Talk:−1 (number)

From Wikipedia, the free encyclopedia

This article is part of WikiProject Numbers, an attempt at creating a standardized, informative and easy-to-use resource about numbers. Suggestions for improving multiple articles on numbers and related subjects should go to Wikipedia talk:WikiProject Numbers.

Before the fanatic deletionists start ranting about slippery slopes, let me say that I have no intention on writing articles on any other negative numbers. On the other hand, if someone else writes an article on another negative number, I shall read with interest, and maybe even add to it. PrimeFan 20:02, 2 Feb 2004 (UTC)

---

Isn't the action of "raising a number to the power of -1" defined as "the same thing as calculating its reciprocal"? Dysprosia 09:35, 10 Feb 2004 (UTC)

I guess the way I expressed it was inelegant. Can you reword it more elegantly? PrimeFan 20:36, 10 Feb 2004 (UTC)

It's not that, it's just I'm usually wrong on matters such as this - but things are changing! I think I'm right, though, so if I'm wrong, then the correcty-person can change the article :) Dysprosia 23:48, 10 Feb 2004 (UTC)

You are correct, and you have made the article more elegant. Your addition on the rules of exponentiations is much appreciated. PrimeFan 18:36, 11 Feb 2004 (UTC)

---

I don't understand why − 1 is equal to "FF" in hexadecimal. It is equal to − 1 in hexadecimal. It is CONGRUENT to FF mod 16, which is something different. Revolver 21:26, 30 Aug 2004 (UTC)

It is equal to 11111111 (or FF) to your computer if it holds it in a signed byte, 1111111111111111 (or FFFF) in a signed word, 11111111111111111111111111111111 (or FFFFFFFF) in a signed double word, you get the idea. Since two's complement is used by almost all computers, from pocket calculators to CRAY supercomputers, it merits being mentioned in this article, though it was agreed early on that showing the signed byte was enough to get the point across. Anton Mravcek 16:29, 31 Aug 2004 (UTC)

Thank you for clearing that up. There weren't any links to anything originally, so I didn't know what the terms meant. (I had never heard of "two's complement", or "signed word", e.g. these are terms used mostly by programmers, not in math.) I hope my rewording is accurate. Revolver 02:11, 1 Sep 2004 (UTC)

[edit] The Möbius Function

The Möbius function is worth mention in this article because -1 is one of its only three possible return values. That's worth mentioning in the article. (Well, technically it has an infinitude of return values, but they all boil down to -1, 0 and 1). Anton Mravcek 16:29, 31 Aug 2004 (UTC)

Well, I find that a very weak justification, but, whatever. I have nothing against the Mobius function, but the function by itself isn't particularly relevant...dozens of other functions and formulas use -1 prominently, should we include them?? The sign function only takes on -1, 0, and 1, should we also include it? What about Euler's formula (e^(pi.i) = -1)? What about the index of a CW circle? What about...?? You get the point. Revolver 02:06, 1 Sep 2004 (UTC) Revolver

The Euler formula sounds interesting, and so does the CW circle. Maybe you should add them. After all, it's much easier to take stuff out than to add it in. 141.217.70.75 18:44, 1 Sep 2004 (UTC)

No, I won't add them, because none of them are relevant. They don't pertain directly to the number -1. Revolver 06:39, 2 Sep 2004 (UTC)

Three important transcendental numbers put into a simple equation happen to yield -1 and you don't think that's relevant to -1????!!!!!??????!!!!! Wow. As for the CW circle, you got me curious about that. I want to know what a CW circle is. Anton Mravcek 19:48, 2 Sep 2004 (UTC)

Euler's formula is more relevant to e, pi and i then -1. It's a statement about the exponential function, and -1 happens to be a nice output. The equation is at heart a statement about the exponential function, not the number -1, and to really understand why the equation is true requires understanding the exponential function and getting a grasp on that, not getting a grasp of the number -1.

A CW circle is a clockwise circle, i.e. trace a circle exactly once in CW direction and finish where you started. The index (or winding number) is basically the "number of times" you went around a point inside the circle, with CCW being positive direction. So, a CW circle traced once has index -1. It's nothing special, you can do it for any integer. Revolver 05:52, 3 Sep 2004 (UTC)

BTW, it doesn't have an infinitude of return values, just 3. Revolver

Yes it does have an infinitude of values, just as I was saying below. They range from -1^1 to -1^+infinity, and of course also 0. PrimeFan 21:27, 1 Sep 2004 (UTC)

No, it does not. The range is {-1, 0, 1}, this set has 3 members, not an infinite number of members. The fact that -1 = -1^1 = -1^3 = ... and 1 = -1^2 = -1^4 = ... doesn't matter. -1^1 and -1^3 are the same number, not different. You don't get to "count" them more than once just because they're expressed differently. Revolver 06:39, 2 Sep 2004 (UTC)

The k in -1^k is the number of factors of the number in question, k = 1 for prime numbers, k = 3 for sphenic numbers, etc. In some number theory applications, the distinction is useful. For the Mertens function it is not. Anton Mravcek 19:48, 2 Sep 2004 (UTC)

I know what k is. I did get my ph.d. in number theory. BTW, you don't quite have it right. k is not the number of factors of the number (this is nother arithmetic function); it's the number of distinct prime factors, and even then, the formula -1^k is only true for square-free numbers. For instance, 60 = (2^2)*3*5, yet μ(60) = 0, because 60 is not square-free (square-free means not divisible by a square > 1). As for sphenic numbers, I can't comment on the accuracy of the terminology, I haven't heard it before, and a look in On-line Encyclopedia of Integer Sequences gave only a technical name, not sphenic. (That doesn't mean it's not correct.) Given this is the definition you mean (product of 3 primes, which is not the same as having 3 prime factors, I corrected this at the sphenic number article) then the Mobius function returns -1 for sphenic numbers. But the Mobius function still only has 3 elements in its range, not infinitely many. This is what I explained above. Revolver 05:52, 3 Sep 2004 (UTC)

Just speaking for myself, I'm tired of arguing this minor though elegant point about the infinity of values this function has.

Again, it doesn't have infinitely many values. It's not a matter of opinion. I'm tired of arguing this.

The important point here is that there is something special to -1 being one of the three solutions. If I had come up with this function instead of Möbius, I probably would've chosen something like {19, 20, 21} instead of {-1, 0, 1}. PrimeFan 20:57, 2 Sep 2004 (UTC)

Of course there's something special about -1, in the trivial sense that it's the only reaonsable definition to make ({19, 20, 21} would destroy all arithmetic properties of the function. Still, this is the case for almost all functions...if you change them, they don't work anymore. So, just being a value that it takes on isn't special by itself. Revolver 05:52, 3 Sep 2004 (UTC)

That replacing the return values of the function would destroy all its arithmetic properties proves that there is something special to the return values chosen. PrimeFan's choice would, for example, invalidate what he wrote about the Mobius function and heteromecic numbers. Nevertheless, I'm intrigued by his choice of values and have been playing around with them using Mathematica:

PFMoebiusMu[x_] := MoebiusMu[x] + 20

SetAttributes[PFMoebiusMu, Listable]

PFMertens[x_] := Plus @@ PFMoebiusMu[Range[1, x]]

SetAttributes[PFMertens, Listable]

P.S. about the sphenic numbers: the correctness of the term has been discussed on the Talk page for that article. It's an antique term, but an useful one nevertheless. Anton Mravcek 21:19, 3 Sep 2004 (UTC)

geez, Revolver, isnt it alittle early to be worried about this article becoming cluttered? why don't you turn your attention to articles like positive one and three that are in genuine need of pruning? Numerao 20:12, 1 Sep 2004 (UTC)

I'm not sure which deletions you're referring to in particular. Most of the original deletions I made were deleting incorrect information. (E.g., -1 is not a cardinal number, divisors depend on ring, etc.) As for the Mobius function thing, the issue has nothing to do with clutter, it has to do with relevancy. One of the attributes of good writing is not to force the reader to spend extraneous time reading about things they didn't ask to read about. A reader coming to this aritlce wants to learn about the number -1, not the Mobius function or Euler's formula, or Cauchy's integral formula, or the sign function. None of these are really directly related to -1. Here are some things I think WOULD be relevant:

• Cultural history of -1: when it was first used, resistance to the concept, spread of its use, impact (really, this is about negative numbers in general)
• Famous quotes or anecdotes involving -1 (I seem to remember there may be some).
• Remarkable mathematical facts about the number -1, (i.e. not other facts that mention -1), e.g. (-1)*(-1) = 1 and proof of this, etc.

Revolver 06:39, 2 Sep 2004 (UTC)

I too think it would be good to have more info on the cultural history of -1, some famous quotes and anecdotes (there's probably something from Ramanujan). I eagerly await your additions on those areas. Anton Mravcek 19:51, 2 Sep 2004 (UTC)

By infinitude of values, are you referring to the possibilities of k for (-1)^k where k is the total of prime factors of the number in question? If k is odd, then (-1)^k = -1, while if k is even, then (-1)^k = +1. Wow, that's so elegant. Thank you for helping me see that. PrimeFan 21:27, 1 Sep 2004 (UTC)

[edit] Beginning of article

Using a sufficiently broad meaning of 'definition', almost anything can be defined as almost anything; bearing that in mind, can anyone provide a source where, in mainstream mathematics, -1 was defined as the "square of i", without the imaginary units first being defined in terms of their relationship to -1? I seriously do not see it happening.

Although admittedly not having read the whole of the preceding discussion, I must object to the encyclopedic relevance of the Möbius function in its present context. Maybe if we created a "Uses of -1 in Mathematics" section, and listed e^i*pi, Möbius, Legendre symbol etc. Right now the intro has no cohesion. Pietro KC 07:04, 11 February 2006 (UTC)

[edit] Another intuitive explanation?

I like your intuitive explanation, but I wonder whether material like

What would it mean to lay down a stick "negatively many times"? One answer is to say that it would result in a displacement where, if we were to lay it down 3 times immediately after, we would return to where we started.

will be clear to the beginning reader. (To be sure, I don't know; I'm not one, and I don't have one handy.) So I'd like to propose an alternative intuitive explanation for your consideration. It would read something like this:

Imagine, for a moment, that you're in a hot-air balloon. You have the flame going, so your balloon is rising. Let's say that you're rising at a nice steady climb: 2 feet every second. Let's also say that we'll consider up to be a "positive" direction, and down to be a "negative" direction.

Question 1: compared to where you are now, where will you be in 5 seconds?
Answer: you multiply the number of seconds by the speed. (5 seconds from now)(2 feet higher every second) = 10 feet higher. 5 x 2 = 10 -- a positive result.

Question 2: compared to where you are now, where were you 5 seconds ago?
Answer 2: let's think of time in the past as a "negative time" direction. (5 seconds ago)(2 feet higher every second) = 10 feet lower. (-5) x 2 = -10.

Now, let's change the situation. The flame isn't on, and in fact there's a small hole in the balloon, so you're slowly dropping -- 2 feet every second.

Question 3: compared to where you are now, where will you be in 5 seconds?
Answer 3: (5 seconds from now)(2 feet lower every second) = 10 feet lower. 5 x (-2) = -10 -- a negative result.

Question 4: compared to where you are now, where were you 5 seconds ago?
Answer 4: (5 seconds ago)(2 feet lower every second) = 10 feet higher. (-5) x (-2) = +10. A negative times a negative came out to be a positive.

Your thoughts? --Jay (Histrion) (talk • contribs) 16:05, 27 September 2006 (UTC)