[AR] Re: Career advice please.
- From: Norman Yarvin <yarvin@xxxxxxxxxxxx>
- To: arocket@xxxxxxxxxxxxx
- Date: Wed, 17 Mar 2021 11:46:37 -0400
On Wed, Mar 17, 2021 at 12:28:28AM -0400, Henry Spencer wrote:
(Due to some quirks of the sequence of undergrad courses I took, I hit
theoretical chemistry before ever being exposed to complex variables --
the algebra and calculus of complex numbers -- and that was a big mistake.
I literally couldn't understand half the equations. That course was an
ordeal. In hindsight, as soon as I recognized the problem, I should have
found a suitable math text and locked myself away for a weekend to work
through as much of it as possible.) (Don't sweat this exact example --
you won't see complex variables before university, and there's no reason
to take theoretical chemistry if you're not headed into chemistry as a
profession -- but it illustrates the problem.)
Complex numbers are surprisingly useful. To take one example, the
course of trigonometry as commonly taught in high school becomes
superfluous when one knows complex exponentials. Of course sines and
cosines and such are quite useful in engineering, but in a typical
'trig' course one is taught the art of proving that one trigonometric
formula is equal to another by finding a sequence of transformations
(with each trigonometric identity used having to be carefully
memorized), and it's much easier to do that with complex exponentials:
just translate all the sines and cosines into complex exponentials,
and the proof rather than being a puzzle becomes simple and
mechanical, with no arduous memorization required beforehand. I once
ran into a student who had just learned this and was so infuriated at
not having been taught the easy way from the start that he was
proclaiming that sines and cosines should be banned -- which of course
was going too far, but I could appreciate the sentiment.
And in general it is good to progress so that one's reaction to the
formula
e^(i*pi) = -1
stops being "wait, you've got to be kidding me, you start with one
irrational number and take the exponent of it to another irrational
number multiplied by the imaginary number, and you get MINUS ONE?" and
starts being "yeah, that's just a 180 degree rotation in the complex
plane".
The other pointless high school math course is geometry. Euclid is
said to have replied "There is no royal road to geometry" to a king
who asked for an easy introduction to the subject. These days there
is a royal road: analytic geometry, where all the points are given
coordinates and geometric problems are reduced to algebraic and
trigonometric ones. It's so much easier and more powerful that I've
never heard of anyone using Euclid's sorts of proofs in any sort of
practical engineering task. Nevertheless as a student you're probably
going to have to play the geometry game. It won't harm you; it's just
a waste of time. (Likewise, if you tried doing trig proofs the easy
way your teacher would probably disallow it.)
Calculus, in contrast, is how the world works. There's a reason why
Newton invented calculus when he was formulating his laws of
mechanics: he needed to. Pretty much all the laws of physics after
that (Maxwell's equations, Schroedinger's equation...) are likewise
expressed in terms of calculus. It's something not just to memorize
but to get comfortable with.
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