Archive for February, 2011
Exam II. (Math 104.)
Polynomials. Quadratics in particular.
Bad news.
The campus-wide median score slipped
considerably and so did ours.
A lot of damage was done on the page
having problems of the forms
(a.) Complete the Square…
and
(b.) Use the Quadratic Formula…
(… to solve a given quadratic equation.).
I’ve written a little about QF in the past;
one of my most popular posts. And I like
getting attention as much as the next blogger.
Here are some remarks on “completing the square”.
So. Imagine me… imagine *us*…
scribbling on a blackboard and talking.
Let’s look at a quadratic equation
whose answer we *already know*.
Suppose that [ (x = 3) OR (x = -9)].
Then (“obviously”… check it!)
(x-3)(x+9) = 0.
“FOIL”
(or, finding this distasteful, expand
the product-of-binomials on the Left
Hand Side by some “other” algorithm):
x^2 + 6x – 27 = 0.
Now. Suppose it were some bloody
Exam Problem. “Complete the Square.”
Digress.
The trick is to find
a “perfect square trinomial”
having the same variable terms
as *our* trinomial (x^2 + 6x – 27).
We want
(x+?)^2 = x^2 + 6x + ___.
And the thing *here* is that
(x+?)^2 = x^2 + (2*?)*x + ?^2.
It follows that 2*? = 6, and so ? = 3.
Obviously, then, ?^2 = 9.
We’ve established that
(x+3)^2 = x^2 + 6x + 9
has the same variable terms
as our original polynomial.
We’re ready to go. (End digression.)
Start with x^2 + 6x – 27 = 0.
Add on both sides (to “isolate”
the variable terms):
x^2 + 6x = 27.
Here comes the magic…
the step that gives “Complete the Square”
its name. Add on both sides *again*:
x^2 + 6x + 9 = 27 + 9.
(the point here was to get the
“perfect square trinomial”
we computed in the digression;
the LHS has been transformed
into a easily-manipulated form
[abstraction to the rescue!].)
Now just use the “perfect square” property:
x^2 + 6x + 9 = 27 + 9
becomes
(x+3)^2 = 36. So
x+3 = plus-or-minus root-36;
x = -3 plus-or-minus 6;
finally
x = 3 or x = -9
(as we already knew from the
factored form we began with).
With the example in hand, the best
thing is to go off and *do* a bunch
of similar examples. And *then*
consider the “abstract” version
used in deriving the Quadratic Formula.
Never; the less.
Let A, B, and C be Complex Numbers.
(OK… let ’em be elements of your
favorite Field. Rationals, Reals,
and Complexes all work; so do many
other Fields not considered in
School Mathemathics (alas).
The point of using 
the Complex Number field…
is that the “answers” will always “exist”.
Famously x^2 + 1 = 0 has no (so-called)
“real number” answers; this kind of thing
doesn’t happen in “algebraically closed”
fields like the Complex Numbers.)
Suppose further that A\not=0.
The Following Are Equivalent.
Ax^2 + Bx + C = 0
x^2 + (B/A)x = -C/A
x^2 + (B/A)x + (B/[2A])^2 = -C/A + B^2/[4A^2]
(x+ B/[2A])^2 = [-4CA]/[4A^2] + B^2/[4A^2]
x + B/[2A] = \pm \sqrt {[B^2 – 4AC]/[4A^2]
x = [-B \pm \sqrt {B^2 – 4AC}]/[2A].
In other words, when A is nonzero one has
i.e., QF: we have derived
the Quadratic Formula.
(Remark: one need not be able to reproduce
this calculation to “prove” QF. Suppose you’ve
*memorized* it at some point but now you’re not
so sure. To check that *your* formula really is
the true “quadratic” formula, just choose either
the “plus” or the “minus” in the appropriate
place in the code [where i’ve abbreviated \pm
where i haven’t just written-it-out].
Then just “plug in” the whole mess on
Ax^2 + Bx + C and turn the crank until
out pops zero [if your formula and calculations
are correct]. It’s easier to *check*
that the formula works than to *derive* it.
It’s a good exercise, too, in my opinion…
but I don’t think I’ve ever assigned it
[or seen it assigned in a textbook].
)
Here’s a verbal recap.
We began by “dividing away the leading coefficient”.
This is where we “use” the condition A\not=0.
(Of course, it also makes sense to say that we “used”
this condition already in calling Ax^2 + Bx + C = 0
a quadratic equation in the first place…
if A were 0, one would have the *linear* equation
Bx+C=0. (We can handle these *without* memorizing
the “Linear Formula” [the equation is equivalent…
when B\not=0… to x = -C/B]. Instead one learns
“steps” [subtract C from both sides; divide by B].
For that matter, while I’m thinking about it,
neither does one expect students to learn that
“y = Mx + B” equivalences-to “x = (y-B)/M”
[when, dammit, M\not=0… I can’t help myself…];
rather, again, one simply carries out certain “steps”
to *dervive* this result whenever it’s needed.
Formulas-versus-procedures is a major battleground
in the Math Ed Wars so I think about this stuff.
A great deal sometimes.
)
Having “divided through by A” we get
(the so-called “monic” polynomial…
having lead coefficient 1
is a big enough deal to deserve
its own name…) x^2 + (B/A)x + C/A = 0;
subtracting the constant on sides gives
x^2 + (B/A)x = -C/A.
The number (B/A) is in the position of
the “6” in our first example.
Just as it turned out in the example,
where we add 9 to both sides because
9 is (6/2)^2, in *every* instance
of “complete the square” we add
the square of half of the constant-coefficient
(of the monic form) to both sides of the equation.
x^2 + (B/A)x + (B/[2A])^2 = -C/A + B^2/[4A^2]
Factor on the left; take square roots
on both sides (slightly tricky;
don’t forget the \pm!); clean up.
Coffee!
my previous post was about
an earlier version of this drawing.
i’ve added most of the rest of the lines.
if the fine print emerges in your version,
you’ll see
*three vertical
*three horizontal
and
*three upward-sloping
lines, plus the “line at infinity”.
i’ve also added (what i’ll now call)
“polar lines”
for a few more of the points
(specifically, the four points of the
line at infinity… recall that the
“finite points” of the diagram
are the 9 still-blank circles
forming a 3-by-3 square in the middle).
for the *polarity* (a certain
pairing-of-points-with-lines)
i’ve begun to define here,
each point-at-infinity is the
*pole* (or “polar point”)
associated with a *vertical* line
(its so-called “polar line”,
typically called just the *polar*
[for the given pole]).
the topmost point (i hereby declare)
is the Point At Infinity
(the “distinguished” point of
the Line At Infinity).
the topmost point considered as a Pole
has *the line at infinity* as its Polar.
so (assuming the drawing eventually
*does* present some particular Polarity)
the point-at-infinity is a *self-conjugate* point
for the polarity we are beginning to consider.
(quoting meserve, “a point that is on its own
polar is called a self-conguate point
of the polarity” (p. 137).)
it’s not by accident that i chose the *vertical* lines
for the polars of the other “infinite” points
of the system. the notion that
parallel lines of “ordinary” (3-by-3) space
“meet at infinity” (in projective space)
suggests that *points at infinity*
can be associated with
*slopes of lines*… and indeed that’s
precisely what’s been done here.
the three lines of each “parallel class”
(vertical, horizontal, or upward-sloping in the diagram)
come together at some *particular*
point-at-infinity…
and so i’ve placed the points in what
i hope are suggestive parts of the picture:
the verticals with the point on top;
the horizontals with the point to the side;
the diagonals with the points at the corners.
(“easy”) exercise: fill in the missing three lines.
(“hard”) exercise: finish filling in the poles-to-polars
bubbles. (i do *not* claim that there is only one way
to do this). hint.
until that day, i’ve set up a process
involving “photo booth” (on the mac)
and “flickr” (on the web). optionally,
i can involve the mac’s “iPhoto”.
not sure yet if that’s ever going to
be useful… but it organized the
“photo booth” stuff in maybe a better
way than Photo Booth itself.
i’m *writing* this passage in “flickr”
(but i’ll probably *edit* it in wordpress;
in particular, there’ll be a redundant
line ID’ing this as a flickr shot i think;
if so i’ll kill it).
“the medium is (part of) the message”
as i’m given to saying. as for the drawing
itself? also an exercise in media (of course!):
the very-sloppy-looking double line
(mostly along the bottom and RHS
[RHS abbreviates “right hand side”;
very useful in making terse remarks
(typically in chalk or marker on a board
black- or white-) about equations])
shows pretty clearly (if you didn’t know
already) that marker-on-paper is *not*
something i’m very practiced in at this scale.
even this took a couple of drafts
(i’m out of whiteout or id’ve used
it at the first sign of trouble… so
a careful description of the “medium”
would include the information
“no corrections allowed”).
anyhow. as to the “content”.
the 13 circles making up “the big picture”
represent 
the Projective Plane constructed on
the Field of Order 3.
there’s a 3-by-3 square of points.
call these points Finite Points.
the other four are called Points At Infinity
(together, these form the Line At Infinity…
so the “double line” i mentioned earlier
was the “line at infinity” all along).
the P-at-I at the top is associated
with the *vertical* direction,
the P-at-I on the left is associated
with the *horizontal* direction,
and the two P’s-at-I in the corners
stand for the two *diagonal* directions.
as an example of the “diagonal” directions,
i’ve drawn the three “lines of slope one”.
each one passes through three Finite Points
*and* through the Infinite Point at lower-left.
of course the lines-of-slope-one are *parallel*
in Finite Space… in our context, this means
that the dotted lines meet *only* at the lower-left
Infinite Point (accounting for the “association”
of this point with the upward-sloping diagonal
to which i referred two paragraphs up):
the slogan “parallel lines never meet”
is replaced in projective spaces
with “parallel lines meet at infinity”.
(sort of. in the most general setting,
*any* line [or none] can be thought of
as “the” line at infinity… so it’d be more
accurate to replace “parallels never meet”
with “there *are no* parallel lines”.)
this is probably the best example of the
the concept of Ideal Points: one creates
new elements to include into a set to make
certain nice things happen. another example:
i’ve gone on (in my exuberance) to draw
*another* little P_2(F_2) inside the lower-left
(infinite) point.
when the drawing is finished, the points of
the Big Picture that correspond (in the
“obvious” way: unreoriented-blowup-and-shift
[a “homothety” if i understand correctly])
to the points of the “shaded” line will be
precisely those points whose Little-Picture
lines-“inside”-of-points *pass through*
the point-at-lower-left itself.
this “little” P_2(F_2) isn’t *necessarily*
found at this spot… i *put* it there.
there are many other ways…
in some other lecture, i’ll want to
look at *how* many…
to set up a points-to-lines correspondence
like the ones i keep on drawing over and over
(and this is true even *after* selecting a
“line at infinity” as i’ve done here).
enough for today; actual work is still
a long busride away…
i didn’t swipe this
when i drew this
… but of course the *idea* has been around for centuries.
i did *learn* a lot from _the_fundamental_concepts_of_geometry_
(bruce e. meserve). it’s a cheap dover reprint
and a masterpiece of clear exposition.
*everybody* oughta have one of these…
just like a bible, a shakespeare, and a dictionary. i’ll bet
the matching algebra book is good too… but i’ve got
*lots* of good algebra books.
here’s MEdZ #1 in both editions: last year’s “mini”
and this years “digest” sizes. the 2011 edition
features, along with “the hip-pocket vocab”
(a glossary for math-for-humanities), some
remixed drawings from some “micro” zines
(also from last year), along with some (new,
brief) handwritten commentary.
the seven-sections pictures look way better
cut together (so here *that* is).
this last one’s previously unpublished.
anyhow. it exists. there’s a even a (proof) copy in circulation.
mostly it’s just masters, though. i’ll be running off the first
big print run in the next few days i think. numbered & dated.
tonight we looked at a *great deal*
of material-from-the-syllabus.
too much by almost any standard,
i think. and obviously, missing a day…
half a week, really… doesn’t help.
and yet. it’s built into the course.
never mind snow days: if you’re getting
most of your ideas about What Algebra Is
*or* How To Do It, *from this presentation*,
then you’re almost sure to be left behind
(& very quickly) in the nature of the case.
this is College Math: “way too late
and much too fast”.
(meanwhile. *read*, dammit! and *talk* to each other.
for hecksake! do you think you’ll be young *forever*?)
okay. everybody gets their expectations
pushed-and-pulled around (and otherwise distorted)
and somehow we work out some way of getting along;
some way of talking about what it all might mean.
if there’s scribbling-on-the-board involved…
if there’s *symbolism*… then that’s what i call
“Doing The Math”. okay okay. don’t hate me for it.
quadratic equations, *again*? well, yes.
maybe you’ve missed something.
it’s much better if *you* take a piece
of chalk: *what* were you saying, again?
why have the deepest-thinking sages
returned to suchlike issues again and again
throughout all recorded time? (no… really:
why?)
because, yeah, duh: “theology”, so called.
“ontology”, forsooth.
it’s mostly transparent, though, if we agree
that “appeals to authority” are even more
contemptable than “outright lies”.
funding, funding, funding. everybody
talks about the weather. fuck the
god damn weather.
philosophy. feh.
quadratic equations.
*every* quadratic equation…
Ax^2 + Bx + C = 0…
can be solved (in the appropriate “domain”)
by the famous “Quadratic Formula”
(cf: QF Lore [a popular piece
from my blogging heyday]). To wit.
let D = B^2 – 4AC.
(D… or \sqrt{D}… i forget…
is the *discriminant* of our function…
we *were* talking about a function, right?
let’s see. let f(x) = Ax^2 + Bx + C, where
A, B, and C denote “numbers” [i.e., elements
of the Domain of Discourse] and “x” is
an “indeterminant”.)
we are not data.
*obviously* there isn’t-and-can-never-be
any “quadratic formula” for life itself:
“if you act *this* way, life will
work itself out in *that* way!”…
and it’s halfpast time we stopped
thinking that “math’, all by itself,
could ever fool anybody into thinking
that it’ll even ever’ve been a good
idea to *try*…
still. dammit.
suppose A \not= 0.
(otherwise, our equation
isn’t quadratic at all
[rightly so-called] but
merely linear; refer to
some already-well-understood
theory, duh). then 1/A
is a number. multiply
both sides of Ax^2 + Bx + C = 0
to obtain (the “monic” equation…
leading coefficient equal to 1…)
x^2 + (B/A)x + (C/A) = 0.
one “easily” applies the technique
called “completing the square”
at this point (add-and-subtract
the square of “half the middle coefficient”;
“regroup” the pieces and rewrite
the appropriate bit of code
as a “perfect square”).
clean it up and show
(at some appropriate level
of rigor) that the equation
Ax^2 + Bx + C = 0
(by the godlike authority
of faith-in-perfect-clarity
[work it out!]) is equivalent
(when A \not= 0; when the domain
of discourse allows the relevant
operations) to
x \in { (-B +\sqrt{D})/(2A), (-B -\sqrt{D})/(2A) }.
nobody wants to talk about the set theory here.
wait. that’s false. *i* want to talk about
the set theory here. can i get a witness?
but i’m here of course. in a short while i’ll go see
who *else* made it through the ice and snow;
naturally i’ll try to give ’em something extra.
but “time listening to owen rambling about math”
doesn’t seem to be *widely* understood as
“something extra”. so we’ll see.
there’s a new edition of MEdZ #1.
an 8-page digest. much easier
to read. order now.
so here’s my latest version of
the two-dimensional projective space constructed on
the field of three elements.
and the story. there are thirteen “windows”.
through each window, one sees a “line”.
each line is associated with four windows;
these in their turn, upon “looking through”,
show the four lines through the original window.
i went through essentially the same explanation
(on an earlier drawing of a different space) here
(and left a bunch of footnotes).
my post, “some finite projective spaces”
consists of a bunch of photos of zines with pictures
of projective spaces in them; some of these could
be considered “drafts” of this version of P_2(F_2).
i’m trying to make stuff simple where possible.
there’s been some progress since *this*:
the livingston review 