# Archive for the ‘Ring Theory’ Category

## use the source luke

Posted by vlorbik on November 15, 2009

the gospel i preach: simple things first.

basic arithmetic, for example, turns out

to possess a bottomless depth… and will

reveal new secrets to any investigator

that can bring it new eyes.

the easiest way to get new eyes is of course

to find a student.

[

or rather to have students

provided for you as i did at indiana u,

ohio dominican, columbus state cc, ohio state u,

and capital u at various times. finding students

outside ivory tower walls is a new challenge for me.

i’m recruiting. and at reasonable rates too but so far

i haven’t really got anything you could dignify with

the name of “a plan”. or for that matter any students.

]

but the easiest way to find the depth

is to lend one’s own eyes to old texts.

and start simple.

better to understand arithmetic,

for example, one could and should

investigate (as soon as possible)

**modular arithmetic**.

the most familiar example

(“clock arithmetic”) uses the

hour-numbers marked on a clockface.

replacing “12” with “0” gives

the commonest mathclass

representation of “the integers

mod 12”:

{0′,1′,2′,3′,4′,5′,6′,7′,8′,9′,10′,11′}.

i’ve “primed” the numerals

representing the elements

of Z_12 (the set of integers-mod-12)

here to emphasize a point…

namely that they are *not*

integers. 3’+5′ = 8′ all right

like anybody would expect but

we also have… what would just

look *horribly wrong* without

the “primes” 8’+7′ = 3′

(because starting at eight

o’clock and waiting seven hours

takes us to three o’clock…

we “throw out multiples of

twelve” to do arithmetic mod 12.)

it turns out you can multiply too.

very important this. turns out if

you use a *prime* number in place

of twelve you get something called

a **finite field**. manymany

very simple examples to be found

of phenomena whose depth remains

unsounded. the ever-growing subject

of “crypto” more or less *begins*

with the study of these fields

from what i’ve been given to understand.

but that’s not the subject of today’s ramble.

which is counting permutations. and starting simple.

there’s one permututation of one thing.

A

that’s it.

there are two permutations of two things.

AB, BA

pretty simple still.

and uninstructive.

things get much more interesting

when three things are considered.

ABC

ACB

BAC

BCA

CAB

CBA

.

this is the key example in understanding

what goes on. but the key *exercise*

is “write out the permuations of

the elements of {W, X, Y, Z}”.

(one will have been given a solution to

“write out the permutations of

the elements of {A, B, C, D}”

already in the body of the text.)

ABCD BACD

ABDC BADC…

ACBD .

ACDB .

ADBC .

ADCB

how does it work? well.

the “reason” that the

six **permutations** of

the elements of the set

{A, B, C} are six

(ABC, ACB, BAC, BCA, CAB, CBA)

is that 6 = 3*2*1.

(where “star” denotes “times”:

5*3 =15 etcetera).

the point here being that we can

imagine writing our letters

one-at-a-time into three “spots”

“_ _ _”

with *three* choices for the first letter

and *two* (we are, thus far implicitly,

but now by explicit decree, not allowing

ourselves to repeat letters [and moreover

each letter *must* be used]) choices

for the next. finally, there’s one letter

left… our only “choice”. the “3” (A, B, or C),

the “2” (don’t reuse the one you picked!)

and the “1” (whatever’s “left over”)

in 6 =3*2*1 are now all accounted for.

the same reasoning tells us than in the case of four letters

one will have 4*3*2*1 = 24 possible permutations.

one-fourth *of* ’em will start with “A”;

one-fourth with “B”; so on. six each.

so one might write down

A______B______C______D______

A______B______C______D______

A______B______C______D______

A______B______C______D______

A______B______C______D______

A______B______C______D______

first. (the underlines denote

*blank space*… the very model

of “harder than it looks” in my book).

then “go down the A’s column”.

there are six entries…

and B, C, and D

(the “unused” letters so far)…

will each have the same chance

of appearing next: twice each

as it turns out. so fill in two B’s,

two C’s, and two D’s.

AB_____B______C______D______

AB_____B______C______D______

AC_____B______C______D______

AC_____B______C______D______

AD_____B______C______D______

AD_____B______C______D______

likewise for the B’s…

“go down” the column

distributing “missing letters”

(in this case A, C, and D).

and the C’s… finally the D’s.

we’re now at

AB_____BA_____CA_____DA_____

AB_____BA_____CA_____DA_____

AC_____BC_____CB_____DB_____

AC_____BC_____CB______DB_____

AD_____BD_____CD_____DC_____

AD_____BD_____CD_____DC_____

finally go down each column one more time.

there are two of each “string” so far.

in the first, AB, add the remaining letters

first in their “natural” order (ABCD)

then in the opposite order (ABDC).

continue with this pattern:

two missing letters in each case,

first in the natural then in the opposite

order. go through the whole list.

voila. all twentyfour permutations

of four things. in alphabetic order

no less.

i’ve taught this procedure to dozens.

one-at-a-time, too. somehow seeing me

do it in class doesn’t do a thing *for* ’em

sometimes. the student has to move the pencil

(and wants an expert standing by

to bail out at the first sign of trouble).

sure it’s tedious. but it’s just incredibly necessary.

you can’t do *anything* with certain sections of

very common textbooks until students can rattle off

lists like this.

one way or another. certainly there’s no need

to learn *my* algorithm if you’ve got one

of your own that works. i’d love to have you show me.

better if you show the whole class.

there is nothing for “knowing what you’re talking about”

quite like “being able to write it down”.

Posted in Notations, Permutations, Ring Theory | 4 Comments »

## this will all make sense later

Posted by vlorbik on August 20, 2009

whenever you’re teaching, say,

“intermediate algebra” and talking about,

say, adding fractions… whatever,

sooner or later somebody will say

“my *old* teacher showed me…[this]”,

where [this] is, typically a sketch of,

some very-likely-flat-out-wrong or

in-any-case-insufficiently-general or

*would*-work-if-not-radically-misapprehended

other way to go.

and we’ve got, as it were, a new

stumbling block. because, although

it’s easy enough to learn to rein in

one’s initial reaction

(“well, first of all i don’t believe you;

*no* math teacher would’ve told you *that*…

but more to the point, so what?

it obviously hasn’t been working out

for you since you’re still being

“remediated”. maybe *this* time

you’ll take it *seriously*”)…

the student isn’t getting “buy-in”.

because this “new” thing that *you’re*

presenting is *felt by the student*

as an *attack* on the “trusted source”

that they remember as having taught them

*something*… something that will have,

at least, made some sense to them at one time.

so they’ve circled their mental wagons

around their wrong idea hoping to keep

your good idea out as some kind of

*act of faith* (in their trusted source).

or so i conjecture.

i was telling essentially this story

one day in the barracks and was overheard.

“i like the way you put things”

said i-forget-who (a near-stranger)

gladdening my heart and giving me

some confidence that here was,

anyway, some blog fodder.

because we teachers need to talk about

issues like this and i’m just the guy

to jump right in and participate

whenever i can. i’ve *always* felt

this way. i’ve *also* felt that

formalizing such discussions,

say under the heading of

pedagogical content knowlege,

is in vain and a grasping for wind.

because why? because there’ll be

*no* “right answer”… anyway no

right answers of the kind we encounter

in, well, say, adding fractions.

where certain very precise claims like

can be displayed as it were

like a “text of our sermon”,

an anchor to insure that our discussion

never drifts off into mere opinion;

a way to get our *feelings about*

the material off to the side

and compare *our* ideas to

what i’ll go ahead and here call

“the truth” or “the fact of the matter”

or something like that.

nope; it’ll always, like in *most*

academic subjects, be about *words*…

and *opinions* about them…

things that flat-out *don’t have*

“literal meanings” in the sense

that equations do. what’s weird

is that we usually like to pretend

that words *can* and even usually *do*

have strict, this-and-only-this

“meanings” in ordinarly conversational speech.

i was kidding around yesterday

with some funny-when-taken-too-literally prose.

typical math-major stuff, really.

people say “always”, for example,

to mean “sometimes”… and are clearly

understood in their context.

this happens all the time (see?).

along comes a math major who thinks

(or pretends to) that “one non-example

refutes any claim to universality”

and expects (or pretends to) that

*displaying* a counterexample will

cause whoever is making the sweeping assertion

to *modify* their claim rather than,

say, reply with, “oh, you’re being

too literal” or some variant.

“acceptance is the answer to all my problems”.

all? really?

you’re just being too literal.

well, maybe *you’re* not being literal *enough*.

we keep wanting to be able to say things that

are *true*, as it were universally, and moreover

we keep wanting these things to be *believed*.

but because we’re working with ill-defined concepts

(not mathematical symbols), this is,

strictly speaking, impossible.

there’s *always* (okay, i don’t mean this)

some man-behind-the-curtain “hidden assumptions”

preventing us from settling arguments

in, say, religion and politics:

we *say* we’re talking about *school* classes,

for example, when maybe we’re “really”

talking about *social* classes

(but are prevented by blindspots and taboos

[in this case, by generations of redbaiting]).

math-heads keep noticing this; it annoys the others.

oh, and we saw _revolutionary_road_ a little bit ago.

good picture. but did the cinema really need another

madman mathmajor? couldn’t he have just been a philospher

or something? geez.

Posted in Movies, Rambles, Ring Theory | 4 Comments »