Talk:Negative and non-negative numbers

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[edit] Through 2003

Added rules for division (in quite simplistic terms, since this is the same as multiplication -- same sign -> positive result, different signs -> negative result).

I get the point that the article is meant to be understood easily, but can't we just refer to things by their names? Using words like "dividend" and "divisor" (for division) or "factors" (for multiplication) makes much more sense to me than exhaustively mentioning "if you add a positive number to a negative number"... you get the idea. ;) --doshell

This situation cannot be understood as repeated addition, and the analogy to debts doesn't help either. The ultimate reason for this rule is that we want the distributive law to work:

(3 + (-3)) · (-4) = 3 · (-4) + (-3) · (-4).

The left hand side of this equation equals 0 · (-4) = 0, while the right hand side equals -12 + [(-3) · (-4)]; for the two to be equal, we need (-3) · (-4) = 12.

I didn't understand the above, so I just cut it and pasted it. I hope the sections on arithmetic with negative numbers are correct, as well as clear, now. Someone really ought to check me, because in my haste I could easily make a non-negative number of errors :-) --Ed Poor 20:58 Dec 5, 2002 (UTC)

Makes sense to me. Follow the brackets carefully, Ed. negative * negative always did make sense to me as a repeated addition when I was a kid. 2 * -3 means "two lots of -3", -6, and since this can be also written as -3 * 2, it seemed logical to interpret this as "-3 lots of 2". hm. years since I thought about this stuff... -- Tarquin 10:26 Dec 6, 2002 (UTC)

Removing:


Multiplication of a number by -1 changes its sign. This is called negation, and may be expressed by placing a minus sign in front of a number or a quantity in brackets:

-1 × 5 = -5
-1 × -8 = -(-8) = 8
-1 × (3 + 4) = -(3 + 4) = -7

In fact, negation is equivalent to multiplying a number by -1:

-5 = -1 × 5

This equivalence can be used to simplify multiplication involving negative terms:

-6 × 3 = (-1 × 6) × 3 = -1 × (6 × 3) = -1 × 18 = -18 (if you have a debt of $6, and then your debt is tripled, you end up with a debt of $18.)

Multiplication of two negative numbers yields a positive result:

-3 × -4 = (-1 × 3) × (-1 × 4) = (-1 × -1) × (3 × 4) = 1 × 12 = 12, or more simply,
-3 × -4 = -1 × (3 × -4) = -(-12) = 12

since negation was something I remember had to be proven in analysis, I'm not entirely sure how correct it is to just blankly state it. Restoring Axel's version for now, until he's back to maybe take the best of both & merge. -- Tarquin 11:15 Dec 6, 2002 (UTC)

Doesn't this just follow from 0*x = (1 + (-1))*x = x + -1*x = 0, so that -1*x is guaranteed to be the additive inverse (i.e., negation) of x, denoted by -x? Chas zzz brown 11:32 Dec 6, 2002 (UTC)
That's nothing. I'm waiting for the AE/BE argument to start about whether it should be math or maths... Maybe we should just use mathematics all the time to be safe. ;) --Dante Alighieri 11:18 Dec 6, 2002 (UTC)
Yup, you're right, Chaz. It's hard to determine how axiomatic to be in covering what the lay readers takes to be a very basic topic. -- Tarquin

Hold on. I really don't think it makes much sense to merge Positive number and Nonnegative into Negative number. They're not the same thing, after all. I don't expect to read about positive numbers in an article called "negative number". Evercat 13:03 21 May 2003 (UTC)

There is no doubt they are not the same thing. How about the title negative and positive number? -- Taku 13:08 21 May 2003 (UTC)
That would be better. Perhaps Negative and positive numbers is grammatically better. Still, I rather prefer seperate articles for them, all linking to one another... Evercat 13:11 21 May 2003 (UTC)
Since wikipedia is an encyclopedia, I think it makes more sense one article talks about negativity of number. Currently the article is nothing more than a bunch of definitions and properties, but we certainly want to discuss when the concept of negative is introduced, notations and other stuff. I don't think positive number article can grow more than a mere dictionary entry. (I don't mean to impose my will but just trying to justify why I did. We can discuss this.)

-- Taku 13:17 21 May 2003 (UTC)

noo!! the example at the bottom uses two-complement!! if the leftmost bit is used to express the sign (wich it seldom does in processors!) it cannot express -128 but only -127. there is also two zero's, -0 and 0, wich makes some operations quiet odd: -3+4 = 0, wich is wrong..? :P

I thought the example I put is quite typical. If I remember correctly, char of C can express -128 to 127 because there are 255 distinct numbers. There should be only one zero. -- Taku 19:15 21 May 2003 (UTC)
That's in two's complement. In one's complement, a negative number is represented as the complement of the value. Thus, the top bit is "1" if the value is negative. A weird thing about one's complement is that there are two representations for zero (all zeros and all ones). One's complement is much less common today, but it's still important historically -- Dwheeler 19:30 21 May 2003 (UTC)
This seems quite interesting. If you can, don't hesitate to add this scheme (called one's complement?). The article certainly doesn't have to be limited to one mechanism. -- Taku 22:00 21 May 2003 (UTC)
A more detailed discussion is already in Integral data type, and this article ("Negative and positive numbers") links to it.

Negative and positive numbers... hm... so that's like numbers except 0.

Zero, the square root of zero, the cube root of zero, zero squared. ;) --Dante Alighieri 19:25 21 May 2003 (UTC)

Dante, you little sound sarcastic, but really I didn't notice numbers except 0, but then do you have any idea how to name this article? Topics like representation of negative and positive numbers in computers look weird if they are located in negative number article. -- Taku 21:51 21 May 2003 (UTC)

Why not put all this information on number? -- Minesweeper 22:01 21 May 2003 (UTC)

Good point. Why not? Any objection? -- Taku 22:02 21 May 2003 (UTC)


All this detail about how to add and subtract negative and positive numbers would be a burden in "Number". However, cross-links sure make sense. Having this as a separate article makes it easier to reference specifically the issue of + vs. -.
Then what about negativity or even the concept of negative number. If possible, we certainly want to add about the history of negative numbers. -- Taku 22:27 21 May 2003 (UTC)
Yes. It sounds like there's many good reasons to leave this as a separate page. -- Dwheeler 22:30 21 May 2003 (UTC)

I would like to rename this to negativity because I knew negative and positive numbers sound like any number but zero, which is not the intent of this article. Any objection? -- Taku 22:57 21 May 2003 (UTC)

To me negative number would make more sense than negativity, for one thing because the latter does not make it perfectly clear that mathematics is the subject. Negativity (mathematics) seems overly complicated. Michael Hardy 00:08 22 May 2003 (UTC)

But what about "I don't expect to read about positive numbers in an article called "negative number" by User:Evercat. He has a point. It seems little weird the article negative number has a lot of mention about positive numbers. But the trouble we invented a concept positive number after invension of negative numbers. Without the concept of negative number, we don't have positive numbers. Then a compromise, how about negative and non-negative numbers? Sounds strange? -- Taku 02:24 22 May 2003 (UTC)
I think it's fine where it is. The discussion of where zero falls is natural for an article called "negative and positive numbers". Evercat 14:24 22 May 2003 (UTC)

They're called signed numbers! -- Toby Bartels 04:14 6 Jun 2003 (UTC)

I revert new move since there seems no agreement with it yet. -- Taku 04:21 6 Jun 2003 (UTC)

I was bold since (unlike some page moves) it could be undone if somebody didn't like it (as you don't). But I'd like to hear your opinions of disagreement too! -- Toby Bartels 04:42 6 Jun 2003 (UTC)

First of all, I have never heard of signed numbers. I mean is it really a popularly accepted term? Do you have evidence? If you do, I have no trouble to restore your contribution myself. -- Taku 04:48 6 Jun 2003 (UTC)

I hear it often enough -- though this is hearsay. There's some evidence in the article itself, where people other than me used the term. But I should provide some documentary evidence of use outside of computer science too, so I'll go look some up. -- Toby Bartels 09:58 11 Jun 2003 (UTC)

I really don't see why this page exists at all. Initially it was about negative numbers. What was wrong with that?? Then it became negative and positive numbers, until someone pointed out that it was a bit silly that it excluded zero (ohh year that was me). Now it's about er .. what ? er... numbers. Content should be moved to either number or integer where negative numbers can be discussed in context. The stuff about binary representation of negative numbers is already well covered in Computer numbering formats. The use of links where appropriate should suffice. Mintguy 16:39 6 Jun 2003 (UTC)

You seem to be correct about the computer representation (although Computer numbering formats needs to be broken up). Signed numbers are a separate concept from simply integers, since one may consider signed or unsigned numbers of other sorts (like rational, real, cardinal, etc). This article could talk about the common issues, while Integer would deal with the specific properties of Z (like its special position among rings). -- Toby Bartels 09:58 11 Jun 2003 (UTC)

Agreed. Could we have a simple page title back, ie negative number? -- Tarquin 18:56 6 Jun 2003 (UTC)

While I like Signed number, I also see no reason why Negative number and Positive number can't also exist separately. And maybe when all the material specific to those articles, to Integer, and to Computer numbering formats is taken out, there'll be very little left of Signed number (or whatever you want to call it), in which case it can be folded into Number. -- Toby Bartels 09:58 11 Jun 2003 (UTC)
Though strage title, I think having a separate article about the concept of negative numbers in math or its representation does make sense. I don't think negative and positive is part of number. Breaking up the article to two articles doesn't make sense. Any article in wikipedia is an encyclopediac article, which means we want to discuss not just what it is, but also more about history, significance in society and so on. Unfortunately there are a lot of overlaps between Computer numbering formats and other wikipedia articles. Rather than moving stuff here to it, it should be more reasonable to move stuff from there to here as we break up the article. -- Taku 21:30 11 Jun 2003 (UTC)

Computer numbering formats. Actually It is a really good written article but the trouble is that the article is rather isolated from the rest of well-cultivated wikipedia articles. The stuff about binary represention is vital because the article should not be limited to that in math but that in general cases. Besides, in the future we might want to add portions for example history of concept of negative and positive. Actually I don't have much trouble to rename this to simple negative number but then what about positive number then? Are people suggesting split it off into two articles? Honestly I really don't like a current ugly title but I don't know a better one. Actually it is rather silly to discuss a lot about naming because unlike dictionaries, in encyclopedia articles, the article tends to be more general, thus, sometime the title also tends to be complex. For example, political status of Taiwan or something (I don't remember the current name). -- Taku 22:04 6 Jun 2003 (UTC)

I don't think the title of this article is as important as its contents: the discussion of 1-complement, 2-complement etc. does not belong here, only a link. After all, that is a discussion of numerals for negative numbers in the binary system, not of negative numbers themselves. What we desparately need however is a history section. AxelBoldt 15:04, 29 Sep 2003 (UTC)

[edit] Classification of 0

Mathematically, 0 is neither positive nor negative. However, in naive English it is common to use the word "positive" to include 0. Any comments about this?? 66.245.1.229 19:30, 6 Nov 2004 (UTC)

It may or may not be naive, but it would certainly be confusing and misleading to call zero a positive number. If I say that "I have visited Paris a positive number of times" I would mean I have done it at least once. --Henrygb 22:32, 17 Nov 2004 (UTC)
People don't use their languages correctly many times. But I think it is unnecessary to mention such misuses in too much detail. -- Taku 01:38, Nov 18, 2004 (UTC)
Did you mean to say "naive English" or was that supposed to be "native English?" As a native English speaker, I've never heard anyone refer to zero as a positive number, except when discussing the mathematical classification, in which case they were simply wrong. -- Foof 03:04, 6 February 2006 (UTC)

It's increasingly common in mathematics to distinguish, in general, between positive and strictlyt positive objects, abolishing the slightly awkward term non-negative (for example, a complex number is usually neither negative nor positive nor 0. In order of increasing generality, the possibilities are:

  • a linear order with 0. The usual terminology is positive, negative, zero, as in the article.
  • a partial order with 0. There are now elements that are incomparable to 0, and being non-negative no longer means being positive or 0. That's why for complex numbers, the longer term "nonnegative real number" is sometimes used.
  • several partial preorders with 0. That's the tricky one. It's not at all uncommon these days, and there is usually no good way to say "an element that is nonnegative in every individual preorder".

To illustrate (and to give me some practice with tables, but don't tell anyone I wasn't perfect before), consider the space R2 (that's just maths-speak for tuples of real numbers):

Element "old" terminology "new" terminology "new expanded" terminology
(1,1) positive positive strictly positive
(1,0) ? positive varies
(0,0) zero zero or positive or negative zero or positive or negative
(0,-1) ? negative varies
(-1,-1) negative negative "strictly negative"
(1,-1) ? ?

Now, it turns out that in such general cases, it usually turns out that there are many useful theorems about the "new" positive elements; sometimes there are useful theorems about the "new expanded" strictly positive elements, including or excluding the (1,0) case depending on which object you deal with. The set of "old positive" elements is usually far less interesting, and when it is interesting, there is virtually always a set of preorders such that it becomes the "strictly positive" set, and the positive set will be interesting then, too!

In short, many mathematicians, including myself, think it is an unfair accident of history that "positive" excluded the zero case. It is also questionable etymologically (it is quite possible to put zero apples on a table. It's much harder with -1 apple, particularly if there aren't any on it to begin with).

I definitely think that this should be discussed in an article linked to from positive. It is also worth mentioning that 0 is "positif", in French, and that this practice has spread through adoption of French terminology.

Finally, since this is something that people argue about a lot until they finally go find a mathematician who is subsequently annoyed at being asked again, it's a convention. Mathematicians tend not to feel strongly about which convention you use, though they do feel strongly about wasting a lot of time because you used a nonstandard convention without telling them. Still, it is a convention, and if you prefer another one, just state so clearly and move on.

RandomP 00:30, 1 May 2006 (UTC)

[edit] bad jokes and other non...negatives

no comment on the following :

  • "Division is similar to multiplication"
  • "If both have different signs"

some comments on the folllowing :

  • nonnegative can be defined as desired, but "non negative" has (imho) the meaning of "not negative" and is thus invariably defined once "negative" is defined. For example, an imaginary number is not negative.
  • in the context of "nonnegative matrix" I think one should include not only links but also comments to what is commonly called a positive matrix (for which the associated quadratic form takes ony nonnegative values)

I don't want to impose my ideas and thus don't make changes since this might be controversal, and I risk to be too axiomatic: I would call nonnegative all elements that are not less than zero (in any group equipped with a partial order), so this is not always the same than "positive or zero"; and suggest to specify "nonnegative reals" or "nonnegative integer" etc. in order to get the "usual" (particular) meaning.

But if someone feels an inspiration, I strongly suggest to make the adequate changes. — MFH: Talk 13:16, 28 September 2005 (UTC)

[edit] First usage of negative numbers

From the current article:

"Negative numbers were not well-understood until modern times. As recently as the 18th century, the Swiss mathematician Leonhard Euler believed that negative numbers were greater than infinity..."

This seems unfairly closed-minded. The convention that −1 < 1 is natural if you want an ordered group, but some uses of negative numbers demand a different ordering: see negative temperature. Melchoir 01:08, 11 February 2006 (UTC)

[edit] Diophantus

Diophantus's rejecting 20x+4=0 as a meaningful equation is cited as an evidence of knowledge of negative numbers in Greece. This is absurd, since it is a clear evidence to the contrary. It's like saying that somebody rejecting square root of negative numbers is an evidence that he knows imaginary numbers. deeptrivia (talk) 03:18, 17 February 2006 (UTC)

off the current topic slightly. can anyone prove the existance of negative numbers? i dnt mean prove as in negative temperatures i mean prove lik u would prove the quadratic equation by using completing the square or prove the sum to infinity for a geometric series.

The short answer is yes, but the longer answer is long indeed. After all, what do you mean by "existence"? One construction of negative numbers is given by the "Formal construction of negative and non-negative integers" section of this article. If you want a thoughtful explanation of what it all means, I think you'll get an excellent response if you ask on Wikipedia:Reference desk/Mathematics. Melchoir 17:50, 9 June 2006 (UTC)

[edit] Minus numbers?

Terminology is important. It's time for a rant.

If I Google "negative number" I get 2,470,000 hits, and all of them are coherently talking about negative numbers. If I Google "minus number" I get 53,100 hits. Even of those, I grow suspicious: out of the top 10 hits, only 4 are actually talking about negative numbers; another 4 are using "minus" as a verb, and the other 2 are incomprehensible.

So I Google "negative numbers" with an s, and this time I get 3,060,000 hits, and all of them are coherently talking about negative numbers. But "minus numbers" gets 17,000 hits, and even then the very top item is an incomprehensible PDF technical sheet in all caps; below that is a subject-line of some student asking "dr. math", and further below we find such gems as "NBA Plus Minus numbers for the last 30 days!". On the next page there are three more "plus/minus" phrases.

I conclude that virtually no one says "minus numbers", including the British; that even in the rarity when they do use the phrase it's even odds on what they mean; and of that tiny minority who actually use it to mean "numbers less than zero", they're either double-talking pedagogues or just confused.

If we search Wikipedia itself, it gets even better: all of the bolded phrases at the top are used throughout the project, even "non-positive numbers". "Minus numbers" turns up nothing.

Even if we assume that all the searches are lying to us: I've read mathematics books at all levels; I've read research articles written from all over the world; I've even read the literature for elementary school teachers. They all say "negative number", and more importantly, none of them says "minus number". If anyone has a reliable source that says "minus number" for a number less than zero or naught, please cite it and educate me. Until then, there is no need to encourage or even acknowledge confusing and truly obscure terminology. Melchoir 05:48, 23 March 2006 (UTC)

The usage is not all that obscure. The first page of a Google Book search turns up several usages, some of which seem like they could be called a reliable source:
  • [1] Practical Statistics Simply Explained by Russell A Langley - Mathematics - 1971 - 399 pages. Page 61 - "Remember that a minus number multiplied by another minus number gives"
  • [2] Basic Ac Circuits by Clayton Rawlins, John Clayton Rawlins - Technology - 2000 - 541 pages. Page 400 - "There is no real number which when squared results in a minus number."
  • [3] Statistics Explained: A Guide for Social Science Students by Perry R Hinton - Mathematics - 1995 - 256 pages. Page 31 - "if you calculate a z score and it turns out to be a minus number, all this means is that the score is less than the mean."
  • [4] Conduct of Monetary Policy (pursuant to the Full Employment and Balanced Growth Act of 1978,... by Finance, and Urban Affairs United States. Congress. House. Committee on Banking - 1980 - 199 pages. Page 157 - "... argue that in a noninflationary situation with lower interest rates it should not — we should, consistent with price stability, have a minus number in M1."
This should show that in fact the phrase "minus number" is sometimes used to mean the same thing as "negative number". Its popularity may be due to having one less syllable.-R. S. Shaw 06:44, 23 March 2006 (UTC)
Wow, I'm surprised to see that in a technology book published in 2000. Well, again on Google book search, "minus numbers" gets 139 pages while "negative numbers" gets 14000 pages; they're not even on the same level. The relative authority of the books that show up in those two search results is also pretty evident to me. Melchoir 07:34, 23 March 2006 (UTC)

I saw (and corrected) a claim on the decimal page that +5 means "plus five" and -8 means "minus eight". I think that this should be mentioned on this page, just to tell people that it is incorrect. Also, I'm going to add a discussion of -.

[edit] Superscript notation

The article claims

In order to avoid confusion between the concepts of subtraction and negation, often the negative sign is written as a superscript:

I've not come across this before, so I'm a bit doubtful. I've seen the notation where a bar over the number represents negation, and I've seen various people write (well, define) negative numbers like this:

....99999 is -1 ....99998 is -2 ....99990 is -10

(particularly if you use some other symbol to mean "nines all the way to the left", this notation makes some things more consistent; it's also the equivalent of the two's-complement notation used by most computers).

But I can't see I've seen the negative sign as a superscript before, and if it's used "often", I should have. Is this specific to some education setting?

RandomP 14:09, 23 September 2006 (UTC)

I've seen it before, but it isn't done "often" in my estimation; it's rare, or at most occasional. I first saw it 30 years ago; it's used for negative numbers in the APL programming language. The APL documentation as I remember flogged the raised sign as a wonderful thing invented by Iverson for APL, but in my opinion it was mainly used because the APL syntax needed a separate symbol in order to be able to parse its expressions (which are unusual). I've never seen the raised minus in any context not connected with APL (except this article).
I think the usage in the article should be reduced to a single example, and the "often" changed. -R. S. Shaw 18:59, 23 September 2006 (UTC)
I believe the raised sign is pretty common in early education, where the target audience is easily confused. Melchoir 19:11, 23 September 2006 (UTC)
I think it looks really odd and should be changed. With proper use of brackets and/or multiplication symbols I don't see how confusion could arise. --CompuChip 09:53, 24 November 2006 (UTC)

I agree with Melchoir, the superscript notation is usefull in early education where "#--#" could easy confuse a person (I've found that "#-(-#)" doesn't help much). In many contexts, a shorter dash for negation verses subtraction is used (like on calculators). The superscript notation also serves to keep the signs distinct (so they don't appear to be the same dash).

[edit] A question on terminology

I am a native Dane, but teach math in English at highschool level. I have a problem with terminology.

In Danish, "-5" and "-x" are read aloud as "minus fem" and "minus x", not "negativ fem" and "negativ x". How's that in English?

Many students would read "-5" as "negative 5", but that's nonsense to me as 5 is not negative. I.e., I understand "negative as a property, and 5 does not have that property. Am I right?

Also, many students would read "-x" as "negative x", but again, I'd understand that as "a negaitve x" (i.e. x<0), and that's of course something else. Am I right? Or am I at least right to the extent that "negative x" would be ambiguous?--Niels Ø 14:07, 2 December 2006 (UTC)

Almost everyone says "minus x". A small number of people say "negative x" because they think it sounds cool or because they are acting in Hollywood movies. --Zerotalk 14:40, 2 December 2006 (UTC)

Thanks for the reply! Are there perhaps other opinions? How about "-5", is that also nearly always "minus 5"? When many of my students (being taught all over the World, and in many different languages, before I get them) say "negative 5" and "negative x", is that a primary school thing, or what?

And should some of this go into the article somehow?--Niels Ø 15:21, 2 December 2006 (UTC)

In my experience (American), both "minus 5" and "negative 5" are common, with "minus" more frequent, I'd guess mainly because it is a syllable shorter. "Negative 5" makes perfect sense to me, essentially being the name of the number 5 units less than zero. While "negative" is essentially always a property, "minus" seems more like the operator to me. "-" is always "minus" in "7 - 5", and "7 - ( - 5 )" would be "7 minus negative 5".
With an unknown, "-x", the situation is different because the "-" in that context is always an operator, never part of the name of a number. Thus with a variable it is almost always "minus", never "negative". For "y - (-x)" one might use "minus" for both, or maybe "the negation of" for the second "-". -R. S. Shaw 06:16, 3 December 2006 (UTC)

I'm a graduate from an American university and in my experience, "negative 5" is much more commonly used than "minus 5." At least, no mathematics professor I've ever had has ever used the term "minus" for anything but subtraction. Occasionally, a non-professional might use the term "minus" for that purpose, but very informally. Five away from zero, to the left, is NEGATIVE (not minus, unless you're in the 4th grade), five away from zero to the right is POSITIVE (not plus). -Laikalynx 03:06, 21 December 2006 (UTC)

It looks to me that the "-" is as part of the number as the "5." You wouldn't normaly break up other symbol combinations (like 23 becoming "two three" instead of "twenty-three"), so why seperate the negative sign. Also, in many contexts, negative (negation) and minus (subtration) use a different sign. — Jaxad0127 06:21, 24 January 2007 (UTC)

There cant be a -X. Say that was supposed to mean -9. The -9 is the variable. So that would be negative negative 9. There is no -(Random Variable Here) —The preceding unsigned comment was added by 65.80.7.142 (talk • contribs) 1:56, 9 July 2007 (UTC).

If -X was supposed to be -9, then X would be 9, not -9. Negating variable names is quite common and the basis for subtraction itself. — Jaxad0127 04:10, 16 July 2007 (UTC)

[edit] math

is 0.1 a negative # —Preceding unsigned comment added by 24.176.17.147 (talk) 20:46, 16 January 2008 (UTC)

[edit] math

is 0.1 a non-negative #

[edit] math

is 0.1 a non-negative number —Preceding unsigned comment added by 24.176.17.147 (talk) 20:53, 16 January 2008 (UTC)