## 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”.

## jd2718 said

So lots of nice bits.

One note: A kid playing monopoly often learns to work with 10 as the modulus. Reinforced by the numbers we regularly use. A 9 brings you from a RR to one spot before a RR, and everyone knows that, right? Clocks are a lot harder, and interfere with learning, imo. Unless you really are attached to the dozen.

Jonathan

## kibrolv said

never thought of monopoly as an example

in this context. good one!

remark: the *monopoly board itself*…

as opposed to the railroads-and-corners

“visual” approach used in calculations…

is mod 40

and there may be distractors here.

particularly in *writing things up*.

mod five is a also familiar for that matter.

“how many one dollar bills would you need…”

(… to change a lightbulb.

none. we’re converting to yen.

buddabing.)

nothing sacred about the dozen here…

and the examples that once hung on

classroom walls aren’t as common

as they used to be. skill in calculation

may always have been as hit-or-miss

as now. anyhow it can’t be assumed

anything but real shaky now and

subtract-twelve-from-X

is *way* harder for at least some people

than subtract-ten.

there’s still some virtue in the *circle*

for me. “mod 360” or “mod 2\pi”

provide still more examples

(with appropriate modification

in the case of the irrational

modulus, natch… one no longer

speaks of “remainders”…)

i tend to write the objects of Z_n

(actually of course… ain’t it cute…

)

for *any* small n in a circle for

certain computations.

clockwise too at that.

of course twelve also has the virtue

“lots of factors”. so the clockface

can be divided in sixths and quarters

and so on. again like in trig.

but for actually presenting to beginners:

for sure mod ten… “what’s the last digit?”

makes for *the* essential this-is-easy

motivating example.

and then the *exercises* will use moduli

like three through seven that’re easy

to work with at a level of detail that

soon becomes tedious for larger examples.

specifically the students should be required

to write out full multiplication tables for,

say, Z_5, Z_6, and Z_7. my comment

on prime moduli should then become

much more vivid.

here’s mod five (if the type gods smile).

*.1234

1.1234

2.2413

3.3142

4.4321

exercise. the bottom row

“counts down” to one.

do Z_6 and Z_7;

notice that this pattern

repeats. explain it.

you may choose to use

the concept “negative

number” (but are urged

instead to say “additive

inverse”). or not.

pascal’s triangle mod n looks cool written out

for small moduli. good exercises too i imagine.

## jd2718 said

It is a good exercise (Pascal’s triangle) – I’ve assigned it, but disguised the assignment as even/odd or with reference to remainders – when I’ve assigned it the point was getting a feel for the relationships in the triangle, the feel for numbers in general was a happy side-effect.

Maybe we’ll do some of that in class in a week and 3 days. I like handing out crayons, esp to 17 year olds. Makes the triangle come alive, and they tend to look with younger, opener eyes when they are clutching crayolas.

## Vanessa said

thanks for the click ~ cross-disciplinary too, not just for math

## cut & paste | the livingston review said

[…] testing testing boolean functions11/14/09 in media’s rays ) cross-products11/15/09 use the source luke permutations11/16/09 never get outta the boat four 4’s11/17/09 there is no frigate like a […]

## Vlorbik On Math Ed | the livingston review said

[…] 11/13/09 testing testing boolean functions 11/14/09 in media’s rays ) cross-products 11/15/09 use the source luke permutations 11/16/09 never get outta the boat four 4’s 11/17/09 there is no frigate like a […]