[ExI] Complex numbers (was: new entry from symphony of science)

[Darren Greer]

John wrote:

>The short answer is that the square root of negative one is essential if
mathematically you want to calculate how things rotate. It you pair up a
Imaginary Number(i) and a regular old Real Number you get a Complex Number,
and you can make a one to one relationship between the way Complex numbers
add subtract multiply and divide and the way things move in a two
dimensional plane, and that is enormously important. Or you could put it
another way, regular numbers that most people are familiar with just have a
magnitude, but complex numbers have a magnitude AND a direction.

Thanks. That's exactly the sort of answer I needed. Much better explained
than Wikipedia. which goes into a long explanation about electricity that
I'm certain is correct but lost me after a paragraph. My prof does promise
me that we will eventually delve further into complex numbers and that this
was really just an introduction, but I was fairly interested in it. I'd
heard of them in high school, but had never given them much thought.

>remember with i you get weird stuff like i^2=i^6 =-1 and i^4=i^10<

Yes, I think that's what drew me to them in the first place. I wondered why
you'd even bother squaring a number such as the square root of negative one
when the square roots cancelled each other out and you ended up with plain
old negative one, or when you cubed it you ended up with -i. Things are
getting most interesting in my classes anyway. I took advanced math in high
school, but we are in a place now where we have to step away from that
island of real-world logic that I was always comfortable with, and swim out
to a depth where the math as its own rules and internal logic that can at
times be counter-intuitive. I'm actually doing surprisingly well in all my
courses. I think I'll send a transcript to my old high school physics
teacher, whom I eternally frustrated with my poor performance. One of my
fellow students brought one of my novels into class yesterday for me to
sign, and I scribbled down Heron's Formula under my signature. :)



Darren



Many thought the square root of negative one (i) didn't have much practical
use until about 1860 when Maxwell used them in his famous equations to
figure out how Electromagnetism worked. Today nearly all quantum mechanical
equations have an"i" in them somewhere, and it might not be going too far to
say that is the source of quantum weirdness. The Schrodinger equation is
deterministic and describes the quantum wave function, but that function is
an abstraction and is unobservable, to get something you can see you must
square the wave function and that gives you the probability you will observe
a particle at any spot; but Schrodinger's equation has an "i" in it and that
means very different quantum wave functions can give the exact same
probability distribution when you square it; remember with i you get weird
stuff like i^2=i^6 =-1 and i^4=i^10<



2010/11/25 John Clark <jonkc at bellsouth.net>

> On Nov 24, 2010, at 10:52 PM, Darren Greer wrote:
>
> I'm having trouble understanding how the square root of negative one could
> have a practical application beyond abstract mathematics. Or even in
> abstract mathematics, for that matter.
>
>
> The short answer is that the square root of negative one is essential if
> mathematically you want to calculate how things rotate. It you pair up a
> Imaginary Number(i) and a regular old Real Number you get a Complex Number,
> and you can make a one to one relationship between the way Complex numbers
> add subtract multiply and divide and the way things move in a two
> dimensional plane, and that is enormously important. Or you could put it
> another way, regular numbers that most people are familiar with just have a
> magnitude, but complex numbers have a magnitude AND a direction.
>
> Many thought the square root of negative one (i) didn't have much practical
> use until about 1860 when Maxwell used them in his famous equations to
> figure out how Electromagnetism worked. Today nearly all quantum mechanical
> equations have an"i" in them somewhere, and it might not be going too far to
> say that is the source of quantum weirdness. The Schrodinger equation is
> deterministic and describes the quantum wave function, but that function is
> an abstraction and is unobservable, to get something you can see you must
> square the wave function and that gives you the probability you will observe
> a particle at any spot; but Schrodinger's equation has an "i" in it and that
> means very different quantum wave functions can give the exact same
> probability distribution when you square it; remember with i you get weird
> stuff like i^2=i^6 =-1 and i^4=i^100=1.
>
>  John K Clark
>
>
>
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>


-- 
"In the end that's all we have: our memories - electrochemical impulses
stored in eight pounds of tissue the consistency of cold porridge." -
Remembrance of the Daleks
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