What are imaginary numbers good for?
Like Greg Mankiw, I am also stumped by this question.
Especially so if I am asked by a layperson.
Are you, too? (Referring to my readers who are into engineering, math and physics.)
Thursday, August 09, 2007
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8 comments:
I discussed this some time ago with a friend. He thinks that the best answer will be to say that its simply a mathematical artifact.
The problem comes from trying to give an intuitive explanation that directly relates to our daily lives. Some of the more abstract ideas in mathematics and physics are difficult to appreciate and relate to. Probably the best method will be to dig deeper in. This is what Roger Penrose is trying to bring to the general audience in his book "The Road To Reality".
But recently, I came across a discussion on the sum of divergent series that seems to justify complex numbers in a more elegant and slightly more accessible fashion.
I forgot to leave the relevant link. Voila...
http://cornellmath.wordpress.com/2007/07/28/sum-divergent-series-i/#more-108
Six words:
1. Complex analysis
2. Hermitian matrices.
I see only four words. And they still don't make any sense to most people.
Heck, I also think your descriptions are vague. :)
Regrettably I skipped complex analysis in college... But I would say it's a mathematical construct. Probably started when mathematicians said, "Hmm... is the square root of 4 is 2 (and -2), then what is the square root of -4?" Then someone came along and defined the i, and mathematicians took this definition and investigated how far we can go with this definition. Obviously, very far, unlike defining a group with binary operator a+b=47, which probably won't be a very useful mathematical construct.
At the fundamental level, in math we define concepts and objects (i.e. axioms) and see how far we can go. Some definitions are useful, some are not.
i was never good at it, had the same questions as the little kid unanswered.
all i got out of it was to learn how to avoid them.
btw nice blog, love the way you separate takchek from zogang
To elaborate on Elia's four words:
Complex analysis is required, amongst all things, to do some contour integrals. Contour integrals are basically integrals from -infinity to +infinity (along the whole of the real line). Complex analysis provides an easy way to do this using something called the Residue Theorem. Contour integrals pop up everywhere. In quantum mechanics, we use them to calculate the probability of certain measurements being made. In statistics in general, we need to integrate over many probability distributions from -infinity to +infinity.
Complex numbers pop up everywhere in physics. To find the resonance frequency of a driven, damped, harmonic oscillator, to calculate the superposition of waves, the Schroedinger Equation and Hermitian matrices in quantum mechanics,...
All mathematical objects are constructs.
hmm this is not going to help anyone outside EE much(?) but...i spent a good 6 months pondering about its use on electronics during ns...maybe this is of use in general...
in short, we are dealing with electrical signals, ie. some function of time. it aids the imagination to view these signals as composed of more basic building blocks. Fourier transform is pretty good, Laplace transform works even better. the building blocks are generally sine and cosine functions, with the laplace one including exponentially decaying/growing sines and cosines too.
dealing with trigo functions directly really sucks, but expressing it in terms of an imaginary number in its exponential form saves lots of math trouble.
in the laplace transform, this usage of imaginary numbers also makes it a convenient way of specifying the building block. it all fits together very nicely, and brings convenience too.
so i'd say its all abt simplifying the math (and avoid trigo), and presenting the stuff from a different perspective so things become easier.
the difficulties with imaginary numbers are: 1. the name is misleading, and confusion arises. 2.usage of mathematical technique without regard to why its a useful TOOL of expression and analysis in the first place 3. we learn the how, but are hardly taught the why
mostly i'd say that its use in the applied sciences is convenience, simplicity and most imptly a tool for expression.
its not abstract. its perspective.
[sorry can't help the usual rants.]
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