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Originally Posted by Xei

That's an assertion, not an argument. The real line in the physical realm is nonsensical. Period. (Not true, of course).

It's a reiteration of what needs to be taken into account. I already made the argument.

Originally Posted by Xei

And yet again this is exactly the same as the real plane.

Like you say, if you walk the distance from 0 to i, you have not travelled i units. You have travelled the norm of the difference, which is 1.

If you walk the distance from (0, 0) to (0, 1), you have not travelled (0, 1) units. You have travelled the norm of the difference, which is 1.

These two are literally in exact analogy. In fact they are exactly the same statement.

Again, (0, 0) and (0, 1) are points, not measures. They are like addresses. i on the y-axis is a measure away from the origin. Having 1 for a y-coordinate is completely legitimate. Having i for a y-coordinate is fantasy.

Originally Posted by Xei

The existence of complex numbers does not somehow entail the existence of complex distances any more than the existence of ordered pairs entails the existence of ordered pair distances. Your belief that they did was a mistaken one. That has been thoroughly explained now and there is no argument left here. Complex numbers have real norms, just like ordered pairs.

The norms are real, as in actual and legitimate. The complex numbers are not. This has been thoroughly explained and now there is no argument left.

Originally Posted by Xei

'Number' is a new and undefined term.

Up to now we have been talking about what makes a mathematical object 'real'. You defined this to be conditioned on the object corresponding to something physical. (3, 1) and 3 + i are 'real' by this definition.

Now you're talking about numbers, which is something different. You need to give a precise definition of what you mean by 'number', and then we can easily determine what kind of thing is one.

But non-numberhood will not imply non-reality.

"Number" is difficult to define, but it is essentially an amount (of something) that exists or can exist in reality and is involved in reality consistently and logically. 3 + i is not a logical concept. 3 is.

"Number" is new and undefined?

Originally Posted by Xei

They don't 'happen on the complex plane', they happen on the real plane. You define a rotation operation on the real plane and that makes it the complex plane; the complex plane is a special type of real plane, 'complex' referring to the way in which the objects in it act, not what they are, which is just ordered pairs. I think you're only rejecting this because of some vague, mistaken stereotype that 'complex' means something weird and magical. If we called it the 'real plane with rotations' instead I don't think you'd actually have a problem with it.

When it is "made" the complex plane, the plane that has come about is fiction. Rotations are real, but not when they happen on what is/becomes the complex plane.

I want to add here that I am arguing my perspective, but I often do that to test it. My most landmark mind changes have happened only after long, intense series of debates where I rigorously tested my beliefs. That is how I stopped being a Christian and later stopped being a Democrat. The use (necessity?) of imaginary and complex numbers in electronics throws a monkey wrench into my perspective, so I do have some doubt about my overall position.

Originally Posted by khh

I
I believe what tkdyo said (or meant) is that a signal that's purely imaginary would be 90 degrees behind or in front of (depending on the sign) a signal that's purely real. But i*i = -1, which equals 180 degrees, as
-sin(2π∙f∙t) = sin(2π∙f∙t + π)

I'd say that the representation of inductance and capacitance as the imaginary part of impedance is an explicit use, and (as long as you're not exceeding the operational zone for which the components are designed) the math and the actual measured signals do agree. In fact I don't know how to analyse a system containing both inductors and capacitors without the use of complex numbers.

Also all parts of the complex power S = |S|e^jφ = P + jQ for a component can be measured directly from a signal if you have sinus input. There is a wikipedia article on this too, but it's not so good. This is a particularily good illustration I believe, because it can be directly observed that the real part of the power, P, is usable by the component while the imaginary part of the power, Q, is not.

I guess you could argue that while this is a convenient way of writing it, the same could be achieved using xy- and polar-coordinates. My guess is that it would be possible (I don't know for certain, but it would make sense for what I've shown at least), but in that case I'd ask what makes that a more "real" representation of the phase-shift of a sine than imaginary numbers.

I thought it might be something like what you said at the end. If it is the case, my view is that what would make the imaginary figures less real is that I cannot conceptualize sqrt(-1) as anything logical beyond its value as a hypothetical. Does the existence of -1 (represented by the squaring of i) as 180 degrees have anything to do with the fact that the x-coordinate of -1 on the unit circle is at 180 degrees?

Originally Posted by tkdyo

UM, that sounds very interesting about the multiple dimensions and definitely sounds like its worth the time to look up. I remember we did use i in a basic relativity class I took, but Im in no position to try and teach anything about that as it was only one class lol.

That's an even more difficult area to tackle, but I am determined to do it. I am too philosophical not to learn relativity as well as I can.

Originally Posted by Universal Mind

I thought it might be something like what you said at the end. If it is the case, my view is that what would make the imaginary figures less real is that I cannot conceptualize sqrt(-1) as anything logical beyond its value as a hypothetical. Does the existence of -1 (represented by the squaring of i) as 180 degrees have anything to do with the fact that the x-coordinate of -1 on the unit circle is at 180 degrees?

Yes. But I guess it's easier to say that it follows directly from Eulers formula e^iφ = cos(φ) + i∙sin(φ).

Originally Posted by Universal Mind

Again, (0, 0) and (0, 1) are points, not measures. They are like addresses. i on the y-axis is a measure away from the origin. Having 1 for a y-coordinate is completely legitimate. Having i for a y-coordinate is fantasy.

Okay? i is (0, 1), it doesn't have i as a y coordinate.

The norms are real, as in actual and legitimate. The complex numbers are not.

No, this is just an assertion again. All of the arguments so far have been mistaken, including the mistaken thing about complex norms being complex numbers.

"Number" is difficult to define, but it is essentially an amount (of something) that exists or can exist in reality and is involved in reality consistently and logically. 3 + i is not a logical concept. 3 is.

You've just introduced more undefined terms; I don't have any idea what it means for a mathematical object to be 'logical'. The most obvious interpretation is 'consistent' but complex numbers are consistent.

But in any case I think this is just a tangent. The issue wasn't about whether or not imaginary numbers are 'numbers', which is just semantics really, it's about whether or not they're 'imaginary', that is to say, well grounded mathematical objects, which is important.

When it is "made" the complex plane, the plane that has come about is fiction. Rotations are real, but not when they happen on what is/becomes the complex plane.

I want to add here that I am arguing my perspective, but I often do that to test it.

Then I think it's time to change it. All you're left with now is falling back on assertions like the one in this quote; all of the arguments have gone, it's just an ad hoc rejection of the conclusion. What you're basically saying here is that you're fine with all of these concepts, unless they give you complex numbers, in which case you automatically rule them out. You gave a definition of real and I established fairly unambiguously that complex numbers satisfy it.

As I said in one of my posts, I think your aversion was based on an understandable misconception of the basis of complex numbers; basically one which is antiquated. When they were first used it was a purely symbolic leap in the dark which happened to generate the correct answer to the problem at hand (solving cubic equations with real roots)... I suppose it could even be the case that they could have been inconsistent and formally manipulating them only worked in that specific context for some underlying reason.

Interestingly all of the above also applies to calculus, when it was first formulated by Leibniz and Newton. In fact there's an almost comically striking symmetry. Calculus was written off as rubbish by the British empiricist philosopher Berkely (although he granted that the formal manipulation seemed highly useful), and in fact he was in a strong sense correct to do so: it doesn't really make any sense to divide an infinitely small quantity by an infinitely small quantity, and end up with some specific number. Complex numbers on the other hand were written off as rubbish by the continental rationalist philosopher Descartes, and he was also in a sense correct to do so. In fact Descartes was the one to coin the term 'imaginary'; the only difference is that by a historical accident, we've kept the term imaginary, but we don't refer to infinitesimals as 'impossibles' or something.

But that was many centuries ago. Although these pieces of mathematics were fantastical, intuitive leaps at the time, new philosophical and mathematical understandings have since shown that they are indeed well grounded. For calculus, the formal basis is Analysis, based upon Weierstrass's definition of a limit. I have already explicated the formal basis of complex numbers; Gauss was one of the chief architects of this. The philosophical insights were largely similar to each other, essentially being based on a strict delineation of what mathematics actually is.

Whilst writing this I checked out the Wikipedia article and found that the history section is amusingly pertinent. Apparently even Euler made the same mistake with the square roots thing. So really your raising of these problems speaks well for your philosophical criticality. It's just that the problems were actually only problems ostensibly, and have since been thoroughly addressed.

Complex number - Wikipedia, the free encyclopedia

The impetus to study complex numbers proper first arose in the 16th century when algebraic solutions for the roots of cubic and quartic polynomials were discovered by Italian mathematicians (see Niccolo Fontana Tartaglia, Gerolamo Cardano). It was soon realized that these formulas, even if one was only interested in real solutions, sometimes required the manipulation of square roots of negative numbers. As an example, Tartaglia's cubic formula gives the solution to the equation x3 − x = 0 as
The three cube roots of −1, two of which are complex

\frac{1}{\sqrt{3}}\left((\sqrt{-1})^{1/3}+\frac{1}{(\sqrt{-1})^{1/3}}\right),

At first glance this looks like nonsense. However formal calculations with complex numbers show that the equation z3 = i has solutions –i, {\scriptstyle\frac{\sqrt{3}}{2}}+{\scriptstyle\fra c{1}{2}}i and {\scriptstyle\frac{-\sqrt{3}}{2}}+{\scriptstyle\frac{1}{2}}i. Substituting these in turn for {\scriptstyle\sqrt{-1}^{1/3}} in Tartaglia's cubic formula and simplifying, one gets 0, 1 and −1 as the solutions of x3 – x = 0. Of course this particular equation can be solved at sight but it does illustrate that when general formulas are used to solve cubic equations with real roots then, as later mathematicians showed rigorously, the use of complex numbers is unavoidable. Rafael Bombelli was the first to explicitly address these seemingly paradoxical solutions of cubic equations and developed the rules for complex arithmetic trying to resolve these issues.

The term "imaginary" for these quantities was coined by René Descartes in 1637, although he was at pains to stress their imaginary nature[15]

[...] quelquefois seulement imaginaires c’est-à-dire que l’on peut toujours en imaginer autant que j'ai dit en chaque équation, mais qu’il n’y a quelquefois aucune quantité qui corresponde à celle qu’on imagine.
([...] sometimes only imaginary, that is one can imagine as many as I said in each equation, but sometimes there exists no quantity that matches that which we imagine.)

A further source of confusion was that the equation sqrt(-1)^2 = sqrt(-1)*sqrt(-1) = -1 seemed to be capriciously inconsistent with the algebraic identity sqrt(ab) = sqrt(a)sqrt(b), which is valid for non-negative real numbers a and b, and which was also used in complex number calculations with one of a, b positive and the other negative. The incorrect use of this identity (and the related identity 1/sqrt(a) = sqrt(1/a)) in the case when both a and b are negative even bedeviled Euler. This difficulty eventually led to the convention of using the special symbol i in place of sqrt(-1) to guard against this mistake[citation needed]. Even so Euler considered it natural to introduce students to complex numbers much earlier than we do today. In his elementary algebra text book, Elements of Algebra, he introduces these numbers almost at once and then uses them in a natural way throughout.

In the 18th century complex numbers gained wider use, as it was noticed that formal manipulation of complex expressions could be used to simplify calculations involving trigonometric functions. For instance, in 1730 Abraham de Moivre noted that the complicated identities relating trigonometric functions of an integer multiple of an angle to powers of trigonometric functions of that angle could be simply re-expressed by the following well-known formula which bears his name, de Moivre's formula:

(\cos \theta + i\sin \theta)^{n} = \cos n \theta + i\sin n \theta. \,

In 1748 Leonhard Euler went further and obtained Euler's formula of complex analysis:

\cos \theta + i\sin \theta = e ^{i\theta } \,

by formally manipulating complex power series and observed that this formula could be used to reduce any trigonometric identity to much simpler exponential identities.

The idea of a complex number as a point in the complex plane (above) was first described by Caspar Wessel in 1799, although it had been anticipated as early as 1685 in Wallis's De Algebra tractatus.

Wessel's memoir appeared in the Proceedings of the Copenhagen Academy but went largely unnoticed. In 1806 Jean-Robert Argand independently issued a pamphlet on complex numbers and provided a rigorous proof of the fundamental theorem of algebra. Gauss had earlier published an essentially topological proof of the theorem in 1797 but expressed his doubts at the time about "the true metaphysics of the square root of −1". It was not until 1831 that he overcame these doubts and published his treatise on complex numbers as points in the plane, largely establishing modern notation and terminology. The English mathematician G. H. Hardy remarked that Gauss was the first mathematician to use complex numbers in 'a really confident and scientific way' although mathematicians such as Niels Henrik Abel and Carl Gustav Jacob Jacobi were necessarily using them routinely before Gauss published his 1831 treatise.[16] Augustin Louis Cauchy and Bernhard Riemann together brought the fundamental ideas of complex analysis to a high state of completion, commencing around 1825 in Cauchy's case.

The common terms used in the theory are chiefly due to the founders. Argand called \cos \phi + i\sin \phi the direction factor, and r = \sqrt{a^2+b^2} the modulus; Cauchy (1828) called \cos \phi + i\sin \phi the reduced form (l'expression réduite) and apparently introduced the term argument; Gauss used i for \sqrt{-1}, introduced the term complex number for a + bi, and called a2 + b2 the norm. The expression direction coefficient, often used for \cos \phi + i\sin \phi, is due to Hankel (1867), and absolute value, for modulus, is due to Weierstrass.

Later classical writers on the general theory include Richard Dedekind, Otto Hölder, Felix Klein, Henri Poincaré, Hermann Schwarz, Karl Weierstrass and many others.

Originally Posted by Xei

Okay? i is (0, 1), it doesn't have i as a y coordinate.

The imaginary axis values are spaced i apart vertically. i is where (0, 1) would be on the
Cartesian plane, but it is labeled with something imaginary. The points can be labeled the way I presented (0, i), but the more formal way to present it is 0 + i. A point 4 horizontal units and 2i vertical units from the origin is generally called 4 + 2i, but another form would be (4, 2i) due to its mimicking of the real/Cartesian plane. I used the less formal (0, i) to illustrate my reasoning.

Originally Posted by Xei

No, this is just an assertion again. All of the arguments so far have been mistaken, including the mistaken thing about complex norms being complex numbers.

You've just introduced more undefined terms; I don't have any idea what it means for a mathematical object to be 'logical'. The most obvious interpretation is 'consistent' but complex numbers are consistent.

But in any case I think this is just a tangent. The issue wasn't about whether or not imaginary numbers are 'numbers', which is just semantics really, it's about whether or not they're 'imaginary', that is to say, well grounded mathematical objects, which is important.

I have been showing disagreement with the complex plane by pointing out its representation and saying that it is not fully based in reality. I have discussed representing it in the physical realm for the purpose of illustrating its purely conceptual and unreal diagram nature. I know the imaginary units on the imaginary axis are not really distances. That is exactly my point. The idea is that the complex plane looks like this...




It's just a conceptual diagram, a game with ficitious ideas. There is no such two dimensional reality apart from such an illustration. If you agree with that, then I am confused on what we are even disagreeing on at this point. A lot of semantics involved.

Originally Posted by Xei

You've just introduced more undefined terms; I don't have any idea what it means for a mathematical object to be 'logical'. The most obvious interpretation is 'consistent' but complex numbers are consistent.

By "logical" I mean it follows the laws of logic/reality. -1 has no true square root. It's just a concept with a symbol. It is based on absurd premise. That is why there is no such amount. They are also inconsistent with previously established rules, such as the product rule concerning radicals. That is not proof alone, but suggestive evidence.

Originally Posted by Xei

But in any case I think this is just a tangent. The issue wasn't about whether or not imaginary numbers are 'numbers', which is just semantics really, it's about whether or not they're 'imaginary', that is to say, well grounded mathematical objects, which is important.

Then I think it's time to change it. All you're left with now is falling back on assertions like the one in this quote; all of the arguments have gone, it's just an ad hoc rejection of the conclusion. What you're basically saying here is that you're fine with all of these concepts, unless they give you complex numbers, in which case you automatically rule them out. You gave a definition of real and I established fairly unambiguously that complex numbers satisfy it.

As I said in one of my posts, I think your aversion was based on an understandable misconception of the basis of complex numbers; basically one which is antiquated. When they were first used it was a purely symbolic leap in the dark which happened to generate the correct answer to the problem at hand (solving cubic equations with real roots)... I suppose it could even be the case that they could have been inconsistent and formally manipulating them only worked in that specific context for some underlying reason.

Interestingly all of the above also applies to calculus, when it was first formulated by Leibniz and Newton. In fact there's an almost comically striking symmetry. Calculus was written off as rubbish by the British empiricist philosopher Berkely (although he granted that the formal manipulation seemed highly useful), and in fact he was in a strong sense correct to do so: it doesn't really make any sense to divide an infinitely small quantity by an infinitely small quantity, and end up with some specific number. Complex numbers on the other hand were written off as rubbish by the continental rationalist philosopher Descartes, and he was also in a sense correct to do so. In fact Descartes was the one to coin the term 'imaginary'; the only difference is that by a historical accident, we've kept the term imaginary, but we don't refer to infinitesimals as 'impossibles' or something.

But that was many centuries ago. Although these pieces of mathematics were fantastical, intuitive leaps at the time, new philosophical and mathematical understandings have since shown that they are indeed well grounded. For calculus, the formal basis is Analysis, based upon Weierstrass's definition of a limit. I have already explicated the formal basis of complex numbers; Gauss was one of the chief architects of this. The philosophical insights were largely similar to each other, essentially being based on a strict delineation of what mathematics actually is.

Whilst writing this I checked out the Wikipedia article and found that the history section is amusingly pertinent. Apparently even Euler made the same mistake with the square roots thing. So really your raising of these problems speaks well for your philosophical criticality. It's just that the problems were actually only problems ostensibly, and have since been thoroughly addressed.

That is fascinating. I know there has been a lot of stuff, and even impressive stuff in the area of technology, that has been accomplished with imaginary and complex numbers (I think the British use "complex" for all of them while Americans call them "imaginary" when there is no real value in the expression.). I see that, and I see their use, but I am a huge skeptic. I see them as useful hypotheticals. You've made a ton of good points worth thinking about, but I still have not swallowed the realistic validity of sqrt(-1). I might have a different view later.

I think we have gone about as far as we are going to go for now. I am going to rest it here. You can have the last word if you want to.

Originally Posted by Universal Mind

By "logical" I mean it follows the laws of logic/reality. -1 has no true square root. It's just a concept with a symbol. It is based on absurd premise. That is why there is no such amount.

Why are you assuming that a real number coordinate system isn't just a conceptual diagram?

My main point is really that with both mathematical and philosophical clarity, your issues with complex numbers (which refers to any number a + bi rather than imaginary which only refers to bi, by the way) should be solved.

Philosophically, different number fields are abstracted from different things, and having a single physical basis in mind can lead you astray, as it led the Pythagoreans astray.

And mathematically, the complex plane is really a very benign formal construction, a fact which is somewhat obfuscated by the language and the notation. Something I must remark on looking at your final response is that in the careful modern approach, we do not simply add a new entity 'i'; rather, we literally use i as a shorthand notation for points on the plane. Namely, we write (a, b) (where a and b are both real) as a + bi; they are literally exactly the same, it just makes shunting the symbols a little simpler and clearer. Don't think of i as a special entity, just think of it as a marker. You could just as well write a' + b".

That is my summary and I hope it turns out to be edifying.

Beyond this discussion, I really recommend you look into the maths of complex numbers even if just on a provisional basis, because being familiar with what something is and what it does is usually important for the 'meta' considerations, and because a lot of the most beautiful mathematics is found here. In fact it contains what is generally considered to be the most beautiful result in all mathematics, e^i*pi + 1 = 0; that justifies enquiry for aesthetic considerations alone.

I also really really recommend you look at the theory of abstract algebra. It will be totally different from anything you've seen before; it's really awesome and I think you'll like it. It will also provide far greater insight into what maths actually is, especially through ideas like actions and isomorphisms. Groups and rings is the place to start; I have notes for the former on my laptop which I can email you if you'd like.

Originally Posted by Wayfaerer

Why are you assuming that a real number coordinate system isn't just a conceptual diagram?

See post #55.

Originally Posted by Xei

Beyond this discussion, I really recommend you look into the maths of complex numbers even if just on a provisional basis, because being familiar with what something is and what it does is usually important for the 'meta' considerations, and because a lot of the most beautiful mathematics is found here. In fact it contains what is generally considered to be the most beautiful result in all mathematics, e^i*pi + 1 = 0; that justifies enquiry for aesthetic considerations alone.

I also really really recommend you look at the theory of abstract algebra. It will be totally different from anything you've seen before; it's really awesome and I think you'll like it. It will also provide far greater insight into what maths actually is, especially through ideas like actions and isomorphisms. Groups and rings is the place to start; I have notes for the former on my laptop which I can email you if you'd like.

I know a good bit about complex numbers and have taught them on a high school level (algebra II and pre-calculus), but there is always more to learn. Abstract algebra sounds like fascinating stuff, and I would love to look into it. I might request your personal notes after my initial research. Thanks.