Difference between revisions of "Talk:Real number"

Revision as of 21:42, August 17, 2008

Moved comment from Anjruu

The introduction is actually a blatant misrepresentation of the truth. Real numbers are no more or less "abstract" than imaginary numbers. The author, one must conclude, was trying to allude to the fact that there cannot be an imaginary number of objects, but there cannot be a negative number of objects either. Or perhaps s/he was trying to say that the imaginary numbers cannot be represented on the number line, which is irrelevant, since they can be represented on the complex plane, an equally valid, and more complete, numerical device. Finally, perhaps the author simply meant that the square root of a negative number does not exist, which is wrong. It does exist, in the guise of an imaginary number. While this may seem cyclical, simply because a number is "unusual" does not mean that it is abstract and non-existent.

Illustrations of this fact, and the incorrectness of the introduction, can be seen when solving differential equations which contain a bifurcation parameter, where the method of eigenvalues must be used. There, an complex, "imaginary" solution to one of several equations often contains eigenvalues and thus a corresponding eigenfunction. Therefore, the real-life existent solution to a real-life existent problem depends on an imaginary number. This exemplifies why complex numbers are just as valid as real numbers, and are not merely an "abstract concept," as the introduction erroneously points out. Or to be more precise, ALL of mathematics is an abstract concept, but there exists an isomorphism to the "real world"; an isomorphism that holds just as strongly in the domain of complex numbers as in the subset of complex numbers, the reals.

The "real" world

Re: this sentence from the intro:

They are called "real" because they actually exist as quantities in the real world, i.e. as measures, weights, temperatures. In contrast, the imaginary numbers are just an abstract concept, but do not exist concretely in the real world.

The bulk of that statement simply isn't correct, which is why I removed it earlier. Historically, it was thought that imaginary numbers didn't "really exist," but for the last 200 years or so it's been understood that they do; I believe Cauchy and Gauss were heavily involved in putting that question to bed. Furthermore, there are indeed physical applications of complex numbers. Anything with a magnitude and phase (think alternating current, any harmonic oscillator, communications signals, etc.) is conveniently represented by complex numbers, and they are absolutely essential in quantum mechanics.

I second this motion: this sentence should be erased. Real numbers are as real - or imaginary - as complex numbers. --DiEb 15:41, 15 August 2008 (EDT)

EDIT: I've altered the sentence in question to reflect my comment.--Recorder 15:30, 15 August 2008 (EDT)

Again: Real vs. Imaginary

The statement The term "real number" is in contrast to the imaginary numbers. (RSchlafly) is IMO a false dichotomy and shouldn't be put in the article. I like the sentence: In classical physics, measurements of real world physical objects that can vary smoothly and continuously, like speed or temperature, are treated as real numbers. as it doesn't presuppose the existence of real numbers, but - analogously - you can state: In physics, measurements of real world physical objects that can vary smoothly and continuously - and exist from an angular compound and an absolute value - like current or voltage, are treated as complex numbers. --DiEb 16:30, 16 August 2008 (EDT)

I don't understand your point. The terms real number and imaginary number are in common use. Are you denying that? Are you saying that they are not in contrast? What do you want? RSchlafly 18:35, 16 August 2008 (EDT)

Of course, I don't deny the terms. And I acknowledge that the terms - which can be understood in their historical context - are contrasting. But I don't like the idea to think of the real numbers as more real than the imaginary numbers - especially not because of an etymological reasoning: To quote Kronecker: "Die ganzen Zahlen hat der liebe Gott gemacht, alles andere ist Menschenwerk"--DiEb 16:47, 17 August 2008 (EDT)

Huh? Real numbers are those people concretely use. You can express any amount of dollars with real numbers. What would 3 + 2i dollars mean? Or the length of a segment or a curve: it is "real" and expressed in real numbers. Imaginary numbers are just imaginary constructions to help our calculations. They don't refer to "real" concepts people use. Thus, the historical meaning of real vs. imaginary is still valid for normal people - and homeschooled students who use Conservapedia to learn. I absolutely agree with RSchlafly. SilvioB 16:53, 17 August 2008 (EDT)

Actually, you can express any amount of currency with rational numbers. That doesn't make the real numbers any less "real," does it? Using your logic, I could say "Pi is a real number. What does pi dollars mean? Nothing. Therefore, the real numbers don't exist." Again, imaginary numbers are not "just imaginary." They exist just as much as any other number, and have several practical applications.--Recorder 17:21, 17 August 2008 (EDT)

For currency, you use a subset of the rational numbers, i.e., finite decimal fractions (even less, generally: . To speak of dollars (or even $) is as surreal as to speak of 3 + 2i dollars. Real numbers which aren't rational, are just a figment of our imagination, invented to help our calculations, too--DiEb 17:25, 17 August 2008 (EDT)

I realized that and that's why I gave a second example, that of the length of a segment. Do you people really consider real numbers and imaginary numbers equal from the point of view of normal people? Let's not forget who is the target of this project. SilvioB 17:30, 17 August 2008 (EDT)

Presumably the purpose of articles like this is to provide accurate information, not to confirm misconceptions caused by terminology. Plenty of people use complex numbers regularly, and just because many people don't isn't a reason to have an inaccuracy in the article.--Recorder 17:42, 17 August 2008 (EDT)