Talk:Real number

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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)

As someone who scored in the 99th percentile on my math SATs as a teen, I seem to recall that "imaginary" has nothing to do with imagination. The square root if minus one - termed i - is called imaginary because it represents a parabola which never touches the axis to leave roots.

The "real" numbers have nothing to do with existence either. It's just that rational numbers can be expressed as the ratio of two integers, while real numbers cannot.

To some I say don't be misled by the connotations of the terms. To others I say do not deliberately mislead our young readers by adding patent nonsense to the math articles. There is nothing obscure, arcane, or hard to understand about the basic math concepts taught from primary up to high school levels. --Ed Poor Talk 16:40, 23 August 2008 (EDT)

Without knowing much about maths, I'm pretty sure that's not right, Ed. The term 'imaginary number' comes from Descartes' work La Geometrie (pg. 380), where he uses the term to distinguish between those possible roots of an equation which are actually possible according to his understanding, and those which are not, or imaginary. --AKjeldsen 18:13, 23 August 2008 (EDT)
Historically, you're absolutely right, mathematically, less so... Mathematicians often are not at their best when naming new inventions (elliptic functions springs to my mind), so we end up with somewhat unappropriate labels. And then, there's the tendency to overload words, as 'normal' (must be the most favorite mathematical term ever). But that's alright, as there's always the definition to rely on (and these tend to converge over time to a final state...) A problem arises when someone tries to interpret the label, using the -well- normal meanings. Much of the Sokal hoax relied on this...--DirkE 18:14, 25 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: \mathbb{Z} / 100. To speak of π dollars (or even $\sqrt{2}) 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)
In the physical world, it is not possible to find a line segment of length πcm (or even of length 1 cm), you may find approximations. For normal people, this may be of no interest. So, for them, we can skip the hole real is more real than imaginary stuff. We shouldn't mislead more interested pupils by this entry: in mathematics, you go from the simple to the more complex, but not from the real to the surreal. --DiEb 17:52, 17 August 2008 (EDT)

The Four Number Systems

Ok, this is something I remember getting taught in advanced maths a couple of years back.

At the beginning of modern arithmetic there was only one number system consisting of integers. You could add/subtract/multiply/divide, but you couldn't work with parts of numbers, which became a problem if you wanted to, say, divide 3 by 2.

To overcome this a second number system involving fractions and decimals was developed. Once again this worked for some time, but as maths and engineering grew people came to realise that they could not use trigonometric ratios or square roots with many numbers. They found that many of their answers could not be expressed as a fraction, and that there was seemingly no pattern to their decimal digits.

This lead to the development of a third number system, using surds to represent irrational numbers. This worked well for some time, and as all the numbers could be plotted on an ordinary axis (although a computer would be needed to plot the irrational ones) they were referred to as real numbers. However, there were still some numbers which could not be represented, those involving the square root of a negative number.

Finally, a fourth number system was developed, which incorporated imaginary numbers. Unlike the numbers found in the previous three systems these could not be represented on a normal graph, which lead to the development of the complex plane.

The moral of the story is that "real numbers" are "real" because they can be represented on a standard graph, "imaginary numbers" are "imaginary" because they cannot.

Now, looking at Wikipedia it seems this story is a rather simplistic summary of history, but then again my maths teacher was only using it to introduce us to the imaginary number system. Nonetheless it does show that imaginary numbers do actually exist, even if they may seem somewhat absurd to us. NormanS 18:54, 23 August 2008 (EDT)

Periodic Decimal Fractions are Ratios

Just to eliminate this [proof needed]tag:
  • Any decimal representation which repeats or recurs, like 1.86292929292929..., because these can be shown to be fractions[proof needed]

Now, here's an outline of the proof:

  • wlog, let 0 < x < 1 be a periodic decimal representation, i.e.,
  •  x = 0.\overline{a_1 a_2 ... a_n}. Then
  • x = \frac{a_1 a_2 ... a_n}{10^{n}-1}  as
  • x =a_1 a_2 ... a_n \cdot 10^{-n} +  a_1 a_2 ... a_n \cdot 10^{-2n} + a_1 a_2 ... a_n\cdot 10^{-3n} + ... and
  • 10^{-n} + 10^{-2n} + 10^{-3n} + ... = \frac{10^{-n}}{1- 10^{-n}}
--DirkE 17:46, 25 August 2008 (EDT)
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