ago have become part of what is perhaps the most universal “language.”  Whether you do science or finances, in Kentucky or in Siberia, you know precisely what these symbols signify.

  16 Responses to “Where and When Did the Symbols “+” and “–” Originate?”

  1. Very interesting. Thanks for the book reference and the essay. Maybe I should refer people to books, too, instead of just reading them all myself. o_O

  2. Hmm… Somehow, the Sumerians had the signs “plus” and “minus”… They had two mathematical systems, by the 10′s and by the 60′s 60 was called “sar”). They knew that a circle had 360 degrees, that an hour had 60 minutes and even that a minute had 60 seconds. They used to reckon various things in maths, in geometry and in would use them in architecture…

  3. The use of ./. for minus in Germany seems to be falling slowly into disuse. It was much more common 30 years ago, particularly when handwritten. Nowadays I find that many young Germans do not even know it.

  4. Thanks! I’m an Australian living in Norway and always wondered why they were using a divide sign went they meant minus (e.g. during sales it is common to see ÷20%). Now I know! They use “:” for divide, so there is no confusion (unless you come from outside Norway!).

  5. Interestingly your article doesn’t actually use “minus” (“−”), it’s using the en dash (“–”), and this is immediately obvious when looking at the text due to the fact that the en dash is on a different height, and has a different length, than the plus sign: the modern day minus sign is (in proper fonts) designed to exactly match the plus sign’s dimensions (“−” vs “+”) so that they are visually aligned, both vertically and horizontally.

  6. @Strumpkin The ‘therefore’ symbol is still used occasionally in math, mostly in the formal proofs you mention. It’s present in Unicode at code point U+2234, so it’s available in any encoding that has access to those code points (such as UTF-8), and you can produce it in HTML as &x2234;. On a Macintosh, you can find it inside the Character Viewer panel in the Mathematical Operators section, but I’m not sure where to locate it on Windows. Also, if you’re writing in LaTeX, the \therefore command produces it in math mode. (Flipping it upside-down gives you the ‘because’ symbol, U+2235 in Unicode or \because in LaTeX.)

  7. @speaks2 Do you have a source for that. According to wikipedia notation was not yet invented and the equations were all written out in words.

  8. What has happened to the three dots that represented “therefore”? They were on the corners of an invisible equilateral triangle with a horzontal base. The “therefore” sign was much used in solving equations on a one step at a time basis (Required to show you understood the method). I do not seem able to find a “therefore” sign on my computer or in any system.

  9. Hi, I’m from Vietnam, I’ve been always fascinated about the links between math, physics, chemistry.I’m only 15 , but I just feel like I’ve caught some sense of the impermanance of arts, maths…I don’t know I just find it’s pretty amazing to bare in our mind those blur but vivid connection everytime I solve a problem.I just want it so bad to know more!!
    PS: Sorry for my bad English, I’ve tried my best

  10. You forget Al Khawarimi who invented the rules of an equation “Al Jabr and al mukabala” which is based on + and – and equality symbol “=” >

    ax + b = c one of the equations examples .

    Thanks for the effort :)

  11. @Ivan Savov I love coming across words like “betoken” in old English texts because it shows how similar the language once was to its cousins Dutch (“betekenen”) or German (“bezeichnen”)

  12. Thank you for the excellent article.

    I find it the following phrase fascinating:
    “There be other 2 signes in often use: [...] + betokeneth more [..and...] – betokeneth lesse.”

    The first cool thing is the verb “betoken” (to “denote”).

    The second cool thing is that the author (Robert Recorde) seems to think of + and – as verbs, which makes a lot of sense.

    How did you get white-on-black latex work? Very nice.

  13. “÷” is still used for subtraction in some areas. I’ve seen taught in Scandinavian elementary schools, though most people do use the more globally standardized “-”.

  14. The article skip the Arabic period. I read they were famous for Algebra.

  15. Very well written. I appreciate it when people have concern for things as trivial as the origin of symbols, especially when they can find things that are bigger than that in the process.

    Very good read, thank you.

  16. I learned recently that accountants (at least in Germany) use ./. for minus, like this:

    10 ./. 4 = 6

    I was stunned when a banker calculated something for my mortgage using that symbol. He in turn was surprised that I didn’t know it.

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