“modular arithmetic” was introduced to me
somewhere around 1968 as “clock arithmetic”
(among other things). it turns out—
very interestingly, as it turns out—
that one can do Additions and Multiplications
very much like the familiar operations on
(the integers), using any of the sets
(the last-named, of course, gives
“clock arithmetic” its name).
suchlike “arithmetics”… having an addition
and a multiplication (with the addition assumed
commutative and with the multiplication “distributive
over” the addition) are (for some reason) called
rings. the rings we care about today are
called fields; examples include whichever
“clock arithmetics” you care to name that have
a prime number of elements.
fields are “good” because we can *divide* in them.
(not just add, subtract, and multiply). the “prime”
criterion ensures this by ruling out the possibility
of solutions to “xy=0”
having “x” and “y” both *non*-zero.
for example, “on a mod-12 clock”, one has 3*4=0,
so we have a zero-product formed by two nonzero factors.
this is a “bad” thing and doesn’t happen in fields.
once we can do divisions, we can carry over most of
the “analytic geometry” from intro-to-algebra into
any of the settings
and much else besides.
the value for me in replacing the so-called real field
with the finite fields is inestimable:
one can show *every single instance* of a certain phenomena
directly, without appeal to such weird notions as “continuity”.
in the example at hand, we have
“projective space over ”
(which the colorful image calls ).
each line has four points. four of the lines…
including the “line at infinity”… have been
color-coded (and given equations as names).
the “finite points” here have z=1;
the line-at-infinity has z=0;
the “point at infinity” has x=z=0.
the rest is commentary.
*At least when I was at school we were correct in writing “Answer = “, even though the teachers hated it!*/— howardat58, upthread.
i’m more likely to’ve encouraged this behavior
than to’ve “hated” it.
but “Answer” is a pretty awkward variable-name
so, given a chance, i’m also likely to’ve made it
as plain as i could find a way to do that what i’d
*really* like to see is a clear
A = …. messy expression to be simplified
right at the beginning and *then* the
A = Simplified-Version-i.e.-“Answer”
bit at the end. which gives a presentation
clearer than one is likely to find on the
blackboards unerased by the previous
because “define variables (with units) precisely”
is a *major* sticking point for *many* students
and i’m not just talking about Remedial Algebra.
one of my favorite-ever calculus tutees
refused my excellent advice on this subject
but without it, we simply *cannot* organize
our presentations coherently.
she finally… same calc ii student here…
couldn’t endure my continual insistence
on keeping equations balanced as she
wrote out her calculations. we broke up
the attitude seems to be “it’s all just
ritual-process calculation anyway
until i can get the Answer”, whereas
of course one seeks to instill instead
something like “the Answer is itself
a collection of equivalent statements
(leading to the value of a variable)”.
“scratch” work is *obviously* the enemy of clarity
once one is made to *grade* the work.
and not just clarity of *presentation*.
having calculated out some expression,
let’s say correctly, one is in the position
of having to *do something* with the result.
but without the whole A = Answer format…
a “proof”, if you will… one is left with a
bunch of area-on-the-page with certain
code-strings (and scattered english)
bearing no particular *stated* relation
to one another at all.
and if Answer = “the thing i want to see”
i’m very likely to give ’em full credit.
but that won’t make it good work.
I have briefly mentioned that the alternative to explicit instruction may be described as ‘constructivist’ teaching. I don’t want to become bogged-down in this – I am aware that constructivism is actually [a] theory of learning and not of teaching and I have no problem with it in this regard; we link new knowledge to old etc. If it is true then, no matter how we teach, our students will learn constructively. However, some educationalists clearly do see implications for how we should teach.
i don’t want to become bogged-down in this either.
and yet i have been, deeply, many times, for years.
not so much these days. i just, you know, despair
of anything useful being said or done and check out.
all educational philosophies are useless in practice
until particular special cases are to be discussed
in carefully constructed contexts… so all we readers
ever seem to get is atrocity stories and suchlike
ill-disguised partisan politics.
“carefully constructed contexts” would include, for
example, a lot more attention than i’m usually able
to find about who the heck “we” are supposed to be.
this annoying pronoun is used as if it’ll mean all things
to all people. but it usually means nothing to me.
(in the passage at hand, i take “we teachers”
readily enough, so this isn’t a good example of
what bothers me… but hints at it. in electoral
politics, “we” can mean we-voters, we-americans,
we-patriots… and, often enough, two or three meanings
must be inferred to make any sense out certain
passages at all.)
angels dance on pinheads and owen leaves the room.
i pulled my (dover edition of) cantor’s epoch-making
contributions to the founding of the theory of transfinite numbers
(one can evidently download it here)
yesterday to show tony from church;
he’d noticed my (prominently displayed)
copy of god created the integers
(hawking’s anthology of great math by math greats)
and mentioned “infinity” a few times in
my hearing, so it seemed like a natural.
and maybe it is… anyway, one does *not*
need a lot of high-tech “advanced math” to
read cantor’s stuff… and be just as mystified,
most likely, as most of the mathematicians
of cantor’s time (and many long after).
but i *should* have broken out the fourth dimension (w’edia),
by rudy rucker (w’edia).
tony’s *also* mentioned “the fourth dimension”
(as a concept) and *this* thing is bound to be
a whole lot more accessible than cantor.
i don’t know this particular rucker book at all well…
but i used his infinity and the mind in a class
long ago and’ve read some of his stories and whatnot.
rucker’s one of math’s best “popular” writers ever,
with a “transreal” SF-like vibe all his own.
i’d post more but my mouse is acting up again. damn it.
the PDF introduction to eli maor’s trigonometric delights
is still up at princeton press. when i plugged it here
back in ’09, along with a mini-lecture about the number “e”
for my then-precalc-class blog (Math 148: Precalculus),
it seems they were offering the whole thing.
apparently it’s out of print. great book.
i can’t get the mouse to behave
and goddammit we the living were
supposed to be in charge but if
i twitch what little is left of
my, admittedly imaginary, “free”
“will”, in this fucking digital
ratmaze, for, one, second, longer.
well. no. not well.
that would be, hello.
make it stop i beg you.
hang up and drive.