Archive for April, 2011


a recent post in KTM asked why
{6 \choose 2} = {6 \choose 4}.
i know that one.

the “graph” in the upper right is K_4…
the “complete graph on 4 vertices”…
and has *6* (so-called) edges.

its “complement” (upper left) has *none*
of the edges. obviously there’s one
(so-called) *complete* graph having
all six possible edges and one “empty”
graph having *none* of the edges.

move down to the next couple lines:
the graphs having exactly *one* edge
match up in one-to-one fashion
with the graphs having *five* edges
(because “including five (edges)”, in this context,
is the same as “leaving out *one* (edge) out”.

so on for two-edge graphs.
each is the “complement”
(via the inclusion-exclusion principle
as hinted at a moment ago)
of a *four*-edge graph.

posting; gotta go to class.

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i’ve finally put the “points at infinity”
where they’ve belonged all along.
this’ll probably be the last draft
as far as “where do the points go”.

the color scheme is still almost
entirely up in the air. all i know
so far is that nobody not already
interested in maths will look
at the damn thing in b&w.

:)


i did this before rearranging the points.
now i suppose i oughta do it again.
anyhow there sits the plane on
the field with *two* elements
in red right inside the plane
on the field with *four* elements.


another microzine 8-pager
(one side of a single sheet
of typing paper, cut & folded).

pages 6 & 7 are this new version
of P^2({\Bbb F}_4)^*. i’ve used
the (more “obvious”) ordering
00 01 0a 0b
10 11 1a 1b
a0 a1 aa ab
b0 b1 ba bb
on the “finite points”;
last time i posted a drawing
of this space
i used the weird
aa ab ba bb
a0 a1 b0 b1
0a 0b 1a 1b
00 01 10 11
.
it seemed like a good idea
at the time. part of the point
is that there *are* different
ways to go about putting in
a co-ordinate system.
but mostly i just wanted
to see how it would look.

belling the cat

calculator thread in walking randomly.

john d.~cook on sliderules.


i’ve just edited in a subscript-backslash-zero
to the top line…
which now reads S = ({\Bbb Z}\times {\Bbb Z})_{\backslash 0} =...
to repair an *earlier* repair, done sloppily.

somewhere along the line i tacked on the “—{(0,0)}”
*without* adjusting the “zee-cross-zee” ({\Bbb Z}\times{\Bbb Z}).
a beginner-like blunder, i confess. onward! *more* mistakes!
(just get down in the dirt and *calculate*, by golly.)

anyhow, owners of Math Ed Zine #0.4—\Bbb Q by name—
should please to adjust the appropriate page in their issues.

which, being interpreted, means that
the set of *rational numbers* (Q) can be
represented as the collection of *lines through
the origin* (in the usual (x,y)-plane),
having *rational slope*. The slope condition,
for a given line, is equivalent to the condition
that there be an *integer* pair lying on the line
(nonzero; it gets to be something of a pain…).

the algebraic process whereby S…
nonzero-integer-pairs…
“maps onto” Q
is called “factoring by a relation”.
the relation in this case is called “tilde” (~).

tilde is defined by
” (x_1, y_1) ~ (x_2, y_2)
MEANS THE SAME THING AS
x_1 * y_2 = x_2 * y_1″

(“cross-multiplication” is in effect;
tilde is the relation we want “because”
{{y_1}\over{x_1}} = {{y_2}\over{x_2}}
when x_1 y_2 = x_2 y_1).

oh heck. there’s that infinite-sloped line.
belongs to S/~, too. OK. modify the \Bbb Q.
let’s call it {\Bbb Q}^\infty, say. okay.
that’s it.
.

tutor room first day

two hours fly by like nothing… working exercises
with one student at a time is generally the best part
of the whole lifetime-math-teacher trip.

there’s quite a bit of homework for 151 students
to work out this week… and then to digitize.
putting the answers into the computer interface
can be expected to present *lots* of problems.

but this level of coding is easy enough for *me*,
so part of how i was able to help with
in a couple of instances was translating.
handwritten work leading to x \not= 0,\pm  { {3\sqrt{2}}\over{2} }
might be entered as
(-I, -3*2^.5/2)U(-3*2^.5/2,0)U(0,3*2^.5/2)U(3*2^.5,I)
—which is anyway much more trouble to type
than the handwritten version is to write out.
the student bringing this up already had (essentially)
correct code; maybe this is easier than i think.
we’ll see how it goes i guess.

PS
first quiz.
my latest rant mentioned the reason
this was postponed from tuesday to thursday
(or from T to R as i’m likely to use in handwriting):
computer security snafu. so i got a hardcopy on W
and created two slight variants in \TeX.
got those run off in the printing office.

and i spread ’em around around the room
with the allotted 15 minutes left to go
(versions alpha and beta to alternate columns).
took the roll for the first time ever and copied
names into “seating chart” order in my notes;
an old habit. everybody handed in with time
to spare; good.

i haven’t graded ’em yet (more or less of course).
but i glanced at a few of the papers and saw
*lots* of right answers including those on
at least one “perfect” paper. looks like it’ll be
“so far, so good” when i’m passing ’em around.
here’s hoping.

bricks without straw

the procedure for TA’s administering quizzes:
log into the “secure site” with your university password;
print out hardcopies and run off copies (one does *not*
have direct access to the copying *machines*, however…
run ’em off at your own expense or wait your turn
at the department’s copy center).

so far so good if the system actually worked.
but no such luck. when i got the email with
the link to the originals over the weekend,
i went and glanced at the link; okay.
there appears to be a quiz here. but i don’t
own a printer so i waited until i was on campus
to begin creating the actual paper documents.

whoops. locked out of the system. no time
to fix it. gotta postpone the quiz.

naturally there’s some resentment from the class;
naturally i’ll take a great deal of the blame.
(fortunately for me there were others suffering
the same problem or the *administration*
would presumably go ahead and blame me, too.)

in the twentyfirst century economy there’s *usually*
a robot between me and whatever i want.
and the more i want it? the more of my time
the robots will burn up uselessly while i try
to get it. telephones? your service *will* fail
and if you try to get it fixed, prepare to wait
on hold for a long time before you can get
to the menu with no option remotely like
serving your needs. home internet connection?
same thing, but worse (on my model because
i actually *want* to be online but not to talk
on the god-damn telephone). and don’t even
try to get me to *think* about health insurance.

look, i like breathing in and breathing out
as much as the next guy. but when does
this become unacceptable?

(here’s last year’s why i don’t
live at the p.o.
.)