OT: mathematical conundrum

[pamela]

What's coming up next? Fourier transforms maybe. It's all too complex ;-)

[Cursitor Doom]

Maxwell's equations do you?

[GB]

I was puzzled what the conundrum was supposed to be, too.

Cursitor said (3rd post on this thread) "Bugger. You got it right straight away. I was hoping for a lengthy, argumentative thread on this!"

So, argue!

[Robin]

If you say so. But I'll leave you to correct all the sources which talk about the way imaginary numbers extend the system of real numbers to give the complex number system. I was never more than a wannabe practical physicist so I've always deferred to mathematicians when it come to things like that.

[John Rumm]

Indeed - partly a result of the early discoverers of them not really understanding that complex numbers were "real" and not just a mathematical slight of hand.

I find it easier to think of in two dimensions, where the real number line, is just a straight line where there is 0 deviation into the complex dimension.

[harry]

It's a system for manipulating vectors. The i or j is at right angles to the normal.

[NY]

I worked with a guy called John William Taylor, known as Bill, but nicknamed J Omega T by all the electronics engineers who were familiar with the mathematical term j?t which crops up all over the place as sin(j?t) in equations. Sort of nominative determinism: what other field could someone with initials JWT work in, except electronics?

[Cursitor Doom]

Well, since ? is 2*pi*f and f is frequency, I'd imagine no other field (apart from electrical engineering of course).

[polygonum]

Novello!

I for Novello (Ivor Novello)

[Chris Hogg]

Very good. We'll Gather Lilacs, anyone?

Now you've spoilt it!

[The Natural Philosopher]

The idea is that complex numbers are essentially two dimensional whereas ordinary numbers are not.

So they can represent things that have orthogonal components, like amplitude and phase in voltages/currents etc.

[The Natural Philosopher]

Except I learnt it as (?jt)

[ed]

whilst I reckon that j^2 = -1 and j^4 = +1 a better trick question i s what is the sq.root of -1.

Hint: Can -1^(0.5) + -1^(0.5) = 0 ?

Aside: Are imaginary (and complex) is some way less numeric than real numbe rs?

My take is that we start with the natural numbers 1,2,3... but then we find that the operation of subtracting a larger from a smaller number gives us new numbers that aren't in the natural numbers so we need 0 and the negati ve numbers. Division can give us numbers which are integers but also 'new' numbers whic h fall between the integers so we get the rational numbers like 5/23. [De cimal numbers are just special case rationals which are divided by powers o f ten.] There are also real numbers which have indefinitely long numerators and den ominators like 2(^0.5) or e or pi the latter being very close but not equal to 355/113. So then we find that taking the square root of a negative number needs a ne w number j (or i in many texts). I suppose the only thing that makes this conceptually different is that we can't put a pin in the number line and sa y it's "at" the pin (i.e. very close to the pin). However if you make the l eap into 2 dimensional numbers then j has a place at +1 on the 'imaginary' (or perpendicular) axis.

[Fredxxx]

That looks a daft way of presenting the imaginary part. Most signals involve some form of attenuation so would be of the form (a + jwt) to make it obvious which is the real and imaginary parts. I'm pretty sure my father was taught (jwt) BICBW.

[Cursitor Doom]

For doing this plotting in real life we use a Smith Chart, which is an horrifically complicated looking graph specifically designed by its eponymous creator to plot complex impedances in this way. Mr. Smith's chart appears to be the work of some kind of demonic, tormented genius. :)

[Bob Eager]

Oh God, I remember those. I had forgotten until you mentioned them again. Fresh nightmares.

[Robin]

Thanks for that, although believe it or not, I do still recall a *few* things from O-level. That said, the use of complex numbers for probability amplitudes is about all I can now recall from Rudolf Peierls' lectures on quantum electrodynamics :(

[Tim Streater]

Isn't this where we came in?

[Clive George]

Not quite. Maybe reading the post upside-down might help? :-)

[John Rumm]

Rotational kinetic energy or momentum?