Archive for the ‘Notations’ Category

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.)

Message: I Care

I’d been groping for the right notation for Transformations of Graphs since the first day; I settled it over the weekend.

By \langle a,b \rangle I will mean a certain Transformation of the xy-plane (at this point I tend to write “\Bbb R^2” on the board; of course {\Bbb R}^2 ={\Bbb R}\times {\Bbb R} =\{ (x, y) | x \in {\Bbb R}, y \in {\Bbb R}\}, but none of this “set-theoretical” language has made it into my lectures so far). To wit: \langle a , b \rangle := \lbrack (x,y) \mapsto (a,b)  \rbrack.

This definition obviously takes the “maps to” notation (\mapsto) for defining functions for granted—which I’ve sort of been doing all along without pinning myself down with anything as vulgar as a definition. The right-hand side of the latest equation, then … hold it. Does everybody know that the colon-equalsign
combination means “equals by definition”? Well, it’s a pretty handy little trick, let me tell you. OK. Now. The RHS in our latest equation expoits a notation rarely seen in lower-division texts (alas): instead of the ungainly “Let f be the function defined by f(x) = x^2“, we have the straightforward declaration f:= \lbrack x \mapsto x^2 \rbrack (“eff, by definition, is the function that maps ex to ex-squared”).

The more familiar notation gives a “formula” not for f itself but for f(x). A lot of people would have you believe that this distinction doesn’t matter and in certain contexts such people must even be put up with. But it sure matters to me, here and now. Because once I know how to write definitions in the “maps to” style, I don’t need to mention any arbitrary old letter-of-the-alphabet like f when what I’m really talking about is “the squaring function” … and I can just go ahead and write down facts like \lbrack x \mapsto\root 3\of{x+5}\rbrack^{-1} = \lbrack x\mapsto x^3 - 5\rbrack: this is calling things by their right names (“The inverse of the function mapping x to the cube root of xplus-five is the function mapping x to xcubed, minus five”—you just can’t write that sentence in “f(x)” style … only something like “Let f be BLAHBLAH; then f-inverse is LALALA”— but what’s any of it really got to do with anything called “eff”?).

Readers already familiar with all of these ideas—or astonishingly quick on the uptake—might notice that, so far, it might appear that I don’t actually need the “maps to” notation for my purposes.After all (for example), one has \lbrack x \mapsto \root3\of{x+5} \rbrack = \{ (x, \root3\of{x+5}) \} (recall that a function is a set of ordered pairs)—and the “ordered pair” version is almost as concise as the “mapping” language. But here’s the real payoff: the “arrow” notation carries over seamlessly when the domain is, say, {\Bbb R}^2 (ordered pairs of numbers as opposed to individual real numbers)—and this is the application we actually wanted: \lbrack (x,y) \mapsto (-x,y) \rbrack denotes the “reflect in the y-axis” transformation. Note that \{ ((x,y),(-x,y))\} is harder to scan (anyway, so it seems to me); also T((x,y)) = (-x,y) invokes that pesky “T” and anyhow you try getting students to believe that T(x,y) simply won’t do as a LHS.

So. Whenever we say “reflect in the y-axis”, we can write \lbrack (x,y) \mapsto (-x,y) \rbrack. And I’ve been saying so all along. What’s new here is that I’m proposing to call it \langle -x, y \rangle. This has the drawback that it “freezes” the variables x and y: wherever “angle brackets” are in effect, x and y must mean “the first and second co-ordinates of a certain ordered pair” (note that, by contrast, \lbrack (x,y) \mapsto (-x,y) \rbrack = \lbrack (a,b) \mapsto (-a,b) \rbrack; the variable names here can be changed without changing the actual set of orderded pairs itself).

And this “freezing” is indeed somewhat unfortunate. But I’m more than willing to pay that price, to have a quick-and-dirty way to spell “shift left by three”: \langle x-3, y \rangle is sure as heck gonna be a lot easier to calculate with.

Radical Notations

The Square Root Function (let’s call it “Squirt”) is—of course— a certain bijection on the non-negative reals: symbolically, \sqrt{}: [0,\infty) \rightarrow [0,\infty) (“squirt maps the interval zero-to-infinity to itself”). Specifically, if you want to get all technical about it, \sqrt{\null} := \{(x,y) | x, y \in [0,\infty); y^2 = x\}. Conventionally, one writes \sqrt{x} in place of \sqrt{\null}(x); the argument (x) is called the “radicand” and is said to be “under the radical”. We’ll observe this convention throughout the rest of the discussion. Squirt is a continuous function and enjoys the property that \sqrt{ab} = \sqrt{a}\sqrt{b} whenever both sides of the equation are defined (functions with this property are said to be “multiplicative”). All of this ought to be completely uncontroversial.

Now, it’s perfectly possible—of course—to extend squirt to a certain discontinuous nonmultiplicative nonbijection (f, say) on the Complex Number Field. But, and I only wish that this were uncontroversial, we sure as sunrise shouldn’t call f “The Square Root Function” (or denote it by \sqrt{\null})—continuity, bijectivity, and multiplicativity are all very useful properties and shouldn’t be given up at the careless stroke of a keypad.

Yesterday, for my sins, I was made to write out somesuch ghastly nonsense as \sqrt{-5}\sqrt{-7} = -\sqrt{35} right in front of my 104 students. Obviously, I couldn’t bring myself to do it without complaining about it—these people mostly seem to trust me and I’d like to try to deserve it. But whenever I’m made to differ with the text it not only undermines their faith in me personally, but also gives support to the all-too-common idea that Mathematics Is Management: that our ways are arbitrary and meaningless and subject to change at some the whim of some unseen authority figure.

I suppose I know why this section of the textbook is there. We’re about to develop QF—the famous “Quadratic Formula” (assume A\not=0; one then has Ax^2 + Bx + C = 0 if and only if x = {{-B \pm \sqrt{B^2 - 4AC}}\over{2A}}; I’ve spelled it out mostly out of sheer joie de symbolisme but also to take the opportunity to beg other teachers to adopt the convention that Constants Get Capitalized). When the radicand in QF (also known as the “discriminant”, B^2 - 4AC) is negative, the solutions to Ax^2 + Bx + C = 0 are non-real; students of Algebra need to learn about such solutions. The moment has come: the (so-called) Real Numbers are no longer enough for our purposes. This is, not only well and good, but dearer to my heart than I like to admit in public.

But for pity sake, now that we’re letting these struggling beginners in on this earth-shaking idea (that confused great mathematicians for hundreds of years), why make it any harder than it has to be? Why not just admit what every professional knows: that the symbol ``\sqrt{\null}'' as applied to a negative number is slang and should never appear without the symbol ``\pm''?

Let me be as clear as I know how. I understand why the textbooks (and the furshlugginer graphing calculator) get this wrong: publishers and computer manufacturers are capitalist pigs, not only indifferent to the truth but actively hostile to the truth. What I don’t get is this: where are the mathematicians? How can you go to work every day and allow this kind of thing to go on in your department at your university, in the name of “mathematics”? What the devil do you think tenure is for?

Turning our attention to beginning algebra courses. First of all, the text I’ve been using most recently is called Intermediate Algebra; “intermediate between pre-algebra and actual (university-credit earning) algebra” is the most charitable spin I can put on that. Anyhow, here again, set theory is used so clumsily that it’s hard not to attribute malice to somebody along the line (Hanlon’s Razor notwithstanding).

Consider, then, an abomination like
\{ x | x is a natural number less than 3\}.

If we’re going to go around calling a perfectly inoffensive set like \{1, 2\} out of its name in order to make some point about our notations, we’d be much better off to actually pretend we believed these very notations were actually good for something and write instead
\{x | x \in N, x \, \langle\, 3\}.

Or how about
\{x| x is a real number and x is not a rational number \}? Doesn’t it just make you want to, I don’t know, hurl the chalk at something? Actually, I have to admit that I’ve copied this display on several blackboards in my time … but only to illustrate a point (namely, that it was created by enemies of mathematics and that one of course really means
\{x| x\in R, x\not\in Q\}).

The symbols in question (N, R, Q, \in, \not\in) appear in this very section and are used in the exercises.  Though not, of course, in the exercises about “set-builder” notation — no, these have all been carefully contrived to reinforce the reader’s impression that our goal in presenting this material is to make easy things hard by way of the whole ignore-the-point word & symbol mishmosh I’ve just been complaining of.

But then, that brings us to the saddest part of the whole sorry business. We don’t actually need (still less want) these symbols — or the set-builder notation itself! — for whatever follows in the whole rest of the book! And pretty much every 9th-grade-algebra-for-college-students text that’s come out in several (admittedly very short) generations does things in exactly the same way!

I have what feels like a pretty coherent theory of how things got this way (though essentially no idea as to “what, then, are we to do?”).  But I promised myself I’d keep it brief, so I’ll just wave my hands in the direction of The Muddle Machine.