“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

class. alas.

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

*many* times.

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

over it.

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.

fucking.

insane.

make it stop i beg you.

hang up and drive.