Part I
How to get good with math?
Okay. Been there. Done that. Learned a
lot of it. Got a Ph.D. in it. Taught it.
Applied it. Published peer-reviewed
original research in it. Had a good
career applying it. Am using it as an
advantage in the core of my startup.
Broadly, for a career in computing, at
times math can be an advantage, one that
might be significant, e.g., get you
founder's stock in a startup that becomes
successful.
Math and computing can be a career
one-two punch: With some math you might
find an application, maybe a valuable one,
and then with some computing you get to do
the associated programming. Maybe then
you can show up at work one morning, maybe
after doing an all-nighter, and show the
final, useful, maybe quite valuable
results -- done deal, no waiting,
meetings, project approvals, etc.
This is a great time for both math and
computing, no doubt unique in all of
history. We are awash in what is in
historical terms just astounding
computing, and part of that is that a lot
of math is just a few clicks away at
Wikipedia, YouTube, in PDF files from word
processing with TeX, etc.
The first thing in math is arithmetic. Of
course, current computing eats arithmetic
problems much faster than Godzilla eats
fish.
You should know basic arithmetic for whole
numbers and fractions.
Then you should know the basics of ratios,
proportions, percentages, square roots and
exponents, logarithms, compound interest,
areas, and volumes. E.g., on my instance
of Windows 10 Home Edition (that I have as
a result of a sad situation, long story),
the key in the upper right corner of the
keyboard runs (opens, launches -- maybe
computing will think of more silly
synonyms) a version of an old
scientific-engineering pocket calculator
that has a lot of such arithmetic and
math.
Uh, that software is harder to learn to
use than the math it does! If you can
find out how to use such software in less
than a few hours of clicking guesses, you
can also learn the associated math!
Then on to algebra: That subject is
just doing arithmetic with symbols instead
of specific values, and that should be
really easy for anyone who can write math
expressions in a computer language.
Then on to plane geometry: The most
important idea there is triangles,
especially ones with one angle 90 degrees
-- right triangles. Then, sure, the
biggie result is the Pythagorean theorem
-- it gets applied throughout our economy
and has surprisingly far reaching
generalizations. For a proof, take 4 of
the right triangles and arrange them so
that they form a square where each side of
the square is the longest side of one of
the triangles and all the triangles are
inside the square. Then will also see a
square in the middle. Then write out the
area of the squares and, presto, bingo,
get the theorem. There are also 149 or so
other proofs.
For a while, I taught trigonometry (about
triangles) at Indiana University. The
best student in the class was a pretty
girl, and later I dated and married her --
see, math can be useful!
Then there is second year algebra where
learn some more, e.g., about, say,
(x + y)^n
for numbers x and y and a positive integer
n. From that can learn a lot about how
many HEADS might get if flip a fair coin
1000 times and can understand the math
shown in the baseball movie Moneyball.
Also that way can start to understand the
bell curve of Gauss and the powerful
law of large numbers.
Might study solid geometry, that is,
planes, lines perpendicular to planes,
spheres, circles on spheres, etc.
Next up, calculus: As you already know,
in a car the speedometer is the rate of
change of the odometer. The rate of
change of the speedometer is
acceleration. From Newton's law of
motion F = ma, that is, force is mass
times acceleration, in a car you feel the
force as you are pressed back in your seat
when your Tesla does 0 to 60 MPH in less
than 4 seconds! Going around in a circle
is also acceleration, and that's why when
you make a fast left turn the sack of
groceries slides to the right. So, rate
of change -- that is the first half of
calculus.
Given all the speedometer readings, should
be able to reconstruct the odometer
readings, and you can: That is the second
half of calculus and also is the way both
to define and to find the lengths of
curved lines (e.g., that the Webb
telescope is following), areas and volumes
of spheres, cylinders, etc.
How to learn calculus? Long story short,
I was not permitted to take calculus yet
so got a good calculus book and dug in.
Went to a better school and started on
their second year calculus and did fine.
So, I never took first year calculus --
learned it, taught it, applied it,
published research in it, learned math
analysis (that calculus is part of) far
beyond calculus, but never took a course
in it.
How to learn calculus: Get a good book.
At each section, (1) study the text and
examples and (2) work at least half the
exercises, especially the more difficult
ones, and check your work with the answers
in the back of the book. Don't go for
pre-calculus, high school calculus, or
high school advanced placement calculus.
Instead, just get a good book in CALCULUS.
Or get several such books. Then get a quiet
place, good light, big chair, clipboard
with a sharp, soft mechanical pencil, big,
soft eraser and dig in. Since calculus
has not changed much in 50+ years, you
don't need a recent book. Instead just do
an Internet search of used book sites.
I learned mostly from
Richard E. Johnson and Fred L.
Kiokemeister, Calculus with Analytic
Geometry.
It is VERY well written, even polished,
and with an unusually good collection of
exercises. When I used it, it was also
used at Harvard. You may be able to get a
used copy in very good condition for less
than $10.
For on-line sources, my opinion is that
nearly none of them are good. I've seen a
lot of the on-line video sources, and I
never saw a good one. E.g., last time I
looked at Khan Academy, I concluded that
they didn't understand calculus very well.
To learn calculus, or nearly anything in
math, whether you are in a course or not,
essentially you still need to study as I
have outlined. Learning math is not a
spectator sport.
If you have taught yourself to be good at
C++ and Win32, then you should have NO
trouble learning calculus QUITE WELL!
Of COURSE you can teach yourself calculus
and nearly anything in math: To keep up,
that is what college professors and anyone
applying math as a professional do.