Difference between revisions of "Talk:Real number"

Revision as of 19:39, August 15, 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. EDIT: I've altered the sentence in question to reflect my comment.--Recorder 15:30, 15 August 2008 (EDT)