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What are imaginary numbers?

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Re: What are imaginary numbers?

#121

Post by gizmo686 »

shou4577 wrote: post #37

I have mixed feelings about this explanation. On the one hand, the mathematician inside of me is raging "this is neither specific, nor rigorous!" In my opinion, this definition is as close to useless as a mathematical definition can be, since the person who has this (and only this) will be hard pressed to answer any other questions involving complex numbers. For example, what is multiplication by i+1? Without further…

As a math student I can say that if the teacher explains something in a way that

Re: What are imaginary numbers?

#122

Post by d0m »

fexl wrote: post #104
ColinWright wrote: post #88

Earlier quoted context omitted.

This sounds profound, but is wrong on so many levels. In a sense everything about mathematics is arbitrary, but there's a consistency and structure that makes such a statement unhelpful and misleading. Consider. If you're content with the counting numbers then we can construct the negative numbers. These have the specific property that when added to the positive number of the same size we get zero, But most people ar…

Beautiful exposition. Even in the late 1700s, many mathematicians rejected the use of mere negative numbers, viewing them as anomalies which indicated that one had phrased a problem wrong to begin with. On the other hand, Euler understood everything very well and even calmly explained how to take logarithms of complex numbers, which bewildered most of his contemporaries. Someone once said that a lot of confusion coul…

Thanks for this citation. Forward, backward and lateral.. for some reason, this is what make the most sense to me in all these explanations of complex numbers. I guess you could also go upward/downward. And then, in a fourth or even nth dimension.

Re: What are imaginary numbers?

#123

Post by aaronbrethorst »

lmkg wrote: post #83
stiff wrote: post #49

A complex "number" (don't think of it as of a number! think of it like you would think of a vector, group, ring or any other abstract structure) is just an ordered pair of real numbers that behaves in a certain predefined way when being added to another complex number or multiplied by it. For an introduction, to avoid unnecessary confusion, it is best to write such "numbers" as ordered pairs using the notation: (a,b)…

Understanding a mathematical concept doesn't mean being able to perform computations using it. It means having an intuitive understanding for what it can represent and how to use it, and how to interpret concepts that use it. Defining complex numbers as a bunch of arbitrary arithmetic operations on tuples lends nearly zero understanding, no matter how good you get at performing that arithmetic. Understanding complex…

Thank you for explaining in three simple paragraphs why I hated every calculus professor I ever had.

Re: What are imaginary numbers?

#124

Post by scotty79 »

I tend to think that Real numbers is the first kind of numbers that have nothing to do with reality. There are no perfect circles, squares or triangles in reality. Real things can only look like circle if you don't look close enough. I think same goes for sinusoidal waves and everything else.

All fundamental physical laws that contain e or pi or event sqrt(2) seem suspicious to me.

Re: What are imaginary numbers?

#125

Post by gizmo686 »

esrauch wrote: post #68
alok-g wrote: post #34

Can someone please also do this for: 1. Matrices, especially matrix multiplication. Unlike matrix addition, multiplication is defined in a very weird way. I think I understand where it is coming from -- defining it that way allows representing and solving linear equations. More insights, however, would help. 2. Dot and cross products. E.g., the magnitude of dot product in 3D is a.b.cos(theta), while for cross product…

Regarding 4, don't believe anyone who claims it's an intuitive result; it is simply a properly of real numbers that you cannot have nonzero infinitesimals and that any two distinct numbers have a number between them that is not equal to either (infinitely many in fact). You can construct alternate number lines that do allow nonzero infinitesimals and then .9999... actually is not equal to 1 under that number line; th…

There do exist number systems where .999.../=1, however, they are not a strict superset of the reals. If it were, then any operation involving only real numbers would behave identicly to the real number system.

Also, this property is not a mere convention, but rather a nessasary result of what we want the number line to be. For example, assume that X<Y. Consider Z=(X+Y)/2. Z=X/2+Y/2. X<Z<Y. I have just shown, using basic algebra, that for any 2 distinct numbers, their is a third number between them. If that were not the case, then at least one of my steps must have been invalid.

Re: What are imaginary numbers?

#126

Post by zvrba »

From the OP's question:

> When I tried to calculate the square root of -1 on my calculator, it gave me an error.

This was a red flag to me. This person takes the answer from a calculator as the ultimate truth, whereas it (obviously) is not. I wonder how he would have reacted had he had an advanced calculator (HP50g or some TI) and got "i" as the answer instead of an error.

> To this day I do not understand imaginary numbers. It makes no sense to me at all.

Coming from a mixed EE/CS background, I had a lot of math and math-heavy classes in the first 2.5 years (linear algebra, real, vector and complex analysis, discrete math, systems theory)..

What I observed during the 5-yr master's degree was that the people who were seeking "meaning" in math courses were the same people who 1) had the most problems with math exams, 2) had the least ability to apply the math to other domains in order to do something useful.

What worked for me was to accept that math is manipulation of abstract symbols according to some rules, combined with some ingenuity (i.e., when you have to consider external information in addition to the rules).

In the case of complex numbers, you introduce a new abstract symbol called "i" with the property that i X i=-1. Other rules that you (hopefully) learned earlier still apply. So computing (x+y)X(x+z) or (2+3i)X(2-6i) is in essence the same process except that you replace i^2 with -1 when you encounter it.

I actually enjoyed doing math on that level: as a manipulation of abstract rules and invention of new rules (theorems).

In simple EE, inducntances and capacitances are modeled as pure imaginary "resistances"). However, I somehow figured out how to make use of math (not only complex numbers) in other domains. I made use of math in electronics for drawing Bode plots, information theory, systems theory, and even economics! (In economics, I was reading a lengthy chunk of text with some summations, only to realize that, in actuality, integration was described and everything could be summarized to few sentences explaining the meaning of the involved integral. On the exam, I gave MY explanation -- based on the integral -- instead of the lecturer's, and I passed!)

To me the problem seemed to be that lecturers in other subjects resorted to awkward, special-cased explanations instead of showing us how to model the problems at hand with the math we have already learned.

Re: What are imaginary numbers?

#127

Post by zvrba »

joe_the_user wrote: post #97

Aside from people's very worthwhile answers describing the complex number system, I think it is worth mentioning that the use of the term "imaginary" is an unfortunate historical remnant. In experience, a lot of the average student confusion comes from their trying to get their head around the naive meaning of imaginary. Now that modern mathematics understands that all number systems are more or less games with axiom…

Also ironic is that real numbers may not be "real" (i.e., exist) at all. The uncountable part which is THE part that completes rationals to continuity cannot be described or generated in any way since there are only COUNTABLY many different computer programs (or mathematical formulas).

Re: What are imaginary numbers?

#128

Post by [deleted] »

gizmo686 wrote: post #117
yequalsx wrote: post #94

Earlier quoted context omitted.

Additive inverses are unique in any ring. The subtle things I skipped over because of the technicalities involved is how do I know the distributive property holds for negative integers. Indeed, what is a negative integer? How does one get them from the natural numbers? Suffice it to say that this can all be defined in a consistent, precise way and everything works out.

Actually, we only think that it can be described in a consistent way. We have proven it consistent using set theory , but Godwell's theorems tell us that we cannot prove a system to be consistent without using something outside of said system (unless the system is inconsistent). This means that there is some level in our chain of proofs that cannot be proven consistent.

[deleted]

Re: What are imaginary numbers?

#129

Post by jasomill »

lmkg wrote: post #83
stiff wrote: post #49

A complex "number" (don't think of it as of a number! think of it like you would think of a vector, group, ring or any other abstract structure) is just an ordered pair of real numbers that behaves in a certain predefined way when being added to another complex number or multiplied by it. For an introduction, to avoid unnecessary confusion, it is best to write such "numbers" as ordered pairs using the notation: (a,b)…

Understanding a mathematical concept doesn't mean being able to perform computations using it. It means having an intuitive understanding for what it can represent and how to use it, and how to interpret concepts that use it. Defining complex numbers as a bunch of arbitrary arithmetic operations on tuples lends nearly zero understanding, no matter how good you get at performing that arithmetic. Understanding complex…

I'd highly recommend Tristan Needham's Visual Complex Analysis to anyone interested in the subject, for exactly this reason . . . and, subsequently, Henri Cartan's Elementary Theory of Analytic Functions of One or Several Complex Variables as proof that, nevertheless, "algebraic" need not imply "boring" or "computational". As a silly example, I'll never forget Cartan's definition of "2π" as the unique positive real number such that the kernel of the homomorphism "t -> e^it" from the additive group of reals to the multiplicative group of unit-length complexes is the set of integer multiples of 2π.

Re: What are imaginary numbers?

#130

Post by ColinWright »

gizmo686 wrote: post #125
esrauch wrote: post #68

Earlier quoted context omitted.

Regarding 4, don't believe anyone who claims it's an intuitive result; it is simply a properly of real numbers that you cannot have nonzero infinitesimals and that any two distinct numbers have a number between them that is not equal to either (infinitely many in fact). You can construct alternate number lines that do allow nonzero infinitesimals and then .9999... actually is not equal to 1 under that number line; th…

There do exist number systems where .999.../=1, however, they are not a strict superset of the reals. If it were, then any operation involving only real numbers would behave identicly to the real number system. Also, this property is not a mere convention, but rather a nessasary result of what we want the number line to be. For example, assume that X<Y. Consider Z=(X+Y)/2. Z=X/2+Y/2. X<Z<Y. I have just shown, using b…

  > There do exist number systems where .999.../=1,
  > however, they are not a strict superset of the reals.
The hyper-reals of non-standard analysis are, in fact, a strict superset of the reals.

  > If it were, then any operation involving only real numbers
  > would behave identically to the real number system.
Why is that a problem? Seems to me that that's desirable.
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