Why is complex number support so essential to some of you?



#18

This is something I have been puzzling about for a while. For many of you, complex number support is essential and the elegant solution in the 15C makes it the prime choice whilst other machines that I rate highly but do not support complex numbers so well get dismissed.

Since my educational days, I've never touched or gone near complex numbers. Perhaps that was down to the way it was taught. For me it was 1980 GCE A-Level (UK people will understand) and having just looked at my original maths notes, we didn't do much other than the basic methods: +, -, *, /, Argand diagrams, modulus and argument and *nothing* to show real-life usage.

So.... I was wondering if those of you who need complex number support could give me some feel for their usage, the type of solutions you are looking for and the field of use?

If I can understand actual usage of complex numbers more, maybe I can find ways of using them programaticaly or at least round-out my education!

Thanks for any replies here :)

Mark


#19

Aside from the requirement of many of the forum members to solve polynomial equations that have complex roots, imaginary numbers are used extensively by the electrical engineering and aeronautics communities. I, myself am a mechanical engineer... Handling vector quantities necessitates the use of complex numbers for adequate mathematical treatment of the problem (which involves magnitude and direction). You must bear in mind that real numbers as well as imaginary numbers are are only special cases of the complex number set, so the ability to easily handle these complex numbers on a calculator can be of trememdous importance to users.

To really answer your question would take a lot of space and I would much rather refer you to an excellent book entitled: "An Imaginary Tale; The Story of the square root of minus 1", by Paul Nahin ISBN-10: 0-691-12798-0. --- JeffK


#20

Quote:
To really answer your question would take a lot of space and I would much rather refer you to an excellent book entitled: "An Imaginary Tale; The Story of the square root of minus 1", by Paul Nahin ISBN-10: 0-691-12798-0. --- JeffK

Can I crave everyone's indulgence and drag this thread off topic a bit? I'm part-way through this book and hugely enjoying it, but I tripped up slightly on p.5 - Diophantus's solution to finding the sides of a right triangle given its area and perimeter. After showing how Diophantus cleverly reformulates the problem to solving for one variable, Nahin continues:

"The first equation reduces to the identity 14 = 14, and the second to:

1/x + 14x + sqrt(1/(x^2)+196x^2) = 12

which is easily put into the form given above,

172x = 336x^2 + 24"

Now, for the life of me, I can't see how he gets from the first form to the second. I've played around with paper and pencil for ages, but am just not seeing it. In my defence, it's well over thirty years since I studied basic algebra, but he says this is easy!

Can somebody please point me to the basic technique I'm missing, and put me out of my misery? ;)

Best,

--- Les

[http://www.lesbell.com.au]


#21

Quote:
1/x + 14x + sqrt(1/(x^2)+196x^2) = 12

which is easily put into the form given above,

172x = 336x^2 + 24"

Now, for the life of me, I can't see how he gets from the first form to the second.


1/x + 14x + sqrt(1/(x^2)+196x^2) = 12

sqrt(1/(x^2)+196x^2) = 12 - 1/x - 14x

Square both sides (hint: (a+b+c)(a+b+c) = a(a+b+c)+b(a+b+c)+c(a+b+c)):

1/(x^2) + 196x^2 = 196x^2 + 1/(x^2) - 336x - 24/x + 172

0 = - 336x - 24/x + 172

0x = -336x^2 - 24 + 172x

-172x = -336x^2 - 24

172x = 336x^2 + 24

Edited: 6 June 2009, 10:55 p.m.


#22

Egan, you are "da man"!

#23

Ding!

Thanks, Egan - your hint made all the difference, and it's obvious, now. I'll be able to enjoy the rest of the book without feeling quite so daft. . .

Best,

--- Les

[http://www.lesbell.com.au]

#24

Hello!

Quote:
So.... I was wondering if those of you who need complex number support could give me some feel for their usage, the type of solutions you are looking for and the field of use?

For some years I did research work on radar imaging. This involves a lot calculations on electromagnetic wave propagation, where both the signal amplitude and phase were measured and processed. Complex numbers were essential to that kind of computation! An Hp-15 would have been no great help however, as millions of multiplications and additions had to be performed for every image. (I did this work in the early nineties and one of the fastest desktop comupters then awailable, the Sun SparcStation, usually needed a whole night for an 800 x 800 pixel image. The Hp-15 would have taken several thousand years for the same job...)

Since then, I haven't seen complex numbers again, I'm afraid.

Greetings, Max

#25

Mark,

I am an engineer, not a mathematician, and I use complex numbers simply because they afford solutions to many practical problems. Any mechanical or electrical problem in which both magnitude and phase are required can be handled easily using complex numbers. Most common mathematical functions have complex results in certain ranges, such as the square root of X < 0 or the Arc Sine of X > 1 or X < -1. The fact is that all general numbers are complex, with a real and imaginary part, and only special problem cases can be solved using only the real part. For example, the solution to a second order ordinary differential equation is ultimately complex, although the solution can be expressed as two apparently real parts.

Michael

#26

To simplify it almost to the point of lying, aside from simple DC circuit analysis, complex numbers are inseparable from virtually any electronics engineeiring work. Capacitors and inductors that are constantly either storing or releasing energy as a voltage or current changes throughout any waveform shift the phase, and off we go, leaving real-number-only calculations behind. Take that into mechanical engineering, and resonances and so on give you the same kind of thing.


#27

Quote:
To simplify it almost to the point of lying, aside from simple DC circuit analysis, complex numbers are inseparable from virtually any electronics engineeiring work. Capacitors and inductors that are constantly either storing or releasing energy as a voltage or current changes throughout any waveform shift the phase, and off we go, leaving real-number-only calculations behind. Take that into mechanical engineering, and resonances and so on give you the same kind of thing.

Depends entirely on what aspect of electronics engineering. I've been doing practical electronics engineering design for over 20 years and have rarely had the need to use complex numbers. When I have it's usually heavier data set computational stuff as others have mentioned, so in the realm of computers and not calculators.

Engineering school theory is different however, complex numbers are commonly used in learning many theoretical aspect. But as with most electronics engineering theory you learn, it's not so common to actually use in practice.

YMMV of course.

Dave.

#28

I'm an EE, and I think it is our discipline that is responsable for the demand for complex number calculations. I remember programming the basic operations in my Casio FX-602, which of course was far from the comfort that was offered by the 15C or later the 28C. In circuit analysis, complex numbers are essential, as has been said in several replys.
Today I work in non-destructive evaluation, and in eddy-current signal analysis, we need complex numbers all the time. Being able to carry out these calculations on a pocket calculator is quite useful in the field, or sometimes in a meeting. I wrote programs to calculate some of the post-processing treatments when multi-frequency eddy-current is used. It's an EE thing I'd say.

#29

Don't forget quantum mechanics - quantum amplitudes are complex numbers and although all observable quantities are real, to calculate probabilities these complex amplitudes have to be added together (usually by integration). Many operators (matrices) will have complex eigenvalues and eigenvectors too, and this will be true in engineering as well.

A more down-to-earth use is simply to use complex numbers as a convenient representation of 2-D vectors, to make them easy to add and to subtract without fiddling with R<->P conversions.

And so on! There are a *huge* number of applications of complex numbers!

Nigel


#30

Thank you all very much for the informative replies. I'm not into EE or the other fields although one area I work in gets close occasionally in a related way - audio engineering. I do this at the production and testing level rather than circuit or DSP algorithm design levels. Sometimes though when a new product is in development, it helps to try to understand a bit of what is going on under the hood. From what I have read above and the directions it has pushed me to look elsewhere on the net, I can see that complex numbers are a vital part of the mathematics. I doubt I'll ever get to the stage of being able to program my own DSP routines though.

Regarding use as vectors, yes, when being taught CNs it was difficult for me not to think of them in this way and always struck me as being a rather sledgehammer method to crack a simple nut but then perhaps that betrays my superficial understanding of complex numbers.

Interesting to read all your comments - thanks for the help and I hope it didn't seem too basic a question!

Mark


#31

Quote:
Regarding use as vectors, yes, when being taught CNs it was difficult for me not to think of them in this way and always struck me as being a rather sledgehammer method to crack a simple nut but then perhaps that betrays my superficial understanding of complex numbers.

As the EEs have explained above, and elsewhere (as a radio astronomer who does interferometry, I use a lot of Fourier transforms) the complex numbers undergo a lot of multiplication - as long as they are complex, you just go ahead and (complex) multiply and the answer comes out OK. Try thinking about how you'd do that in vector space - perhaps 1e6 rotations! Makes my head hurt!

#32

Would it be too nerdy to say, "I just like complex numbers?"

The book mentioned above is a great read. If you want a little bit of everything with a complex ending checkout the book Prime Obsession.


#33

Whenever you use magnitude at an angle it is a complex number whether expressed as real and imaginary components. These appear through electronics and mechanics. When adding vectors it is convenient to use their real and imaginary parts separately. Being at right angles they do not interact. Sam

#34

Electric Power Engineers use complex numbers extensively in areas such as network analysis for power systems, power system stability, short circuit analysis, load flow, Power factor analysis and correction, generator and transformer design, power system protection; and a whole host of others.

John


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