## Archive for the ‘Notations’ Category

announcing the name change:

Virtual MEdZ.

here in the middle are the seven colors

in “mister big -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 “.

had we but world enough. and time. especially time.

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!

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* ()

with its mindbendingly-wrong “plain english”

equivalent(s). the perfectly-correct

(and altogether-necessary) symbolism

(“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,

.

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 ;

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 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 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:

and , 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.

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

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”,

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

thus

whereas one also has

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.)

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

By I will mean a certain Transformation of the *xy*-plane (at this point I tend to write “” on the board; of course , but none of this “set-theoretical” language has made it into my lectures so far). To wit: .

This definition obviously takes the “maps to” notation () 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 “, we have the straightforward declaration (“eff, by definition, is the function that maps ex to ex-squared”).

The more familiar notation gives a “formula” *not* for itself but for . 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 : this is *calling things by their right names* (“The inverse of the function mapping *x* to the cube *root* of *x*–*plus*-five is the function mapping *x* to *x*–*cubed*, *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 (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, (*ordered pairs* of numbers as opposed to *individual* real numbers)—and this is the application we actually wanted: denotes the “reflect in the *y*-axis” transformation. Note that is harder to scan (anyway, so it seems to me); also invokes that pesky “T” and anyhow *you* try getting students to believe that simply won’t do as a LHS.

So. Whenever we *say* “reflect in the *y*-axis”, we can *write* . And I’ve been saying so all along. What’s new here is that I’m proposing to call it . 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, ; 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”: is sure as heck gonna be a lot easier to calculate with.

The Square Root Function (let’s call it “Squirt”) is—of course— a certain bijection on the non-negative reals: symbolically, (“squirt maps the interval zero-to-infinity to itself”). Specifically, if you want to get all technical about it, . Conventionally, one writes in place of ; the argument () 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 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 *dis*continuous *non*multiplicative *non*bijection (, say) on the Complex Number Field. But, and I only *wish* that this were uncontroversial, we sure as sunrise shouldn’t call “The Square Root Function” (or denote it by )—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 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 ; one then has if and only if ; 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”, ) is negative, the solutions to 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 as applied to a negative number is *slang* and should *never* appear without the symbol ?

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

is a natural number less than .

If we’re going to go around calling a perfectly inoffensive set like 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

.

Or how about

is a real number and 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

).

The symbols in question () 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*.