Where and When Did the Symbols "+" and "–" Originate?

https://blogs.stsci.edu/livio/2013/03/12/where-and-when-did-the-symbols-%E2%80%9C%E2%80%9D-and-%E2%80%9C%E2%80%93%E2%80%9D-originate/

by paulgerhardt • 13 years ago

207 88 13 years ago

blogs.stsci.edu

adamnemecek• 13 years ago

This always amazed me: "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]" [1]

[1] http://en.wikipedia.org/wiki/Plus_and_minus_signs#Alternativ...

ars• 13 years ago

And the symbol for the 4 cardinal directions (like on a map) gets written as: ⊕ (A circled plus.)

otibom• 13 years ago

That's XOR !

recuter2• 13 years ago

I have an uncomfortable memory of my 1st grade teacher showing this to us and me refusing to go along with what I insisted was an irrational superstition. I didn't understand the context or what her old age meant.

Similarly, I was shocked as a kid when I visited the US and saw apartment buildings skip the 13th floor numbering.

People are weird.

ghc• 13 years ago

Similarly, in Japan one pronunciation of 4 is also the word for death, so it's frequently omitted.

arketyp• 13 years ago

I always thought the plus sign had to do with two entities meeting or being placed on top each other, like a road crossing or a pair of sticks, and that the minus sign was just a spinn-off of that sign and also made sense in the same way that Descartes used "..." according to the article.

pavlov• 13 years ago

What fascinates me most about this story is how it illustrates what a young race we humans are intellectually.

If we ever want to be anything more than warring tribes, we must learn to accumulate knowledge together. Universal symbols for addition and substraction are the first small step on that very long road, yet it's only been a few hundred years since we took that step.

On the scale of the universe, we are like a baby who just minutes ago learned to grab onto objects and already is hell-bent on pouring a hot pot of tea on herself using this new capability.

znowi2• 13 years ago

On the scale of the universe we're virtually non-existent :)

xerula3• 13 years ago

On the temporal / spatial scale of the universe we are virtually non-existent. On a scale that ranks everything in the universe according to physical / informational complexity & organization, we appear so far to be quite significant.

The_Yayo• 13 years ago

yes, totally agree, from my basic knowledge of understanding.

rootbear• 13 years ago

And we're still using '*' for multiplication in our programming languages instead of '×' because using anything other than ASCII for programming is just too hard, or too weird, or whatever.

scott_s2• 13 years ago

To be fair, I prefer dots over crosses for multiplication of scalars in when I write it myself. I tend to assume a cross means cross product instead of scalar multiplication. (But, oddly, I do not assume that a dot means dot product.)

sp3322• 13 years ago

We already tried going full-APL but it only proved that it wasn't a good idea. I mean this is impressive https://www.youtube.com/watch?v=a9xAKttWgP4 but I still think the notation is too obscure.

rmccue3• 13 years ago

More likely, it's because we don't have them on our keyboards. Programming languages are usually about reducing friction for programmers, but using the proper multiplication/division symbols would increase that for most people.

meerita• 13 years ago

I remember to read the answer of this in an Isaac Asimov's book: Asimov On Numbers.

jnotarstefano• 13 years ago

Several earliest known uses of other mathematical notations can be found at: http://jeff560.tripod.com/mathsym.html

kijin• 13 years ago

> The asterisk was used by Johann Rahn (1622-1676) in 1659 in Teutsche Algebra

Interesting. I've always thought that the use of an asterisk to denote multiplication was some sort of compromise owing to the lack of a "×" key on a typical computer keyboard. But it seems that there are much older precedents for it.

Meanwhile, using a slash to denote division is explained in the Fractions page.

> The diagonal fraction bar (also called a solidus or virgule) was introduced because the horizontal fraction bar was difficult typographically, requiring three terraces of type. An early handwritten document with forward slashes in lieu of fraction bars is Thomas Twining's Ledger of 1718 ...

chinpokomon2• 13 years ago

Interesting. The close parentheses form of division from 1540 looks like the basis of the long division form of division, i.e. 2)6 with a bar or vinculum, extending from the top of the parentheses over the 6. The long division form finally takes its modern form in 1888. The transformation of that symbol seems somewhat obvious and natural. I wonder if we are likely to see further changes, as modern typesetting should resist the variations that handwriting would introduce. I believe any change in the future will have to demonstrate improved readability before it will be accepted.

gpvos3• 13 years ago

Also interesting: the form for long division that is used in the 19th century in the US: divisor ) dividend ( quotient, is very similar to the long division form that is still used in the Netherlands today: divisor / dividend \ quotient. Looking at Wikipedia, it seems all other countries have gone on to different notations, though.

lmm• 13 years ago

I always felt the origin of the equals sign was wonderful: "I will sette as I doe often in woorke use, a paire of parralles, or Gemowe lines of one lengthe, thus : ==, bicause noe 2, thynges, can be moare equalle."

panacea• 13 years ago

It's what permits those two lines to be essentially irreducibly perfect in describing "the same" that fascinates me. Any 'samer' and they would converge and simply be -.

The opposite of something. It's only when we can separate nothing, that we can begin to describe something.

(And is the universe the mathematical manifestation of everything that isn't nothing?) /stoner

Gormo• 13 years ago

> bicause noe 2, thynges, can be moare equalle

Which is not to say that a higher degree of equality can't be attained; it's just that then you'd have one thing instead of two.

abrahamsen• 13 years ago

Kind of weird that no consensus have emerged for multiplication (·, ×, *, or horizontal juxtaposition) or division (÷, /, or vertical juxtaposition with horizontal line between) yet.

jacquesm• 13 years ago

The / is actually the - from the other symbol and the .'s are placeholders for the numbers to the left and the right of the / so those are in fact pretty close.

mich412• 13 years ago

Everybody uses juxtaposition for multiplication (or dot when some separation is required) and horizontal line for division.

* and / are poor man's substitutes when all you have is ASCII.

× is used for vector products, Cartesian products and products of algebraic structures in general.

The only place I've ever seen × or ÷ being used in arithmetics was primary school. No idea why they use them.

gpvos3• 13 years ago

In primary school in the Netherlands (and maybe elsewhere?), : is used for division.

mseebach2• 13 years ago

> The only place I've ever seen × or ÷ being used in arithmetics was primary school. No idea why they use them.

Perhaps because the other prominent use is on the buttons of every single calculator in the world.

rmccue3• 13 years ago

÷ is the symbol for division since it symbolises a fraction, with the two dots taking the place of the numbers. I'd say this is why, as some people have noted, : can also be used for division in some contexts: it represents a ratio, which is again just a fraction.

Uncompetative3• 13 years ago

What then happens when you have to split a long line of math?

What then happens when you need to multiply 2 by 2?

The answer to the first question is that the following line is begun with a multiplication sign.

The answer to the second question is that bold roman type is used for vectors and often for other forms of tensor like a matrix to overload the semantics of dot and cross to not be scalar products were either of their arguments in italic.

gpvos3• 13 years ago

There's an interesting note near the bottom about the ÷ symbol. Even today in the Netherlands, a symbol like -/- (like a percent sign, but with minus signs on both sides of the slash) is used as a kind of emphasized minus sign, especially in money calculations in a vertical layout where otherwise addition would be assumed. I don't know if this is used elsewhere, but it seems probable to me that it derives from the ÷ symbol mentioned in the text.

contingencies4• 13 years ago

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).

That's interesting. The Silk Road (http://en.wikipedia.org/wiki/Silk_Road) was in full swing at that time. This is pure guesswork, but multiple classical Chinese characters now pronounced yu (予,与,餘,逾,與) meant something close to addition. I wonder if there's a link?

There were a great many languages in operation there, and the Chinese were primarily interested in trade. It wouldn't surprise me at all if there were a regional patois language of business utilizing Chinese idiom that the subcontinent and others were able to borrow from.

Likewise, the borrowing may have gone in the other direction.

Note that some of the above symbols are not so dissimilar from the + symbol, allowing some minor scope for simplification.

On further investigation, http://www-history.mcs.st-and.ac.uk/PrintHT/Bakhshali_manusc... claims the Bakshali manuscript yu was "for yuta" (apparently Sanskrit/Devanagari: युत). Apparently yu is a well recognized word root meaning "to mix" or "(equipped) with" within Sanskrit, yuta being some kind of derived or more expanded/standalone form. (Sources: http://www.hindunet.org/hindu_history/sarasvati/dictionary/9... http://vedabase.net/y/yuta http://dictionary.tamilcube.com/sanskrit-dictionary.aspx?ter... http://www.hinduwebsite.com/sacredscripts/sanskrit_dictionar...)

That yu is apparently the same Sanskrit / proto Indo-European root as yoga, or even the far-western-Barbarian modern derivation in English, yoke. http://dictionary.reference.com/browse/yoke https://en.wiktionary.org/wiki/Appendix:List_of_Proto-Indo-E...

Hrrm... there must be something published on this already. Anyone with academic library access? :) I'm up in the Himalayan foothills, hehehe.

contingencies4• 13 years ago

Had a quick check for Basque/Estonian/Finnish online dictionaries as the major outlying European languages out of further interest. Basque alone seems to use the recognizable root ek... (one would suppose from common root with the Sanskrit for one: eka). Estonian and Finnish seem way closer to Latin languages in most cases .. I checked words like and, plus, sum, total. Lots of readily distinguishable slight variations, but no yu to be found. What about English union, unit and unity? Same root I suppose. Dictionary.com claims union is from Latin ūn(us) (one) via middle French but that unit is only attested to 1642. It seems the Romans had some of that yu going, too.

I guess that's as far as exploration will go with western sources: apparently the Romans weren't big on composing etymologies. Perhaps Indian sources are useful though: Nirukta (Sanskrit: निरुक्त, IPA: [n̪irukt̪ə], explanation, etymological interpretation) is supposed to be one of the six Vedānga disciplines of Hinduism. One of the primary texts there is available at the Internet Archive: http://archive.org/details/nighantuniruktao00yaskuoft

Potentially related tidbits I found there: "Yosa (a woman) is derived from (the root) yu (to join).", "Yutham (herd is derived from (the root) yu (to connect).", "Dasyu (demon) is derived from (the root) das, meaning to lay waste: in him the juices are wasted, or he causes works to be laid waste." Also had some good discussions of soma (a psychadelic draft).

OK, curiosity sated. Obviously there was some yu going on circa Central Asia at the dawn of history. Anything more specific seems like guesswork only. My take is it was to soma takers what PLUR is to early ravers. Party on man!

sesqu2• 13 years ago

For Finnish, you may have missed the now deprecated word for summation, "ynnä", meaning "as one". It's not from Latin or Chinese, though - at least not recognizably so.

yread• 13 years ago

We discussed this article in the office and among other things we found that even today the notation for division (when calculating it by hand) varies wildly:

In Czech Republic we would write

453: 12 = 37.75

In Netherlands:

12/453\37.75

In Russia:

       453 | 12
           | 37.75

all of them are listed on Wikipedia

http://en.wikipedia.org/wiki/Long_division

zapdrive• 13 years ago

In India its usually written: 500 ÷ 4 = 125. Its not a + sign, it a bar stuck between a :

rmccue3• 13 years ago

That's normal division, which is a notation. The GP is talking about long division, which is a technique used to divide large numbers.

The symbol in Australia (and I'd say most of the English speaking world) looks like a long square root sign with a curved vertical part:

  _______

12)12345

Uncompetative3• 13 years ago

      123 ÷ 4/5 + 6 | 7 x 8

    = 123 ÷ 0.8 + 6 | 56

    =    153.75 + 6 | 56

    =        159.75 | 56

    = 2.85267 857142 857142 857142 857142 857142 857142 857142 857142 857142...

i.e. here | acts as a low precedence division operator, ÷ as an ordinary precedence division operator like infix / and the 'closefix' rational notation 3/4 has the highest precedence in my programming language

n.b. in case you were wondering P v Q is used for Boolean Or.

yaddayadda• 13 years ago

The next logical question is, "Where did × and ÷ originate?" From there, I've always wondered how multiplication migrated from "×" to "∗"? And recently, I've wondered if we will migrate "-" to something else? (This last derives from having to annotate a range of numbers including positive and negative. When "-" is used for both range and negative, it causes confusion.)

p.s. I tried to post this to the original poster's blog, but was told "...your comment seems a bit spammy." Really? I'd say their blogger spam settings a more than a bit off.

manish_gill• 13 years ago

I always thought the * was used for multiplication to avoid confusing × with the English alphabet X. We can usually distinguish them while writing equations on paper(I write x in a cursive, for example).

kps3• 13 years ago

* was used for multiplication because there weren't enough bits to fit every symbol everyone would have liked into ASCII, so compromises were made, including preference for symbols that could be 'overloaded' with multiple meanings in different contexts.

qompiler• 13 years ago

Chinese write it as 加, if you start making the strokes you get a + in the upper left. Consider this is over 6000 years old, probably the origin. My theory.

Edit (more evidence): "The plus symbol as an abbreviation for the Latin et, though appearing with the downward stroke not quite vertical, was found in a manuscript dated 1417 (Cajori)."

There is an assumption here that it's an abbreviation for the Latin et. But the downward stroke can be seen in the Chinese word.

aneth43• 13 years ago

And 卖买 are sell and buy respectively.

yaix• 13 years ago

> 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.”

They appeared just at the right time to be carried around the world by conquista and colonialism. Same goes for time notation, etc.

kps3• 13 years ago

They appeared just at the right time to be carried around the world by movable type. Same goes for non-mathematical punctuation.

To put it another way, mass production of text froze the symbols that happened to be in use at the time.

chiph• 13 years ago

One of the interesting things is why are there 24 hours in a day, 60 minutes per hour, etc?

Mathematically, those values are useful because they can be factored in many ways. But who picked them?

I was at the British Museum one time, and they had an horological exhibit (clocks throughout the ages), so I asked one of the curators. He said it was the Babylonians that chose those values.

But since this thread is about notation, I should be asking who picked the colon (:) as the time separator that is generally used. And are there other conventions for this?

aneth43• 13 years ago

Interesting that there is no mention of ancient China, where money and accounting were invented.

The Chinese for buy and sell are 买 and 卖 respectively, the first indicating spending money and the second indicating receiving money.

Granted I don't know what I'm talking about, however I do know Chinese, and I believe these characters far predate the 15th century as do most Chinese characters.

Perhaps someone can elaborate on whether this is a sound theory.

potatolicious• 13 years ago

Both those characters are simplified, which is to say largely different from how they would have appeared throughout Chinese history.

Buy in traditional is: 買 Sell is: 賣

Both of these would be closer to the ancient form of expressing these characters. I don't see "+" nor "-" in either of these characters.

contingencies4• 13 years ago

Fun fact: the one on the left, everything but the top part is a pictograph of a shell, which was used for a kind of status symbol or possibly something roughly equating our modern money in many early societies. When Marco Polo arrived here in southwest China in the Yuan Dynasty (ie. just post Mongol-invasion), shells were still in widespread use. In fact, Chinese literary records document special requests to the emperor to except the region from the national law requiring use of imperial money by the provincial governor they placed in the region, who was an Uzbek and son of the then-king of Bokhara who had ridden with the Mongol horde to take China. One of his father's later successors was one of the Prokudin-Gorskii images (first colour images in the region), over here: http://pratyeka.org/prokudin-gorskii/the-emir-of-bukhara-191...

aneth43• 13 years ago

I had not remembered the traditional version though, so thanks for pointing that out.

Sell is 買 with a 士 radical on top. The simplified version standardized the 買 portion, but it doesn't affect the observation.

pjungwir• 13 years ago

Medieval manuscripts are full of strange squiggles used to abbreviate words or word endings. It's almost like reading shorthand. Here is a short list for Middle English [1] but for Latin there were dozens if not hundreds. So I'm not surprised that + and - originate from that time.

[1] http://isites.harvard.edu/fs/docs/icb.topic453618.files/Cent...

EDIT: In fact, the second symbol on that list, the abbreviation for "and," seems half-way to becoming a +.

Where and When Did the Symbols "+" and "–" Originate?

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