# Archive for the ‘Notations’ Category

## remarks on 3-string guitar

Posted by vlorbik on June 21, 2016

(1) the high three would’ve been easier.

but it’s the “3rd” string—one of
“the high three”—whose tuner (the
gizmo that tightens a given string;
aka, more elaborately, a “tuning
machine”) is actually busted on this
actual guitar (a “rogue”; the logo
didn’t make it into the photo…
SO-069-RA1000-SN, if you must know
[made in china]).

with three strings in the *low* positions—
as shown here—one has good reason to attack
the strings “from above” (also as shown here).
usually if i do that at all… and i do, some-
times… i’ll come at “from above” frettings
with my thumb (as *not* shown here).

usually i use an ordinary “from below” grip
on the guitar, though. in any tuning. (so
only certain low notes get played with my
thumb; more often they get “damped” [so as
*not* to sound when strummed]).

(2) anyhow. message of the day. there are three
“obvious” tunings for three strings (once one
has learned the “standard” EADGBE tuning learned
by every beginner-guitarist): eadGBE, eaDGBe,
and eADGbe, one might call them… or, better,
GBE, DGB, and ADG.

or, still better: 0-4-9, 0-5-9, and 0-5-10.

(the apparently-missing possibility for
“three strings together from a row of six”,
namely EADgbe is accounted for in the “digits”
notation… it’s just *another* 0-5-10).

these number-codes reflect the *process of
tuning* the instrument and so are *very*
easily learned and remembered (once one
has learned a few chords & the “standard”
guitar tuning).

one more revision.
let X=10.
now we have 049, 059, and 05X
(and this last is pronounced
“oh-five-ten”). and typing ’em out
won’t be a problem.

the photo shows a major chord in the
059 tuning (isomorphic to strings 2
through 4 of “standard”): “barre”
all three strings.

anybody that can feel a beat and endure
a moderate amount of pain in their hands
can strum out a song in this tuning:
open strings, fifth fret, seventh fret;
jam around ad lib (these three chords
are, among many other things, the heart
of the blues [for at least one player]:
12-bar “boogie woogie” was the first tune
i ever picked out note-for-note [and, alas,
also one of the last; this’ll change some-
day if the guitar-gods spare me].).

(3) boogie-woogie for absolute beginners.

tune three strings to an open chord.
0 denotes “strum the open strings”
5 is “strum all three, fretted at 5”
7 (exercise)

(for a one-count; how many actual
*strums*—and whether *up* or *down*—
is here left to the player’s imagination
(never let your right hand know what
your left hand is doing).

0,0,0,0,0,0,0,0
0,0,0,0,0,0,0,0
5,5,5,5,5,5,5,5
0,0,0,0,0,0,0,0
7,7,7,7
5,5,5,5,
0,0,0,0,0,0,0,0.

yep. that’s it. the arrangement that won my heart.
it looks pretty naked all spelled out like that.
this must be what they call “over-analyzing every-
thing”. (even when you actually only ever do it
from time to time.) go and do likewise.

Posted in Music, Notations | 1 Comment »

## exercises at random

Posted by vlorbik on December 3, 2009

D1: 0
0:=[]

D2: ^
s^:=su[s]

D3: 1
1 := 0^

T: 1=[0].
P:
1=
0^=
[]^=
[]u[[]]=
[[]]=
[0]
.

D4: 2
2:=1^

T: 2=[0,1].
P:
2=
1^=
[0]^=
[0]u[[0]]=
[0,[0]]=
[0,1]
.

Exercise.
i. Define the symbol “3”
appropriately and show
that 3=[0,1,2].
ii. Do it again another way.
iii. Close the book
and check that
3=[[],[[]],[[],[[]]]].

hint: lay out your
work on the page
systematically to
keep track of parens
and commas better.
(every keystroke is
life or death.)
it’s actually easier
with a keypad since
you can cut and paste
and wipe out spaces
& whatnot. yep.
extra credit.
find the bug in the code.

4=[[],[[]],[[],[[]]],[[],[[],[[],[[]]]]]

Posted in Lectures Without Words, Notations | 2 Comments »

## It’s Been Too Long Since We Took The Time

Posted by vlorbik on November 22, 2009

This just uploaded PDF is the end of my decade-spanning quest for a procedure for putting pages, written and typeset by me on my own computers, onto the web with my own computer. The tables are very ugly. This is because I had to kluge ’em together by hand with no prior knowledge of how it’s really done, just to get something on the page. I was using plain TeX. It appears that LaTeX will have solved many of my problems. So I’ll have to learn the new ways and redo all my tables to reuse the files for my lecture notes from Dominican.

Meanwhile, I’ll experiment with TeX right here in the blog.
$3 \otimes_7 4 \equiv_7 5$.

Posted in Me Me Me, Notations, Scribd, TeX | 1 Comment »

## 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 »

## In Media’s Rays

Posted by vlorbik on November 14, 2009

Finite Sets are most easily displayed
by listing their elements (and, when it’s
convenient so to do, also naming them):

A = {a, b} and T = {a, e, i, o, u}

for example.

We note that A\=a here;
this means that
$A \not= a$
and that typing math
is here considered as
part of the problem
instead of as part
of the solution.

[
The point of having so noted
is that we are, as we so often do,
pretending to begin again.
This is in part the effect of
a lifetime’s classroom work
but in this case is also something
of a conscious choice. Anyhow,
I’ve been going on about handwriting
for a while now and intend to
continue. Hypertext is all well
and good and I’ll drop links
as usual according to my whim
(or careful design… you never
can tell… [until you can]… but
there’s nothing like the body
of the post
when you want
to call your reader’s attention
to something).
]

Make sure that whatever symbols
you use to represent my “A” and my “a”
are easily distinguished. This kind of thing
can sometimes be more trouble than you’d
think. For example,
something of a problem for me,
not in writing down credible versions
of
a\=A
b\=b
or
d\=D
but in… but you’ll have guessed…
c\=C
o\=O
and suchlike U&lc
(upper-and-lower-case [RIP])
pairs distinguishable,
as i tend to write them
(unless i’m being very careful),
only by their relative sizes.

Gedanken Experiment (I).
Think through which letterpairs
will give you trouble
(when you’re not careful).

[
in “script” versions of handwritten
letters there are common flourishes
to distinguish, say, script-c
from script-C; “print” letters
aren’t always so easy.

story making very little sense here
(you have to see the letters as i draw them).
my script-y’s and script-z’s
looked too much alike…
and, as you can imagine,
$x$, $y$, and $z$ come up
*a lot* when you stand up
in front of basic-algebra classes…
so i started writing my z’s
differently. had to start
*crossing* ’em to tell ’em
from 2’s.
no wait. actually this story
makes *perfect* sense.
it gives me an opportunity
to report that students
won’t believe that this
is a good idea when told
or shown or even when
they themselves make
a the mistake you’ve
warned ’em about on
work submitted to you
(yourself; that would
be me in this case).
i’m only *vividly* aware
of this having happened
*once* but i can’t swear
it didn’t happen before
or since then. nobody
believes anything you
tell ’em in this sorry
racket and it’s heart-
breaking. heartbreaking
i tell you.
]

Recall that
A = {a, b} and T = {a, e, i, o, u}
.

The set product (or cross product)
of a given pair of sets is the set of all possible
ordered pairs consisting
of a first (or left-hand) entry
taken from the first set and
a second (right-hand) entry
from the second set. Thus

A x T=
{
(a, a), (a, e), (a, i), (a, o), (a, u),
(b, a), (b, e), (b, i), (b,o), (b, u)
},

and

AxA={(a,a),(a,b),(b,a),(b,b)}.

The “carriage returns” in our display
of AxT are here as a convenience;
it’s just as correct… and in some circum-
stances correcter… to smash ’em.

AxT={(a,a),(a,e),(a,i),(a,o),(a,u),(b,a),(b,e),(b,i),(b,o),(b,u)}.

[
i’ve smashed space too.
this can be well worth it.
a *lot* of trouble lies in those
invisible characters.
and if you can learn to read
code that tight–and prove
it by *writing* code that tight–
you’re way beyond this lesson
and it remains only to endure
my plea that you comment on it.
]

Anyhow. This construction
can be taught to beginners
knowing nothing of set theory
in a single lecture obviously.

And without it, the whole
god-damn “functions as sets
of ordered pairs” thing…
the “let’s cram hundreds of years
of post-scientific-revolution
philosophy-of-mathematics
down their tender little throats
in one big enormous blob
of sticky incomprehension” thing…
is doomed. Because what the
hell is any of this going to have
to do with so-called “graphs”
in the student’s imagination
if the instructors haven’t themselves
ever connected these dots?

Well, quite a bit actually when the student
isn’t already badly damaged when they
encounter the nonetheless-incredibly-mangled
presentations we get in the “standard” treatments…
I was ahead of the instructor sometimes myself
in early days and it did me a world of good.

But the real point.
This should be presented by about 7th grade
and then again and again until everybody can teach it
to a 7th grader easily.
Because it’s the kind of down-to-the-ground
nobody-will-ever-be-righter-than-*i*-am
this-is-the-*one*-thing-i-know-for-sure-so-far
moral fucking certainty
without which math
is just more mental masturbation.

This is sadly lacking in our culture at large;
obviously there’s only so much we can do
in our classrooms about it. And creating
*programs* for the culture at large is a
fool’s errand or showbusiness or politics
not math. But dammit. Give the poor devils
a chance to be right when the teacher’s wrong
and know it in this one arena I beg of you.
It won’t make ’em into little
subversives-like-Kibrolv
and even if it did, they’ll
be just as easily marginalized
as I’ve been fear not of that.

The trick has been to get ’em into “statistics”
with no probability. But just to talk about rolling
two dice, any math-head worth their salt
requires a cross-product (DxD
where D= {1, 2, 3, 4, 5, 6}).

The display alone
11 12 13 14 15 16
21 22 23 24 25 26
31 32 33 34 35 36
41 42 43 44 45 45
51 52 53 54 55 56
61 62 63 64 65 66
and a few hours with
a competent fellow-student
would be more helpful
to a beginning student
than a wilderness of stix classes
as I see it. So. How to keep
students apart? Online classes baby.

Exercise in case you missed the whole rant.
Let $\eta = \{0,1\}$ and $\zeta =\{\pi, \sqrt2, e\}$.
Write out
(including every bracket, brace, paren,
and comma [and no extras]…
but with space on the page
left to your taste and discretion)
copies of $\eta \times \zeta$ and $\zeta \times \zeta$.

Posted in Handwriting, Notations, Rambles, Rants, Sets | 1 Comment »

## notes for chapter zero, modified by hand with considerable grumbling from the TeX code but we won’t be doing this again soon

Posted by vlorbik on November 9, 2009

Concerning Blahblahs

Exercise 1: Let A = {p, q} and B = {x, y, z}. Write out the set of blahblahs from A to B.

Exercise 2: Let X = {0, 1} and Y= {a, b, c}. Write out the set of blahblahs from X to Y.

I hope you’ll have noticed that Exercises 1 and 2 are more than a little bit alike. Specifically, both can be considered as versions of a certain “higher-level” exercise: “Write out the set of blahblahs from a two-element set to a three-element set” (Exercise 0).

Note that I haven’t told you what a blahblah is; part of the point here is that you don’t need to know. (We’ll call them “functions” eventually; forget this for now if you like.) If I type out a solution to Exercise 1 and include it here… as I fully intend to do momentarily… then you can write out a solution to Exercise 2, simply by substituting a, b, and c (respectively) for my x, y, and z, and simultaneously substituting 0 and 1 for my p and q. Then just leave every other symbol in my solution alone. The result you get will be just as good a “blahblah” as mine whatever it is.

Solution to Exercise 1:

{

{ (p, x), (q, x)},

{ (p, y), (q, y)},

{ (p, z), (q, z)},

{(p,x),(q,y)},

{(p,x),(q,z)},

{(p,y),(q,x)},

{(p,y),(q,z)},

{(p,z),(q,x)},

{(p,z),(q,y)}

}

Now, there’s real value in such exercises. For example, “substitution”… copying a line of “code” from a source document (typically a textbook or an earlier line of code, say) to a newly-handwritten one, while replacing certain of its symbols with certain others… is one of the most basic tricks in all of Algebra and every literate person will have to perform some variant of this process at least from time to time.

To digress only slightly, and to introduce what I expect to turn into something of a theme in these notes. I speak of handwriting though of course other media may be used. Handwriting is much the easiest, and so also the commonest, way to produce what usually call “the code” for a given discussion in my experience. I’ve found very few students willing to join me in typing math and I can’t say I blame the others much: it’s way harder. Calculator code I prefer not to go into just now. For drawings of course handwriting wins hands down. Etch-a-sketch and cel phones notwithstanding.

Note that I don’t consider there even to be a discussion until the student actually produces some code (or drawing, or table… something) for us to talk about.

Again because it’s commonest and easiest, I’ll generally “speak” here in terms of oral discussions, as if one were always already working together with a handful of “students”… usually in dialogue with a particular one of them… at a blackboard with plenty of chalk or a tabletop with plenty of (unlined) paper and pencils.

Returning to our Exercises. To produce a solution to Exercise 2 with a copy of Exercise 1 at hand is possible with no understanding at all of functions or even of blahblahs; it requires only what I’ll here call “scribal” skill. You could teach it to a literate foreigner knowing nothing of each other’s languages.

Before we glorify it with the name of Mathematics, though, we’ll need to have… something else. Welcome to the Math Wars. How much “rigor”? How much “understanding”? (And how much homework with how much calculation… and who gets paid and who cleans up the messes… politics.)

At what I’ll here call “University” level, there’s not much dispute: one seeks students that can “work” Exercise 0 (“write out the set of blahblahs from a two-element set to a three-element set”) given only the following definitions.

Definition 1: A blahblah, b, from a set D to a set R, is a set of ordered pairs with the following properties. The first entry of each ordered pair of b is an element of D (the “domain” of b) and each second entry is an element of R (the “range” of b). Each element of D occurs as the first entry of exactly one ordered pair of b.

Definition 1′: A function is a blahblah.

Notation 1: When f is a function from D to R, we can (and should, if we want to be clear about it, though few enough textbooks say so) write $f: D \rightarrow R\,.$ Usually this is pronounced “eff maps dee to arr”; let this be understood as meaning exactly the same thing as “eff is a function from dee to arr”.

But I myself have seldom worked at this level as a teacher. Calculus students, for example, can be counted on to run screaming from anything resembling Exercise 0. Also to crank out Exercise 2’s all day long (given appropriate Exercise 1’s) and beg for more.

I speak here of course not of Exercises 1 and 2 themselves but their moral equivalents. The lazy students I lovingly refer to… as I had better, since I’ve followed their pattern myself all my life… prefer exercises involving, if not symbol-for-symbol substitutions merely, still little more than their equivalent at the level of result-of-calculation: rather than “all the p‘s get replaced with 0’s, one has something like “differentiate twice and `plug in’ the previous answer”. The trick is to find some exercise that shows you how to do what the author wants without understanding the terminology used in the actual sentences.

You can get through a lot of math courses this way believe you me. Freshman Calculus classes are notoriously often examples of this fact, and so students can even get to consider themselves math majors with scarcely any of the down-to-the-ground, from-the-definitions, quote-only-what-you-can-prove this-I-know-for-sure quality that characterizes “real” mathematics.

Real Mathematics occurs at every level of The Art, of course. Such “ostensive” definitions as “Two is this, many” provide all the formalism needed for very precise understandings of the Theorems (if we choose to think of them as such) that are rediscovered whenever anybody anywhere does some Basic Arithmetic.

Which is as far as most people get. Geometry classes are sometimes found in our student’s (typically dimly-remembered) backgrounds. In such cases one sometimes will have had anyway some exposure to “real” math: here if anywhere one is typically introduced to proofs that depend on definitions.

The importance of definitions to our discussions cannot be overstated.

Posted in Notations, TeX | 3 Comments »