Archive for March, 2011

blogging calc i

without wanting to commit myself… here goes.
i’m *undercommitted* this quarter goodness knows.

my calculus blogging from spring ’09
might come in handy (but the Calc III stuff
is mixed in with the Calc I).
for that matter, the common errors page (not by me)
that i cited yesterday is *bound* to come in handy.

heck. learning math on the web?
just like learning anything else,
there’ll be plenty of good info… more than enough
to build a course around…
in the ever-amazing wikipedia. let’s see.

consider this list of calculus topics.
hmmm. it doesn’t refer, specifically and directly,
to the topics we looked at on tuesday…
exponential functions, inverse functions, log functions…
but *does* link to the precalculus page that
*does* treat of these topics specifically (though not directly;
instead it provides links to w’edia pages on each).

continuing in this vein, i’ve just looked (for the first time)
at the exponential function page. hmm.
i imagine myself a beginning student. what do i see?
forest-and-trees issues abounding… there’s an *awful lot*
of material here! but maybe it’s clearer than the textbook
even so. in particular, the article is (very rightly) about
*the* exponential; of course i mean the one with base “e”.
and, right out front, in the first sentence in fact, they’ve got
“the function e^x is its own derivative”.

every calculus *teacher* understands, at least to some extent,
that the importance of the number “e” is very closely tied to
this property… it’s the base that makes exponentiation
“work out conveniently” in “doing calculus”.
most of ’em, if pressed, would probably be able to tell you
that “y = e^x” is the only non-trivial solution
to… the world’s simplest interesting differential equation…
{{dy}\over{dx}} = y. (the trivial one is y’ = 0).

every calculus *book* obscures this point to some extent.
and there are reasons. one has not *defined* {dy}\over{dx}, after all.
*our* text—”stewart”—follows the usual pattern
of “discuss exponentials generally first”
(y = A^x for A a positive
number different from 1) and then singles out the case A=e
as the one having a tangent of slope 1 at its y-intercept.
this can hardly be very motivating for a beginner.
and, anyhow, neither has “tangent” (to a curve at a point)
been defined… so (as far as i can see) *nothing* is gained
in terms of “formal correctness” by focusing on this
particular *detail* of the fact that the exponential
is its own derivative.

okay. there are *better* reasons. textbooks *should*
review exponential-functions-generally (and provide
lots of practice problems). nobody’s going to
understand very much about y=e^x that doesn’t
know anything about its first cousin y=3^x.

still it seems to me that
somewhere pretty close to the moment that
the hugely-important constant “e” is introduced,
it would be helpful to at least some students…
students like i imagine myself to have been,
for example… to have some *succinct*
and *correct* justification (even if its details
can’t be spelled out fully with the concepts
already covered in the prerequisites-so-far).

if i recall correctly, my own experience of
learning-about-e was rather a horrible mishmosh
of formal-correctness and we’ll-learn-about-this-later.
i *did* learn about it later but it was an accident of history;
if i’d merely been a math *major*, it’d’ve been obscure
to me all my life, but since i went on to be a
*graduate student* in maths, i eventually
considered myself duty-bound to make sense
of it all (and had the “mathematical maturity”
to do ahead and do it).

it makes good sense *formally* to consider
the natural-logarithm function (“ln”)… defined
(of course!) as the integral-of-the-reciprocal…
*first*, and define “e” as the solution to
ln(x) =1.

hey, madeline just woke up. more later.

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lindley’s paradox at angrymath.

eric schechter’s common errors file.

i’ve got a post for spring quarter. yay!
less money again and i suppose less
prestige… i’m running two sections
of a calc class. the students see lectures
(not by me) on mondays and wednesdays
and meet me for guided-problem-solving
(and to hand in homeworks and take quizzes;
exams have yet a different schedule;
i grade everything of course) tuesdays
and thursdays. also there are tutor-room duties.

but one-on-one tutoring is in many ways
the *best part* of the whole “math teacher”
thing. not by coincidence, the furthest away
from the show-me-the-money side:
grant applications and suchlike no-math
adminstrivia. pretending to participate
in curriculum design and suchlike policies:
worse-than-meaningless committee work.

there’s a great deal of online activity
associated with teaching at Big State U.
but almost everything has gone pretty well
in my two quarters here so far (this century).
the payroll-related interface was frustrating
to be sure… but there was an old-fashioned
paperwork over-ride that i eventually found
out about by showing up in person ready
to beg for help.

otherwise… filing grades and scheduling tutoring
duties for example… stuff works the first time
and the right buttons aren’t hidden among
dozens of useless ones with similar names.


new drawing today.

you can check my work…
as of course i already *have*:
start with any “cross”, C, and
verify on five cross-diagrams
(by shading one “circle” on each…
namely the one whose position
corresponds to that of C
in the “big picture”) that
the “Point-at-C” belongs
to each of the “Lines-through-C”.

i’ll edit in some algebra; probably soon.
but the point now is that i *don’t* need
to refer to any but purely *visual*
calculations.

somewhat to my surprise, i’ve decided
in middle age to become much more
of a “visual learner”. in my case, this
amounts to “how can i represent results
from abstract algebra as pictures?”.
it still counts.

teacher blog action day

according to EDUSolidarity (moving images
on the screen; yick) or this f’book page,
today is “teacher blog action day”. so i’ve signed on…
and should now post a piece on why i’m pro-union.

“why teachers like me support unions.”

this is harder than it sounds.
it’s my old “imagine an appropriate audience” problem.
everybody *knows* (or they think they do)
the reasons union supporters support unions;
everybody knows how they feel about ’em.
why should *my* testimony change anybody’s mind…
or even excite their interest?

in face-to-face interviews, by exercising incredible
amounts of patience, one can… very rarely in my
experience… sometimes discuss *local* political
issues in what seems to me like a meaningful way.
but almost all of what passes for “talking about politics”
is worse than useless.

this is pretty well-known; anyhow, it’s a cliche
that *religion* and politics are taboo subjects
for polite conversation in some contexts.
both for the same reasons: everybody’s already
picked their side and listens to the *other* side
*only* to try to spot weaknesses that will allow
them to vary their “i’m so right” speeches with
some “you’re so wrong” seasoning.

you’ve got to already have made some commitments
to function as a social being… even if only to the
very common “a plague on both houses” stance
i’m illustrating here. for god’s sake, let’s talk
about something where you won’t be tempted
to try to bully me into saying things i don’t mean.

thus.

“i’m pro-union because i don’t like being pushed around”
is probably about as good as i can do. but if someone
should respond with “how is *another layer of bureaucracy*
going to get you pushed around *less*?” i won’t have
an answer that satisfies even *me*… i have to fall back
on “hope is better than despair” or some such
*attitudinal* thing. anyhow, i do unless i’m talking
about a *particular* workplace at some *particular* time.

together, *maybe*, we can do something about
some particular abuse-of-power by management;
one that we feel powerless to do anything about
by individual action. if your ideological commitments
won’t let you go *this* far, you’re much too far gone
for there to be any value in this part of our conversation
at all. even those who *oppose* unions have to admit
that they can sometimes be effective in changing policies
(even if it’s only about *which* squeaky wheels get greased).

when i worked as an organizer, one of the clearest messages
i got from the veteran organizers during “training” was this.
listen sympathetically while the prospect talks about
their problems with their jobs. then, when you’re
about to ask them to sign on, you can come back to
“we’ve got a plan for dealing with [your issue]:
stick together and *make* ’em listen.”

it’s the “all politics is local” phenomenon.
people talking about *their own lives*
don’t *engage* in the same ways as when
they’re talking about the perfect-world fantasies
that make up almost all “religion and politics”
as encountered in day-to-day life… and almost
the *only* way to get through the fog of
“that’s not how i was taught to believe”
is to speak of *very particular* incidents
from our very particular lives.

so… sorry, internet… i can’t *tell* you
“why i support unions”.

here’s a cliche dump for ya just the same.

because my enemy’s enemy is (sometimes) my friend.
because “love your neighbor” is a lovelier law
than “might makes right”.
because individual human beings
are openly treated as *resources*
by money-machines determined
to respect *only* much larger
entities in making their decisions.
(even the courts now admit that
mere human rights can’t be allowed
to trump *corporate* rights in
our money-uber-alles political system.)
because hope is better than despair;
because co-operation is better than violence.

enough. anyhow, there’s plenty of pro-labor
stuff by me out there i think. i did a blog once.

yadda, yadda, though, i guess. in real life
i don’t pay dues to a union and, as these
remarks suggest, i avoid talking about ’em.
mostly because i never get anything *out*
of talking about ’em but frustration.

much of my working life persuades me vividly
that it’s much harder to get people to change
their minds… about *anything*… than we
usually let ourselves think. even when i’m
armed with *certain knowledge*… in algebra,
say, where the opinion i’m trying to convert
the student to is *universal* among
those-informed-about-the-subject…
students will pick up new ideas only
*very slowly* and typically only then
by *getting them wrong* over and over
until finally finding a way to get rid of
whatever conceptual block it was that
led to our discussion in the first place.

if i can’t get people who have actually
*paid good money* to hear what i think
to believe that, say, “clearing fractions
is helpful in solving certain kinds of equations”…
why, then, how am i going to get ’em to
believe me if i said “your god is an idol”?

never by acting alone, is how. persuasion?
over-rated. come over to our side.
treating people as ends-in-themselves
is more *fun* than treating them as cannon fodder.

january
01/07/11 Blogging 104. Week One. so far so good
01/07/11 Some Finite Projective Spaces photos of zines
01/13/11 Our Medium Is Handwriting drawings of guitar scales
01/18/11 time considered as a sequence of many-named values (modulo twelve) scales for math-heads
01/26/11 today’s morning ramble: music theory oh, no, oot, eeth, rho…
01/28/11 across the universe notations for guitar
01/30/11 Blogging 104: Exam 1. postmortem

february
02/01/11 start one dimension up… increasingly-stylized P_2(F_3)
02/04/11 the skin of our teeth quadratic equations, *again*?
02/07/11 i pubbed my ish drawing skills improving slowly
02/10/11 you could look it up meserve’s _geometry_ plugged
02/15/11 someday i’ll learn how to scan a document “ideal points” etc.
02/20/11 …and then a hockey game broke out… evolution of a drawing
02/24/11 complete the square plus links from sue v.

march
03/04/11 Math Teacher At Play: e^{i\theta} cool trick; new insight

few but ripe


not a projective space though
(there are “parallel lines”, for example).

ten “lines” through ten “points”.
you can get this by using
2-element subsets of {0,1,2,3,4} as points.
the lines are then triples
(like {0,1}–{1,2}–{0,2})
such that each point of the triple
is the symmetric difference
(A \Delta B = (A \cap B') \cup (A' \cap B))
of the other two.

the point associated to {0,1}–{1,2}–{0,2}
in the points-to-lines correspondence
i’ve illustrated here is then {3,4}…
the complement of the union of
the three points of the line.

so you can re-create this at will.
you just have to fiddle out a
nice symmetric version of the picture.

anyhow. pick a big circle.
look at the *three* big circles
matching the dark dots inside.
the three “lines” meet at the
original point.

Somebody’s preparing a lecture or something and mentions the angle-sum laws. There are two of ’em sharing the office with me doing the same class (different from mine but one I’ve taught many times). Says he probably won’t get to it tonight.

Me:
“I could never remember the doggone things until I found out about the “cis” function… e-to-the-i-theta equals cos-theta plus i-sine-theta. I always make it a point to show it to the students this way too… if you can just take this bit on faith, you can remember what’s what forever. After a while it even begins to make sense… cosines are x-coördinates (so we should think of “real parts”) and sines are y’s (and so, “imaginary parts”)… then things multiply out accordingly.”

And the *other* guy seemed to know pretty well what I was talking about. But the guy who brought it up, not. But he’s in too much of a hurry to sit still for any actual explanation, alas (not that I blame him). So I gave up and prepared my *own* stuff or something and did my lecture and all that. But had it on my mind.

So, first, here’s the part I’ve known since, I forget, early grad school days.

Prerequisite: A little trig and a little faith. (Actually, you could take it *all* on faith and still have the best way to memorize the angle-sum laws… but of course there’s no *point* in memorizing the angle-sum laws if you’re not already studying trig.) On faith, you should believe that there’s a Complex Number Field “containing” the Reals and a number, i, satisfying i^2 = -1. This isn’t much of a leap… anyhow, students will have heard rumors about this situation and may even have done a few exercises.

The *leap* of faith comes with the proclamation
e^{i\theta} = \cos(\theta) + i\sin(\theta)\,.
This is admittedly something of a whopper. To *prove* this admittedly-weird equation one requires some Analysis (that’s “Calculus” to you, I suppose). The only proof I’ve worked out in detail involves so-called “infinite series”… oh, what the hell.

[
e^t = \sum_{k=0}^\infty {{t^k}\over{k!}}
\sin(t)= \sum_{k=0}^\infty{{(-1)^k t^{2k+1}}\over{(2k+1)!}}
\cos(t) = \sum_{k=0}^\infty{{(-1)^k t^{2k}}\over{(2k)!}}\,;
Shove in i-times-theta for “t” and turn the crank; equate real and imaginary parts.
]

The function
\cos(\theta) + i\sin(\theta)
is sometimes called “cis(\theta)”, by the way. So the theorem I’ve outlined—the one I ask my students to “take on faith” if they want to learn my favorite trig-mnemonic—can be stated as e^{i\theta} = cis(\theta). OK. Everybody’s willing to suspend disbelief thus far, right? Because, look. The “angle sum” law for the cis function goes like this.

\cos(\phi + \psi) + i\sin(\phi + \psi)=
e^{i(\phi + \psi)} =
e^{i\phi}e^{i\psi}=
[\cos(\phi)+i\sin(\phi)][cos(\psi)+i\sin(\psi)]=
\cos(\phi)\cos(\psi)-\sin(\phi)\sin(\psi) + i[\cos(\phi)\sin(\psi) + \sin(\phi)\cos(\psi)]\,.

Equate real and imaginary parts for the results. To wit:
\cos(\phi + \psi) = \cos(\phi)\cos(\psi)-\sin(\phi)\sin(\psi)
and
\sin(\phi + \psi) = \cos(\phi)\sin(\psi) + \sin(\phi)\cos(\psi)\,,
the “angle sum” laws.

Amazing isn’t it.

(Summary: when you know “cis” and how to work with exponents [“*add* exponents to *multiply* exponentials-with-matching-bases]… and, oh yeah, how to use i^2 = -1 [to multiply “complex numbers”]… you get the [matchings and {s, i, g, n}-signs for] the trig functions “for free”. And you really *oughta* know those things: this isn’t all they’re good for [by a long shot]. )

Now, that’s essentially the lecture I’ve given many times: the part I’ve known for years. But I’ve dusted off some ODE books lately and think I have a better idea than I ever did about what’s “really” going on.

Bump up the prerequisites. To get much of anything out of what follows, the reader should know a little Calculus… derivatives for exponentials and trig-functions… and be willing to believe an existence-and-uniqueness theorem from ODE (“ordinary differential equations”). I plan to get *around* the “infinite series” argument… or maybe just to *hide* it…

Let’s imagine that we *know* about e^x for Real x and want to investigate e^z for Complex z. Write z = x + iy.

Now e^(x+iy) = e^x * e^(iy), of course. And we already understand e^x by assumption. So we can focus in on the function

f(t) = e^(it).

Differentiate f twice: f”(t) = -e^(it).
So f is a solution to the ODE f” + f = 0.

But one already knows (or readily checks) that sin(t) and cos(t) are *also* solutions to the same ODE. Moreover (here’s the “high theory”), since ours is a *second-order* ODE, these *two* (“linearly independent”) functions “span the solution space”: *every* solution can be written as a sum-of-scalar-multiples of these two.

Assuming this is true. There exist constants K_1 and K_2 satisfying
e^{it} = K_1\cos(t) + K_2\sin(t)\,.

Putting t = 0 shows that K_1 = 1 (recall that cos(0)=1 and sin(0)=0); it then follows that e^{\pi i} = -K_1 = -1 (by a similar calculation).

Now consider e^{{i\pi}\over2}. Our sum-of-scalar-multiples equation gives us e^{{i\pi}\over2} = K_2
but we can also calculate (e^{{i\pi}\over2})^2 = e^{i\pi} = -1 (by our previous paragraph). So K_2 is a square-root-of-minus-one: \pm i.

OK. It’s late. Getting rid of the \pm (plus-or-minus) is an exercise. The point, if I still have a reader, is clear, I hope: assuming “good behavior” for Complex solutions to (second-order linear) Differential Equations, we’ve *bypassed* the series (which in some sense are just a complicated expression of the facts that (e^x)’ – (e^x) = 0, (sin(x))” + sin(x) = 0, and (cos(x))” + cos(x) = 0 [along with, now that I think of it, the values of e^0, cos(0), and sin(0)])… to obtain the e-to-the-i-theta-equals-cis theorem.

If I’d’ve known you could actually *do* stuff with ODE, maybe I’d’ve learned about it when I was a sophomore. Or the one time I taught it…