Archive for the ‘Notations’ Category

from blank file-folder (and no idea)
to conceived, drafted, penciled, and inked
before finishing my third cup of coffee:
behold: MEdZ # (1+i+j+k)/2!—
th’ G-mod-H issue!! in which we can see
no less than *four* (count ’em) more-or-less
familiar examples of “modding by a subgroup”.

namely,
(-+) {E,O}—even & odd: the ol’ cosmic yin-yang…
(++) time considered as an endless spiral of half-days…
(- -) the “unit circle” & the “periodic functions”…
(+-) time considered as an endless spiral of weeks.

one can easily look up the (- -) diagram expanded
at arbitrary length in textbooks considering “trig”.
and the helices of endless time are too familiar
to say much more about. the “clock face”, though,
is a bottomless well of shorthand examples—there’s
a car trying to run us off the road at three o’clock—
and so might be worth some further development
if it were useful in, say, fixing notations.

but (-+) one is prepared to go on about at any length.
\Bbb{Z}_2 = \{0, 1\} is one of the most useful finite sets there is,
after all. for example, the 64 hexagrams of the i-ching
amount at some level to \Bbb{Z}_2^6 displayed pleasingly.
in the zine by me about “the 64 things” the hexagrams are
replaced with “subgraphs of K_4” (where of course K_4 is
the “complete graph on four points”; a diagram having
six edges [giving the six “lines” of the ching in that
version]; the 64 things are then subsets-a-six-set
[say {y, b, r, p, o, g} for the “full color” version]).
for example. i hope to continue in this vein later.
hello out there. ☰☱☲☳☴☵☶☷
Photo on 6-20-20 at 9.41 AM

let
B =\begin{pmatrix}  \begin{pmatrix} \begin{pmatrix} 0 & 2 \\ 1 & 2\end{pmatrix}& \begin{pmatrix} 1 & 1 \\ 0 & 1\end{pmatrix}& \\ \begin{pmatrix} 1 & 0 \\ 2 & 1\end{pmatrix}& \begin{pmatrix} 2 & 2 \\ 1 & 0\end{pmatrix} \end{pmatrix} & \begin{pmatrix} \begin{pmatrix} 1 & 2 \\ 1 & 0\end{pmatrix}& \begin{pmatrix} 2 & 1 \\ 0 & 2\end{pmatrix}& \\ \begin{pmatrix} 2 & 0 \\ 2 & 2\end{pmatrix}& \begin{pmatrix} 0 & 2 \\ 1 & 1\end{pmatrix} \end{pmatrix} \\  \begin{pmatrix} \begin{pmatrix} 0 & 1 \\ 2 & 2\end{pmatrix}& \begin{pmatrix} 1 & 0 \\ 1 & 1\end{pmatrix}& \\ \begin{pmatrix} 1 & 2 \\ 0 & 1\end{pmatrix}& \begin{pmatrix} 2 & 1 \\ 2 & 0\end{pmatrix} \end{pmatrix} & \begin{pmatrix} \begin{pmatrix} 1 & 1 \\ 2 & 0\end{pmatrix}& \begin{pmatrix} 2 & 0 \\ 1 & 2\end{pmatrix}& \\ \begin{pmatrix} 2 & 2 \\ 0 & 2\end{pmatrix}& \begin{pmatrix} 0 & 1 \\ 2 & 1\end{pmatrix} \end{pmatrix} \end{pmatrix}

now let
B=\begin{pmatrix} II & I\\ III & IV\end{pmatrix}, i.e., let
II = \begin{pmatrix} \begin{pmatrix} 0 & 2 \\ 1 & 2\end{pmatrix}& \begin{pmatrix} 1 & 1 \\ 0 & 1\end{pmatrix}& \\ \begin{pmatrix} 1 & 0 \\ 2 & 1\end{pmatrix}& \begin{pmatrix} 2 & 2 \\ 1 & 0\end{pmatrix} \end{pmatrix}, etc., so that
I, II, III, and IV (the “quadrants”) denote
two-by-twos of two-by-twos.

we’ll call the smallest matrices in sight “points”.
e.g.,
h =\begin{pmatrix} 2 & 1 \\ 0 & 2\end{pmatrix};
the quadrants are now two-by-two arrays of “points”.

on the graphical models i’ve been going on about all week
the quadrants are (respectively)
I ++
II -+
III —
IV +-
(as is familiar to every calculus student);
expanding on the same logic gives the “trit-code”
for the individual points. for example,
h = ++++
and so on.
one then simply translates the 16 trit-strings
++++, +++-, ++-+, … —-
into “hurwitz units” like
h = {{1+i+j+k}\over2};
voila.

oh, yeah. i forgot to say. the “points” are
matrices *mod 3* (so that 2 = -1). that is all.

Q/{\pm 1} \iso K_4

the (so-called) fundamental quaternion units
can be represented as
\pm 1 =\pm\begin{pmatrix} 1&0\\0&1\end{pmatrix}, \pm i =\pm\begin{pmatrix} 0&1\\2&0\end{pmatrix}
\pm j =\pm\begin{pmatrix} 1&1\\1&2\end{pmatrix}, \pm k =\pm\begin{pmatrix} 1&2\\2&2\end{pmatrix},

with the scalars of the matrices considered
as elements of \Bbb{F}_3—i.e., 2=-1 etc.

“modding out” the \pm1 gives \Bbb{Z}_2^2
the klein-four group { (0,0), (1,0), (0,1), (1,1) }
(with componentwise mod-2 addition).

typing out matrices is sort of tedious. i won’t be doing
\hat{A_4} today.

announcing the name change:
Virtual MEdZ.

Photo on 2014-03-14 at 16.06

here in the middle are the seven colors
in “mister big \pi-oh” (from ohio) order:
MRBGPYO
(mud, red, blue, green, purple,
yellow, orange). i’ve drawn the “line”
(which appears as a triangle) formed by
“marking” the purple vertex and performing
the “two steps forward and one step back”
procedure: one easily verifies that
{G, O, P} is a line as described in
the previous post (“the secondaries”).

all to do with “duality in P^2({\Bbb F}_2)“.
had we but world enough. and time. especially time.

eliminating the middle, man (06/07)
was my third post (in vlorbik
on math ed
, as this site
was first known [to 01/10.
next was MathEdZineBlog, to 01/13;
then a grader’s notes]). my fourth post
was {X : X is full of baloney} (06/07).

these were published as a two-parter
with the (dull, dull) title textbooks
and notations
. the badness of
standard-text “set builder” notation
(the topic of the latter) had, for
quite a while, been a burning issue
for me; my then-recent discovery of the
attack on the “sign of intersection”
(the topic of the former) nudged me
into finally ranting that rant online.

so there’s some evidence that at least part
of why i *began* mathblogging when i did
was the need to announce that i’d been
newly horrified by new depths achieved
by the enemies of clarity in the long-
-established Notation Wars.

much more recently, i was horrified
anew: in the sloppily-ranted (and,
again, boringly-titled) midterm report
of 11/12, i announced my discovery
that The Enemy had come for the set-inclusion
symbol in my favorite intro-to-*real*-math
course (“linear algebra”).

the rest of the notations file is mostly
more about notations themselves than
Notation Wars. i haven’t been very good
about tagging my posts.

along the way, there was capital script-D of f,
(01/09) pointing out (among other things)
that remedial-algebra courses
daring even to *mention* “domains”
and “ranges” really (reallyreally)
ought to also introduce *symbols*
for these objects. (*easily written*
symbols, of course.)

somewhere i may even already have indicated
one high-hope-against-all-lack-of-hope: early
(and correct, and consistent) use of (the standard)
f: D \rightarrow C
notation (for a function f with domain
D and co-domain C). as scary as that
might be. (once the actual students
actually see how useful “careful use
of code” actually *is*, they’ll grab
it when they need it… more and more.
this is part of what’s called “getting
good”.)

anyhow, i guess i’m just circling wagons
right in here. there’s some sign-of-equality
stuff by me in the blog next door, for instance.
and i’m feeling a need to have the evidence
(that Textbook Set Theory, long dying of
a usually-fatal illness, has been mortally
wounded in the bargain and is sinking fast)
much better organized (and, just maybe,
somewhat more level-headedly presented).

but not right away, not now.

this is the limit

so i’m grading linear algebra like i usually do.

where we need to speak… or anyway, to read
and write… frequently about *column vectors*.

much the usual thing (for example) in defining
a linear transformation (called F, say)
on “real three-space” (so F: R^3 —> R^3)
is to *consider R^3 as the space of real-valued
column 3-vectors* and then to supply a matrix
(called [F], say; [F] will be a 3-by-3
matrix in this context) such that
F([a, b, c]^T) = [F][a, b, c]^T.

the “caret-T” here denotes “transpose”…
the idea is that one has something like
[a, b, c]^T=
[
a
b
c
]
;
in laypersons’ language, the transpose
symbol ^T tells us to turn our old rows
into the new columns (which simultaneously
turns our old columns into the new rows).

in the language of the widely-used
TI-* calculator line, one has
[a, b, c]^T = [[a][b][c]]…
and this is starting to look
better and better to me right
in here.

but what one really seems to *want*
here is a quick-and-dirty notation
for expressing (what we will still
continue to *speak* of as)
column-vectors, as rows.

and i’ve noticed student papers using
< x, y, z > = [x, y, z]^T.
this looks like a real useful convention
to me and i’ve adopted it for my own use
until further notice.

angle-bracketed vectors have been useful
to me before. mostly, i think, in the context
of “sequences & series” typically dealt with
in about calc 2 or 3.

LANGLE x_0, x_1, x_2, … RANGLE
(i.e., < x_0, x_1, x_2, … >
… “angle brackets” are special characters
in HTML and so i prefer to avoid ’em)…
in either notation…
represents sequence of objects
LANGLE x_n RANGLE
(which is of course *not* the same
as the *set* R={ x_n } = {x_0, x_1, x_2, …}
[the set of values taken by the function
f(n) = x_n
on the natural numbers NATS:={0, 1, 2, …}]).

this usage actually *extends* a usage,
ideally introduced and maintained earlier
in a given presentation (class or text or
who knows maybe someday even both at once)
using angle-brackets for (finite-dimensional)
*vectors*.

LANGLE 3, 4 RANGLE
now represents the vector that, ideally,
we would represent in some other part of
our presentation as [3,4]^T…
a *column vector*.

i remark here that meanwhile
(3,4)
represents a so-called “point”
in “the x-y plane”… an entirely different
(though closely related) object.

we pause here and take a deep breath.

i’ve pointed at two distinctions:
sequences-as-opposed-to-sets and
vectors-as-opposed-to-points.

many textbooks… and many instructors…
are *very sloppy* about keeping these
(and many other suchlike) distinctions clear.

the *notations* used in *distinguishing*
the distinct situations in each case were
“delimiters”: pairs of opening-and-closing
symbols used to mark off pieces of code
meant to be handled as single objects.

delimiting conventions are vital even in
ordinary literacy (“you see? he” sa)i(d.
and i claim they’re all the more so in maths
(since we get fewer and weaker “context clues”
when the code gets munged [as in the example]).

and students’ll just leave ’em out altogether
if they think… or god help me, know… they
can get away with it. failing that, choose
randomly (itt…oghm,k…).

failing that, “well, you *know*
what i *meant* was”…

too late, too late. here endeth the sermon.

in our next episode of “who stole my infrastructure?”:
the dot product. when they came for the opening-apostrophe,
i pleaded and begged. when they came for the
sign-of-intersection i raved incoherently.
never had a chance, no hope, no hope. doom doom doom.
can somebody pick up the torch, here?
i don’t think i can go on much longer.

vlorbik on punctuation for the twenty-twelve.
more clarity!

midterm report

at the foundation of (an earlier version of)
this blog i ranted and rambled about
a tendency on the part of (lower-division
college math) textbooks to hamper the work
of the teacher by (deliberately!) suppressing
correct technical language.

of course things have continued to deteriorate.
but, by some miracle, i’m still earning
the random crust of bread by helping students
learn to *read* these ever-more-horribly flawed
documents. so far so good, then, i suppose.

anyhow, i’m grading linear algebra again
right in here (nothing *but* linear algebra
for something like a *year* now)…
and i’ve only recently become vividly aware
that this “tendency” has penetrated deeply into
the textbooks at this “higher level of the game”.

specifically, i hereby announce that some
satanic force has somehow (even here) replaced
*the sign of set-membership* (\in)
with its mindbendingly-wrong “plain english”
equivalent(s). the perfectly-correct
(and altogether-necessary) symbolism
x \in S (“x is an element of S”)
is now to be replaced, by the edict of
invisible (and mostly unimaginable)
entities, with “x is in S”.

[
this is a good place to skip ahead.

i’m going to geek out slightly here.
you *don’t* have to be an adept to follow.
i’m hoping to make a point that can be
at least *partly* understood by math laity.

is the set of natural numbers (more here
much my most popular post here and probably
my best-read production of all time) “in” the set
of real numbers? loosely, yes. more precisely,
{\Bbb N} \subset {\Bbb R}.

i can easily imagine myself talking to, say, another
teacher about, say, some “property” (like commutativity-
-of-addition; x+y=y+x [for all x & y]) that applies
in the natural numbers. “how do we *know* it applies?”,
i might say. and the answer might come: “because
the naturals are in the reals, and the *reals* enjoy
the property of commutativity”. “good answer!” i would
then reply, and move on to whatever i *really* wanted
to talk about.

again. are all possible probabilities in the reals?
well, yeah, (duh)! in “code”, one has [0, 1] \subset {\Bbb R};
rephrasing, “all the numbers from zero to one (inclusive)
are in the reals” (but also, more precisely,
“the [closed] unit interval is a subset of the set of real numbers”)…
so. now i’m talking to some grad-school dropout (say): “is
the-interval-from-zero-to-one in the reals?” she asks;
“heck, yes” say i, and we get on with whatever we’re really doing.

is “pi” in the real numbers? sure! \pi \in {\Bbb R}.
“pi is an element of the reals”.

but wait! being-a-subset is an *entirely different* relation
from being-an-element! pi is simply *not* “in” the reals
in the same way that \Bbb N is!

who cares? well, me and a few hundred thousand others or so.
if *you* don’t care? well, that’s why i invited you
to skip this part! read on!
]

the biggest problem from a practical standpoint
(if “how can we make this material better understood”
is a practical question) is simply that students
*hate writing* and at *every opportunity* will
replace “plain english” with (typically very
ill-understood) bits-and-pieces.

*nobody*… no student, no lecturer, no pro
mathematician… is going to write out the phrase
“is a real number” a whole lot more than
twenty or thirty times (in a given sitting-down)
without wanting *some* abbreviation for that
phrase.

and likewise for “is a subset of”… indeed,
*any* sufficiently common phrase *begs for
abbreviation* even in “plain english”

so there it is. mathophiles also… in some sense…
“hate writing”. anyhow, we *love abbreviating*.

“algebra is the science of equations”, i once
heard someone say (explicitly repeating something
he had learned “by rote” from a public-school
teacher during his own schooling… it went on
for another few lines but i didn’t learn that
part from listening to this guy say it three
or four times that one night). and i consider
this to’ve been very well said.

so i’ll hope to return to it.

but first. this history of elementary algebra
at w’edia summarizes the standard dogma of its subject (as i
understand it) well. the evolution from “rhetorical” algebra
(describe *everything* in plain-language words) through “syncopated”
algebra (where “shorthand” symbols [many still common today]
began to replace the most common techical terms… but the
actual *reasoning* was still natural-language based [and so,
by contemporary standards, “informal”]) into
“symbolic algebra” (the “science of equations” as we know it
today: a study of “formal” properties of [carefully-defined!]
symbolic “objects” [“variables” and “equations”, for example]).

what w’edia *doesn’t* make much of… but what matters to me
a great deal… is that the emergence of algebra pretty
closely *coincides* in (so-called) *western* history with
the (so-called) renaissance and the (no sneerword necessary)
scientific revolution.

“modern times”, then, *began* when certain humans (*finally*!)
figured out how collections of symbols-on-paper (representing
certain abstractly-defined-objects), produced according to
various “rules”, could be interpreted to reveal previously-obscure
*laws of nature* (so-called). this *was* the “scientific revolution”.

and how does it work? *equations* is how.
so one of the *first* things in understanding
what’s going on the contemporary philosophical
environment is to find out *what an equation is*
(for all literate people): “the equality meaning
of the equals sign”
.

when we’re being sloppy, we can confuse “=” with “is”…
but when we actually get to work *using* equations,
we have to *much more precise* to get any value from
the procedure at all. plain-english “is” is *always*
in some sense metaphorical (except in empty utterances
like “it is what it is”)… whereas the equal-sign
rightly-used is as far away from metaphor as we know
how to get.

how does algebra work? (equations and *what* else?)
by *the method of substitution* is how.
“in a context such that A=B is taken as ‘true’,
a properly-written piece of code
including (the symbol) B
*does not change its truth-value*
when (the symbol) A is *substituted* for B.”

this “method” *characterises* algebra.
i first became aware of some its awesome power
in about seventh grade.

and *what* else? “doing the same thing to
both sides of an equation”. and what else?
that’s about it. that’s algebra. the rest
is commentary.

now, for *set theory*, two of the main ideas
are caught up in set-inclusion and set-containment:
\pi \in \Bbb R and {\Bbb N} \subset \Bbb R, for example.

and one must be every bit as careful in the use
of these symbols when studying sets as one must be
in the use of the sign-of-equality in studying,
say, polynomial equations (i.e., pretty much,
in algebra).

but about forty percent of the class already don’t
take *equations* seriously. and they’re morally
certain that “sets” are meaningless traps designed
to distract them from “how do i get the *answer*?”.

and the textbook does a great deal to encourage them
to *maintain* this view.

and i don’t like it. please stop.

oh frabjous day

charles (six-winged-seraph) wells
is back at the “discourse” stuff. yay!
abuse of notation & mathematical usage.

composition of linear fractional transformations
compared to two-by-two matrix multiplications.

consider
\lbrack x \mapsto {{Ax +B}\over{Cx+D}} \rbrack \circ \lbrack x \mapsto {{ax +b}\over{cx+d}} \rbrack\,.

in other words,
let f(x) = (Ax+B)/(Cx+D) and
let g(x) =(ax+b)/(cx+d) and
consider the function f\circ g (“f\circ g”,
i.e. f-composed-with-g). recall
(or trust me on this) that
[f\circ g](x) = f(g(x)); i.e.,
functions compose right-to-left
(“first do gee to ex; then plug in
the answer and do eff *to* gee-of-ex”…
first g, then f… alas. but there it is).

so we have
\lbrack x \mapsto {{Ax +B}\over{Cx+D}} \rbrack \circ \lbrack x \mapsto {{ax +b}\over{cx+d}} \rbrack
= \lbrack x \mapsto { {A{{ax +b}\over{cx+d}} + B}\over{C{{ax +b}\over{cx+d}} +D} }\rbrack
= \lbrack x \mapsto { {A(ax+b) + B(cx+d)}\over{C(ax+b) + D(cx+d)}}\rbrack
= \lbrack x\mapsto { {(Aa+Bc)x + (Ab+Bd)}\over{(Ca+Dc)x + (Cb+Dd)} }\rbrack\,.
thus
\lbrack x \mapsto {{Ax +B}\over{Cx+D}} \rbrack \circ \lbrack x \mapsto {{ax +b}\over{cx+d}} \rbrack =\lbrack x\mapsto { {(Aa+Bc)x + (Ab+Bd)}\over{(Ca+Dc)x + (Cb+Dd)} }\rbrack\,.

whereas one also has
\begin{pmatrix} A & B \\ C  & D\end{pmatrix} \begin{pmatrix} a & b \\ c & d \end{pmatrix} = \begin{pmatrix} Aa +Bc & Ab + Bd \\ Ca+Dc  & Cb + Dd \end{pmatrix}\,.

so the matrix-multiplication equation
can be obtained from the function-composition equation
merely by applying an eraser here and there.

(my lecture-note-blogging of winter 09 include some
remarks on \mapsto notation and much more
about linear fractional (“mobius”) transformations.)