Vlorbik's Diner

son of owen's cooking show

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”.

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4 Responses to “use the source luke”

  1. 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

  2. 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…
    {\Bbb Z}_n )
    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.

  3. 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.

  4. Vanessa said

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

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