Talk:Plus and minus signs

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Concatenation[edit]

According to the article, "[Concatenation] is more commonly written as "a"&"b"="ab", using the ampersand." Yet Concatenation and Comparison of programming languages (strings) both say + is more common. I'm removing the sentence. --192.235.8.2 20:52, 31 January 2007 (UTC)

Does Unicode have "superscript plus" and "superscript minus" like it has "superscript 1" and "superscript 2"?[edit]

  • To join the fun with ¹ ² ³ ⁿ / ⅛ ¼ ⅜ ½ ⅝ ¾ ⅞ / ⅓ ⅔ ?
  • Y/N?
  • If yes, what are the code points?
  • If no, someone should create them.
  • Thanks, — M20-2.5 19:04, 17 March 2008 (UTC)
Yes x207a⁺ x207b⁻ x208a₊ x208b₋ See Unicode subscripts and superscripts. Why anyone wants them I don't know when one can superscript or subscript anyway. Dmcq (talk) 12:16, 22 January 2009 (UTC)

Sources[edit]

A personal tripod page is not a very good source. Maybe this book is better, even though it looks like it may have gotten the info from that same page. What about other sources? Anybody have? Dicklyon (talk) 17:31, 7 December 2008 (UTC)

Three uses for minus + computerese[edit]

I think there are three use for minus rather than 2.

xy meaning the result of y subtracted from xx meaning the result of negating the value of x, i.e. x subtracted from 0. -123 the negative number 123. It has the same value as the negation operator applied to 123 but is a different concept. I'll stick this in if nobody has a better idea.

Also a computery side might be an idea too. APL has negative numbers e.g. 123 separate from negated integers and IEEE has -0. Dmcq (talk) 18:43, 21 January 2009 (UTC)

If you have a source that discusses these distinctions, then definitely add such. If not, don't. Dicklyon (talk) 18:53, 21 January 2009 (UTC)
Well the definitions in wikt:minus sign and wikt:plus sign clearly differentiate between the operation and as used as part of a number, also if you look at the article itself immediately after it says there are two uses it gives three different names for the three different uses I gave above though there is a citation mark on that sentence. Dmcq (talk) 22:42, 21 January 2009 (UTC)
  • The difference between the second and third meanings is not at all obvious. As a unary operator, -5 is the negative (or "opposite") of 5, and, in exactly the same way, -x is the negative (or "opposite") of x. If there really is a "technical difference" as the article claims then I think the article should explain what that is. 86.134.90.10 (talk) 02:23, 18 March 2009 (UTC).
There is a difference, it would be hard for APL to distinguish between them for instance if there wasn't one. The citation referenced on the first line of the section might explain it better for you. Sorry I thought I had explained it properly but obviously there's some problem. As it say in the second type it is part of the number, no operator is involved. It has the same sort of status as a positive number. The third type is an operator, if you write it as an function -(5) it might be more obvious. Dmcq (talk) 10:50, 18 March 2009 (UTC)
Yep, I see the citation (incidentally, I just changed the link to point straight to page 9), but I'm unconvinced I'm afraid. To pick up on your example, there is no circumstance in which -(5) does not mean exactly the same thing as -5, and for exactly the same reason. "-5" means "minus five" because the minus sign means "take the negative of". Undoubtedly we should give both -5 (say) and -x (say) as examples, but to claim that the minus sign has two different meanings in these two cases is, in my view, more confusing than helpful. 81.129.130.32 (talk) 13:54, 18 March 2009 (UTC).
It may be more of a technicality in everyday life but mathematically they are quite different, one is the sign part of a number and the other is an arithmetic operator. In computing it does matter because for instance for integers normally there is no positive value corresponding to the maximum negative integer but a person may still want to write the maximum negative integer. Dmcq (talk) 15:28, 18 March 2009 (UTC)
I understand the issue of a computer being able to hold, say, -32768 as an integer but not +32768, but I don't see how this supports the article's claim that the two uses of the minus sign (in e.g. -5 and -x) are "in mathematics... technically different". It's true that the minus sign in -5 is "the sign part of the number", but the purpose and meaning of that minus sign is mathematically indistinguishable from the purpose and meaning of the minus sign in, say, -x. It's reasonable that these are two different uses I suppose, but far from being "technically different", the two uses are, in my view, actually "technically identical" (in mathematics). Computer implementation issues can go in the "Uses in computing" section. 86.134.30.252 (talk) 21:46, 18 March 2009 (UTC)
In mathematics -5 is a mathematical object whereas -x is an expression (mathematics). When x is 5 the expression can be evaluated and the value is -5. Expressions are not normally objects except as objects of study in metamathematics. Dmcq (talk) 22:54, 18 March 2009 (UTC)
I agree that -5 is a number that can be considered as an "entity", but I don't agree that this means the "-" symbol itself has a different meaning compared to, say, -x. To me it's exactly the same meaning, and that's exactly the reason why -5 is written the way it is. -5 means "additive inverse of 5", or "negative of 5", or however you want to put it, and it so happens that the result is another number, a negative one. I think we're just looking at the same thing in two slightly different ways. 86.134.30.252 (talk) 00:06, 19 March 2009 (UTC)
Try thinking how you would explain it to a computer. -5 would produce an entity as you put it whereas --5 would not, so there is some difference between the two minus signs. What's the difference? Dmcq (talk) 11:04, 19 March 2009 (UTC)
There is no difference in the meaning of the three minus signs in your examples. I'm guessing that you may have in mind the difference between "compile-time" and "run-time" parsing of the minus sign. This is an implementation issue that is not relevant to the actual meaning of the sign. (If this isn't what you're getting at then I'm afraid I don't understand your point.) 81.151.230.218 (talk) 11:12, 7 April 2009 (UTC).
If −5 is a single object, why do we evaluate −5^2 as −25, instead of +25? −5 does *denote* a single number, but that doesn't mean the sign should also be part of the numeral; when we write out the expression −5, we're getting that single number by taking the negative of 5, just as by taking the negative of x. Otherwise, we would be evaluating −5^2 as (−5)^2, or +25. Musiphil (talk) 23:07, 10 August 2012 (UTC)

In -5, the minus sign is part of a numeral.

In -x, the minus sign is a unary operator.

In 7 - 5, the minus sign is a binary operator.

They are different. Rick Norwood (talk) 14:46, 19 March 2009 (UTC)

I agree that the minus sign has the above three uses. Why don't we just agree that it (the minus sign) is a heteronym? I think that in the above list, you missed the fact that -5 is not only a number, but could also be an expression. In that case you have (the weird looking) -5=-5, which means in English: The unary operator acting on positive 5 results in the number negative 5. To add to your argument, the fact that there are three different meanings to the minus sign makes the statement -5^2=-25 less confusing to new math students. Gene KleinMortgagemeister (talk) 18:59, 25 July 2009 (UTC)

After more edits than I can feel proud of I've stuck in a bit of text with this example. Dmcq (talk) 06:02, 26 July 2009 (UTC)
I've removed the "technically" bit that the IP editor seems to object to, and I've replaced it with more details on speech. Ironically this gets the spirit of the section closer to the way it was when Rick introduced the word in 2007. That material was challenged here and largely removed here. Melchoir (talk) 09:26, 26 July 2009 (UTC)
  • I still do not agree that there are three fundamentally different meanings of the minus sign. To me, the second and third are identical in meaning. "−5" means the negation/opposite of 5, and "−x" means the negation/opposite of x. None of the above has changed my opinion on this. However, I guess calling them different uses may be acceptable. 86.152.242.40 (talk) 14:38, 14 November 2009 (UTC).
The difference is that for instance −5 inches simply is minus five inches or negative five inches as some teachers insist on calling it. Whereas −x inches means get x and then find its opposite and that gives the number of inches. −5 is a number, not an operation applied to a number. To give the equivalence one would have to write something like −(5)=−5 which in teacherspeak says that the opposite of the number 5 is the number negative 5. Dmcq (talk) 15:25, 14 November 2009 (UTC)
Thanks Dmcq. I'm aware of that argument, but it doesn't work for me. To me, −5 is a number which is arrived at by the operation of taking the negative of 5. The "−" element of the notation indicates the taking of the negative, just as it does in −x. To me, there is no fundamental difference in the two uses. 86.136.194.122 (talk) 20:17, 14 November 2009 (UTC).

In my opinion, the minus sign has two meanings, not three. Although there is a mathematical entity with the value of negative five, our number system happens to denote it using the expression (unary-minus) (numeral-five), because it doesn't have any more convenient way to denote it! This is analogous to how the value satisfying 3x-1=0 is denoted by the expression (numeral-one) (binary-divide) (numeral-three) [usually in vertical form with the divide represented by a horizontal line], and the positive value satisfying x^2=2 is denoted by the expression (square-root) (numeral-two). Mathematically, each of these exists as an atomic concept; notationally, each is represented as an expression. Joule36e5 (talk) 02:56, 24 September 2012 (UTC)

Precedence[edit]

I have made an edit regarding the unary minus and PEMDAS. The unary minus is not part of PEDMDAS, however everyone seems to just go with the flow her and give it the same place as the subtraction operation. Some said: "However in some programming languages and Excel in particular, unary operators bind strongest" Can this be proven? Are there specs for this? I don't work for Excel and there is a good reason for that, but it would be nice to back this up.~~ —Preceding unsigned comment added by 69.65.71.211 (talk) 19:14, 29 July 2009 (UTC)

Reference for excel precedence added. Dmcq (talk) 23:12, 29 July 2009 (UTC)

Teaching[edit]

There seems to be some idea that it is good pedagogy to say subtract, negative or opposite of for the different versions of minus. Also that they should put in raised minus or plus signs before the numbers to show they are negative or positive. I think this is just being stupid and them wanting to cause trouble again and I can't see it helping children but I guess it should be in the article. I just feel so disgusted by it that I can't actually bear to write anything in the actual article itself about it. Flaming teachers can't count but want to stick in more stupidities to cover their inadequacies by making the children unable to do anything is my thoughts on it.

Also perhaps a note about typography might be reasonable saying the unary form has no space but when spacing expressions the binary + and - operations have spaces put round them. Dmcq (talk) 18:10, 20 August 2009 (UTC)

The "minus"/"negative"/"opposite" trend post-1950 is already mentioned in the article. The raised minus sign isn't mentioned in the context of education, but a (critical) reference may be found here: [1]. "We wish this were a joke, but, alas, it is not." Melchoir (talk) 19:14, 20 August 2009 (UTC)

I'm not sure what the objection is here. Right now the minus sign is a "three-valued" heteronym. Are you under the impression that this is NOT confusing? One symbol meaning three different things? Moreover, the minus sign as a negative and the minus sign as the unary minus gets the same answer if arithmetic is limited to multiplication, division, addition and subtraction. However, include exponents and then you do not get the same answer. Tell me that isn't confusing! Mortgagemeister (talk) 22:19, 23 August 2009 (UTC)


An old question in education: is it better to teach students the way things are done in the real world, or is it better to give them training wheels before they have to deal with real world ambiguities. My own experience is that they learn best when you teach them the way things are done in real life, instead of teaching them something (like the little front superscript dash on a negative number) that they then have to unlearn. I tell my students about the three meanings of "minus", but then I explain that because minus has two sylables and the more correct words have more, "minus" has won the day. Get used to it.
There is a similar problem with x-coordinate, more properly called the abscissa. When the "x-coordinate" is really a t-coordinate, why do we still call it the x-coordinate? Because we do. Rick Norwood (talk) 22:30, 23 August 2009 (UTC)
IANATeacher, but I trust that children arrive in school thoroughly pre-optimized for natural language processing. They don't need to be shielded from heteronyms as if they were the plague. Melchoir (talk) 22:38, 23 August 2009 (UTC)

IAATeacher, and I don't want to shield children (or adult students) from heteronyms either. I just want to teach them that the minus sign IS a heteronym. This way when they are shown -5^2, they will not say (or think to themselves) "negative five squared" and be confused when the answer is -25. Mortgagemeister (talk) 02:08, 24 August 2009 (UTC)

  1. So write it as −52. Mathematical typography is more intuitive than people give it credit for!
  2. Confusion isn't all bad, assuming your administrators grant you enough slack in the curriculum to explore it.
  3. To stay on topic, as long as you're not serious about introducing "three-valued heteronym" into this Wikipedia article, we don't really have a problem. Science is filled with both mild and egregious abuses of notation. It's interesting to notice that unary minus is generalized from constants to functions. But binary minus is also generalized from natural numbers to integers and then to rational numbers (and beyond). These generalizations are more conceptually rich and educationally perilous, but I don't see anyone here tracing that back to the notation. Melchoir (talk) 04:52, 24 August 2009 (UTC)


Well - if one writes it as −52 , tell me what you think the result should be? wrt to my "three-valued heteronym", The article already notes the three uses of the minus sign. If it doesn't call it a heteronym, well a rose by any other name and all that. Gotta tell you though, I find the binary minus "generalization" not so conceptually rich and hardly "perilous". I mean if 5-3=2 is it really so deep that 5.2-3.2=2.0? Mortgagemeister (talk) 10:43, 24 August 2009 (UTC)

Obviously the 5 and the 2 are the symbols in the closest proximity, so they are combined first, leading to −25, which cannot be simplified further, so that's our answer.
The article uses the word "three" because given any silly idea expressed by a sufficiently simple English phrase, there exists at least one textbook on Google Books that contains it. I'm not surprised that "minus sign has three" gets 4 hits, nor should you be surprised that "minus sign has two" gets 8 hits. Unfortunately there is no right answer, since the very concept of counting meanings is ill-defined, so we may as well just pick one. But let's not try to run too far with it.
Yes, the fact that 5/7 − 1/2 = 3/14 (and not 4/5) is deeper and less obvious than the fact that if x = −6 then −x = 6 (and not −6). (This is not to say that the latter is shallow or obvious.) Melchoir (talk) 17:59, 24 August 2009 (UTC)

Actually, though Melchoir gets the right answer, the proximity of the 5 and the 2 has nothing to do with it. It's a convention. It could have gone either way. But by now, the matter is settled (except on a few calculators and in a few computer languages, exceptions which the article notes) so that today all mathematicians agree, or do so at least tp the extent that you can get a bunch of mathematicians to agree about anything. Exponentiation, however denoted, is done before either unary or binary minus. Thus - 3^2 = -9, and 10 - 3^2 = 1. The "third" meaning of - , opposite, also obeys the same rule - x^2 is -9 if x = 3 and is still -9 if x = -3. It means the opposite of the square of x. But this is purely arbitrary, a product of the history of mathematical notation. It could have gone the other way.

As for whether - has two meanings or three, those who say two are looking at one unary operation and one binary operation. In short, they see -3 as meaning "the opposite of 3". Those who say that - has three meanings see the unary operation as having two fundamentally different meanings, depending on whether it is part of the name of a negative number or part of an expression. It is best to teach students that there are three meanings of -, else they are confused by the fact that -x can be positive. Rick Norwood (talk) 20:57, 24 August 2009 (UTC)

Do you really believe that the notation just as easily could have gone either way? Is it an accident that 1 + 3x2 is so easy to read? Is it an accident that our delimiters looks like () and [] instead of )( and ][?
Or does this notation specifically exploit the organization of the human visual cortex, which has been honed by millions of years of evolution to be really good at understanding hierarchical spatial relationships with minimal conscious effort? Melchoir (talk) 21:14, 24 August 2009 (UTC)

Morgoth: Our modern mathematical notation is one of the great inventions in the history of writing, but to read its history is to shudder at how often it came near to disaster. What if Newton's notation had won out over that of Leibniz? The notation we have is really good, but we were lucky. Look at all the people who can't agree on the definition of a ring! Or look at the (to me disastrous) notation of big O, little o. The symbol =o is the only example I know of where a pair of parallel lines do not indicate an equivalence relation! Rick Norwood (talk) 23:34, 24 August 2009 (UTC)

Morgoth - it sounds to me like you think that if the unary minus/negative sign convention had "gone the other way" (in RN's words) then it would have been, what? less logical somehow? I don't see that one is any more logical then the other - indeed, I could make a strong argument that -5^2 is more sensibly understood with the minus sign being the negative sign. In fact, most middle schoolers and high schoolers would prefer it that way as well Mortgagemeister (talk) 17:28, 27 August 2009 (UTC)

Um, are you guys talking to me? Melchoir (talk) 18:12, 27 August 2009 (UTC)

No, to Al Pachino. Sorry. Sometimes I get carried away by trivia references. I forget that not everyone is into trivia.

The logic of the order of operations is this: a symbol representing a repeated operation (exponents are repeated multiplication, multiplication is repeated addition) is done before the operation it repeats (exponents before multiplication, multiplicatin before addition). But inverse operations are on the same level. If we made -5^2 positive, then how about 0 - 5^2? How about 0 - x^2 when x is 5? With the other convention, the evaluation of -x^2 would be a real problem. Rick Norwood (talk) 19:04, 27 August 2009 (UTC)

Ah... well, to tell a little more of my story, I agree that the conventional order of operations should be (and is) informed by mathematical considerations. I go on to say that notation should be (and is) informed by the order of operations we wish to imply to the reader.
If we did an experiment asking a large group of people to evaluate one of these expressions for x = 5, chosen at random:
  1. −  x2
  2. − x2
  3. x2
  4. -x2
  5. -x^2
  6. -x ^ 2
  7. -x  ^  2
then I expect the ratio of people who answer −25, versus 25, to decrease as we move down the list. Am I alone in this expectation? Melchoir (talk) 20:38, 27 August 2009 (UTC)

Rick - your point about -x^2 being a real problem eludes me here. In the expression -x, there is no question what the "-" is. It is the unary minus. It is only when you have an actual number that the minus sign "becomes" a heteronym. -5 can be either the opposite of 5 or the number negative 5.

Melchoir (or Morgoth sorry) I'm not sure why adding spaces to the expression should change your ratio. More people would just get the problem wrong maybe, but because they don't know what you are trying to say (like me). But systematically making more people answer 25 then -25? Not following it Mortgagemeister (talk) 01:19, 28 August 2009 (UTC)

Then let's design an experiment with fewer confounding variables. We will invent two new symbols, let's say # and $, and teach the human subject that # n means take the average of n and 10, and n $ means take the average of n and 90. The subject will be given a few simple examples, like 80 $ = 85, and perhaps they will run through a few simple exercises. We will then randomly ask them to compute one of the following:
#50  $
#  50$
I hypothesize that when we compare the two sets of results, the first will be biased toward 60 and the second toward 40. Melchoir (talk) 02:19, 28 August 2009 (UTC)

Melchoir's name is Melchoir. I was making a weak joke, for which I've apologized.

The problem with -x is the substitution rule. We want to be able to replace x by a number and not change the meaning. if -x^2 had a different different meaning from -5^2, it would complicate the substitution rule. Rick Norwood (talk) 12:55, 28 August 2009 (UTC) Rick - So what? There are "complications" right now. Find the value of x^2 when x=-5. You may NOT substitute without changing the form of the expression to (-5)^2 Melchoir: Test your hypothesis and get back to me I guess. Without understanding why you think so, (i.e does it have to do with spaces, the difference between prefix and postfix operations, etc) your example offers no explanation. Mortgagemeister (talk) 13:42, 28 August 2009 (UTC)

True. I should have quit while I was ahead. The rule is arbitrary.
Melchoir: mathematical notation cannot depend on chance. There was a time, a hundred years ago, when there were mathematicians with sufficient clout to change the rules (Klein and Hilbert, for example). That time is past. We live with the rules we've got, or chaos ensues.

Rick Norwood (talk) 15:21, 28 August 2009 (UTC)

Mortgagemeister: Yes, spacing. Spacing, you'll notice, is the only difference between the two forms, and it's the difference we're interested in. The reader who doesn't know any better will tend to think #50  $ = (# 50) $ = 30 $ = 60 and #  50$ = # (50 $) = # 70 = 40. The experiment is designed to cancel out any other biases, such as prefix/postfix. If you really wanted to conduct the experiment, you'd also make sure to include left- and right-handed people, native readers of left-to-right and right-to-left scripts, randomize the names of the symbols, the order in which they're introduced, which one goes high and which low, etc.
Rick: My point is that the rules don't need to be changed. The rules we've got are pretty good. Things that go together are closer together. This allows for the reader to instruct his visual cortex to group by proximity, which is a command it understands. If you use notation that breaks this rule, like -x^2, then you force the reader to rely on learned rules like PEMDAS, which the visual cortex doesn't understand. Melchoir (talk) 16:35, 28 August 2009 (UTC)

Melchoir: The rule that you are quoting, "things that go together are closer together" just doesn't exist. I don't know where you learned it, but it is not a rule. 3+4 x 100 NEVER EQUALS 700. It either is 403 or is "rejected by the reader" due to the excessive amount of spaces. By the way, can't you still read this? Spacing, on the word level, doesn't change any of the rules you have learned to form English sentences.

Your two "forms" above have another difference, other than spacing. Pre fix operation vs Post Fix operation. If in your new system, you are going to rely on "closeness" (now that you have stated your rule I can understand what you were suggesting earlier and I can understand why its wrong) you will have far more ambiguity then if you impose a rule that pre fix always comes before post fix (or visa versa). Mortgagemeister 17:54, 28 August 2009 (UTC)

This conversation isn't working for me, so I will bow out. I appreciate your earnestness, but I fear that this forum is inadequate to communicate across the conceptual gulf that separates us. Have a nice day, Melchoir (talk) 18:55, 28 August 2009 (UTC)

Minus a.k.a. "Take away"[edit]

Another alternate name is "take away". For example: 3 - 2, Can also be read as "three take away two." CaribDigita (talk) 19:43, 22 September 2009 (UTC)

If this is of interest, it belongs in the article on subtraction, not here. Rick Norwood (talk) 19:55, 22 September 2009 (UTC)

What were the + - = signs in ancient Greece? Only Egypt is mentioned. Simanos (talk) 23:22, 28 September 2010 (UTC)

Ed Sheeran[edit]

Ed Sheerans new album is called +, it has a wikipedia entry. Should be connected? — Preceding unsigned comment added by 81.151.38.11 (talk) 17:12, 18 September 2011 (UTC)

No, if you look just under the title of the article you'll see disambiguation links for plus and minus. Dmcq (talk) 18:35, 18 September 2011 (UTC)