my recent example of bad freshman calculus

was, heaven knows, something egregious.

but *this*! this is something else again.

on another day, i might have just bitten down

and let it go with full credit… the very

kind of thing i have to do dozens of times

a night when i sit down to earn my right

to go on calling myself a math teacher.

ordinary unambitious undergrads

will *never* be made in classes like ours

to write carefully. and they’ll take any

attempt to do it as a violation of their

basic civil rights *as* undergrads and

threaten to vote with their feet.

so you’ve got to play it pretty cool.

even though it might break your heart.

just mark it wrong, again.

and tell ‘em why, again.

“CODE IS NOT ENGLISH”

usually doesn’t get points-off

*regardless* of how badly the little

dears have mangled plain sense into

worse-than-meaningless mumbo-jumbo

by using “syncopated style”.

but this example, on this day?

i just couldn’t do it. its author

lost a half a point (out of two).

“f(x) = log_a x = (ln x)/(ln a)

f’(x) = 1/(ln a) * 1/x” …

thus far well and good …

“since the derivative of ln x = 1/x”.

and my first instinct, and indeed my

first move, is to circle the “=”,

write “IS” near it, put a big “X” near

the whole mess so far, and write out

“CODE IS NOT ENGLISH”.

and let it go at that.

like usual.

but, finally, dammit.

if that thing deserves a perfect score

then i’m carl friedrich gauss.

if i *ever* encounter “ln x = 1/x”

*without* letting it bother me,

may my right hand lose its cunning.

take away my mathbooks and send me

to seminary or something; i’ll’ve

quit being a *math* teacher altogether.

the student in question will almost certainly

take my comments for raving lunacy. and so,

i suppose, will some portion of readers of this

post. none of *my* business either way, though,

i suppose. sometimes you’ve just gotta get up

on the net and *vent*. thanks for reading.

by quotiant rule

EXHIBIT A

but this is *senior level* stuff…

“analysis” *not* freshman-calc.

and even *there*, one seldom encounters

such an *explicit* version of the

natural-log-equals-reciprocal fallacy

.

(the “quotiEnt rule”…

or rather, *the* quotient rule…

our subject has fallen into the

“leaving out articles makes it

harder to understand and so is

in my best interest” trap…

has *no bearing* on this [mis]-

calculation [we aren't diff-

erentiating a fraction (or any-

thing else for that matter)].)

here’s the whole sad story.

math is hard and everybody knows it.

what they *don’t* know is that it’s

nonetheless easier than anything else.

*particularly* when one is trying

to do things like “pass math tests”.

there’s always a widespread (and *very*

persistent) belief abroad in math

classes that trying to understand

what the technical terms mean (for

example) is “confusing” and should

be dodged at every opportunity.

we teachers go on (as we must)

pretending that when we say things like

“an *equation*” or “the *product* law”

we believe that our auditors

are thinking of things like

“a string of symbols

representing the assertion

that a thing-on-the-left

*has the same meaning as*

a thing-on-the-right”

or “the rule (in its context)

about *multiplication*”.

but if we ever look at the documents

produced by these auditors in attempting

to carry out the calculations we only

wish we could still believe we have been

*explaining* for all these weeks?

we soon learn that they have been thinking

nothing of the kind.

anyhow. the example at hand.

calculus class is encountered by *most* of its students

as “practicing a bunch of calculating tricks”.

the “big ideas”… algebra-and-geometry, sets,

functions, sequences, limits, and so on…

are imagined as *never to be understood*.

math teachers will perversely insist on

*talking* in this language when demonstrating

the tricks.

well, the “big ideas” that *characterize*

freshman calc are “differentiation” and

“integration”. for example “differentiation”

transforms the expression “x^n” into the

expression “n*x^(n-1)” (we are suppressing

certain details more or less of course).

this “power law” is the one thing you can

count on a former calculus student to have

remembered (they won’t be able to supply

the context, though… those pesky “details”),

in my experience.

anyhow… long story longer… somewhere

along the line, usually pretty early on…

one encounters the weirdly mystifying

*natural logarithm* function. everything

up to this point could be understood as

glorified sets, algebra, and geometry…

and maybe i’ve been able to fake it pretty well…

but *this* thing depends on the “limit” concept

in a crucial way. so to heck with it.

and a *lot* of doing-okay-til-now students

just decide to learn *one thing* about

ell-en-of-ex

(the function [x |----> ln(x)],

to give it its right name

[anyhow, *one* of its right

"coded" names; "the log"

and "the natural-log function"

serve me best, i suppose,

most of the time]):

“the derivative of ln(x) is 1/x”.

anything else will have to wait.

but here, just as i said i would,

i have given the student too much

credit for careful-use-of-vocabulary:

again and again and again and again,

one will see clear evidence that

whoever filled in some quiz-or-exam

instead “learned” that

“ln(x) is 1/x”.

because, hey look.

how am *i* supposed to know

that “differentiation” means

“take the derivative”?

those words have *no meaning*!

all i know… and all i *want* to know!…

is that somewhere along the line in

every problem, i’ll do one of the tricks

we’ve been practicing. and the only trick

i know…. or ever *want* to know!… about

“ln” is ell-en-is-one-over-ex. so there.

it gets pretty frustrating in calc I

as you can imagine. to see it in analysis II

would drive a less battle-hardened veteran

to despair; in me, to my shame, there’s a

tendency… after the screaming-in-agony

moments… to malicious glee. (o, cursed spite.)

because, hey, look. if we all we *meant* by

“ln(x)” was “1/x” what the devil would we have

made all this other *fuss* about? for, in

your case, grasshopper, several god-damn *years*?

did you think this was never going to *matter*?

in glorified-advanced-*calculus*? flunking fools

like this could get to be a pleasure.

but how would i know. i’m just the grader;

it’s just a two-point homework problem.

and anyway, that’s not really the *exact* thing

i sat down to rant about.

next ish: *more* bad freshman calc from analysis II.

3.2.4

Let (n, p) denote an integer and a prime #;

is “x^2 == n (mod p)” solvable for the given

pairs (n, p)?

a. (5, 227) NO

b. (5, 229) YES

c.(-5, 227) YES

d.(-5, 229) YES

e. (7,1009) YES

f.(-7,1009) YES

(The calculations are routine.)

3.2.6

With a TI “grapher”, I’ve just found that

139^2 == 150 (mod 1009).

One simply puts

Y_1 = 1009(fPart(X^2/1009))…

this computes equivalence-classes-mod-1009

of the squares of each of the inputs successively…

and “scrolls down” in the TABLE window

(set to display natural number X-values).

Hence x^2 == 150 (mod 1009)

*does* have a solution.

(In fact, two of them; the other is of course

870 (= 1009 – 139)).

In detail,

139^2 = 19321 = 19*1009 + 150

and

870^2 = 756900 = 750*1009 + 150.

(The point here is that one may easily

*verify* such results by “paper-&-pencil”

methods.)

To find the roots *entirely* by p-&-p methods,

the best line of attack would probably be via

“primitive roots”.

3.2.7

We seek primes p such that x^2 == 13 (mod p)

has a solution.

The case of p = 2 (“the oddest case of all”)

reduces to x^2 == 1 (mod 2); of course this

*does* have a solution. (So we add “2″ to our

list of primes.)

In the case of p = 13, our equivalence is x^2 == 0;

again, this clearly *does* have a solution (and “13″

belongs in our list).

Finally, let p be an odd prime with p \not= 13.

The Gaussian Reciprocity Law now gives us that

(13|p) = (p|13)

(where (_|_) denotes the Legendre symbol).

[Remark:

This happens "because 13 == 1 (mod 4)"...

recall that (p|q) and (q|p) have opposite

signs only when *both* primes are in "4k+3"

form.]

So we need only check the equivalence classes (mod 13)

for “perfect squares”. We know from earlier work that

there are *six* perfect squares (mod 13).

1, 4, and 9 are the “obvious” ones…

4^2 = 16 == 3…

by suchlike computations, one easily arrives at

{1, 4, 9, 3, 12, 10} =

{1, 3, 4, 9, 10, 12};

any odd prime p different from 13 must be

congruent-modulo-13 to one of these “perfect

squares mod 13″.

Summarizing: x^2 == 13 (mod p)

has solutions for p = 2, for p = 13,

and for

p \in { 13k + r |

k \in N ,

r \in {1, 3, 4, 9, 10, 12}

}.

(*)

**************************(end of exercise)******

[

Remark:

Our "list" of primes now begins

L = {2, 3, 13, 17, 23, ...}

.

(Some students had {2, 3, 13})

.

It can be shown

(using "Dirichlet's Theorem on Primes

in Arithmetic Progressions" [DT], section 8.4)

that there are *infinitely many* prime numbers

“for each r” in (*) (i.e., in each of the

“perfect square” equivalence classes mod-13)

.

One is of course not expected to know

about DT already (or to learn it now).

But questions like

“

*is* there a prime that’s congruent to 4 (mod 13)?

“

arise in the present context in a very natural way:

it so happens that 17 == 4 (mod 13), and so

in “testing primes” (in their natural order

2, 3, 5, … , 13, 17, …) one soon learns

the answer: there *are* such primes.

(But then… how *many*? And so on.)

]

4.1.2

Clearly with

2^k || 100! and 5^r || 100!,

one has r < k, and so

to "count zeros at the right

end" of Z = 100! we need only

compute "r"

(the largest power of *10* dividing Z…

the number of zeros in question…

is clearly the same as the largest

power of *5* dividing Z).

But for this we need only consult

de Polignac's formula (Theorem 4.2):

r = [100/5] + [100/25] + [100/125] + …

r = 24.

4.1.9

Let BC(x,y) denote the Binomial Coefficient

BC(x,y) = x!/[y!(x-y)!]

(one usually pronounces this object

"x choose y"; see Section 1.4).

One then has the familiar "Pascal's Triangle"

property: a given "entry" can be computed

by "adding the two entries above it".

The object of study in this exercise is then

the "middle entry" of an "even numbered row";

simple algebra gives us

(2n)!/(n!)^2 =

BC(2n, n) =

BC(2n-1, n-1) + BC(2n-1, n) =

2BC(2n-1, n-1).

This is clearly an even integer.

6.

With a TI “grapher”, I’ve just found that

244^2 == 5 (mod 1009). One simply puts

Y_1 = 1009(fPart(X^2/1009))…

this computes the equivalence-class-mod-1009

of the square of each input successively…

and “scrolls down” in the TABLE window

(set to display natural number X’s).

Hence x^2 == 5 (mod 1009) *does* have a solution.

(In fact, two of them; the other is of course

765 (= 1009 – 244)).

In detail,

244^2 = 58536 = 59*1009 + 5

and

765^2 = 585225 =580*1009 + 5.

(The point here is that one may easily

*verify* such results by “paper-&-pencil”

methods.)

To find the roots *entirely* by p-&-p methods,

the best line of attack would probably be via

“primitive roots”.

*trouble is, is, that that “5″
was supposed to’ve been “150″.
once more, dear friends!
(or close the wall up with our
english dead!)*

i’m going to SPACE next week.

it’s columbus ohio, so the list

narrows down pretty quick:

the “arnold classic”, SPACE,

and, um, let’s see… there

*must* have been somththing

else…

“ameriflora”, maybe, in your

dreams. “miracle mile”, back

from the “miracle” days of

the long-gone late-great

thriving american working class

that shopped (until it dropped)

there. no. never mind.

the chief attraction of columbus

for *me* is that this is where

i *am* (moving around… or

even moving *stuff* around…

is *much* harder than they’d

have you believe…; & of course

the *next* best thing about

columbus is that *madeline*

lives here (and our happy home

*is* our happy home, much to

my surprise). and *staying*

here keeps me this way (happy).

it’s (1) cold (2) cruel

world (3) out there.

the (very existence of)

the billy ireland museum is,

enough to put columbus *somewhere* on

the comics “map”… and there’s already

a better list of local-and-quasi-nearby

talent at the “space” site… so let

me just give a shout-out to ray (!!) t

and “max ink” (still working as far as

i know); one more for glen brewer (even

though i think glen has quit the scene;

his _askari_hodari_ was, for me, very

much a local highlight).

everybody knows about _bone_;

it won’t escape my notice here

that the astonishing paul hornschemeier

lived here, too, when he was getting

started and i met him (and he drew

a cover for my zine gratis… eat

your hearts out). in fact, ghod

*bless* columbus. good night.

i’ve typeface-ized the “formula” stuff

but the point here is the english.

tonight i encountered the passage

Since there are (p-1)/2 quadratic residues & 1^2, 2^2, …, [(p-1)/2)]^2 are all the residues, we need to show that the quadratic residues modulo p are all distinct…

and, after much wailing and gnashing

of teeth, decided that the best spin

i could put on it would be

to *omit* the first “the*

and to replace the second “the”

with “these”:

Since there are (p-1)/2 quadratic residues & 1^2, 2^2, …, [(p-1)/2)]^2 are all residues, we need to show that these quadratic residues modulo p are all distinct…

(which “works” in its context

as the original passage certainly

does *not*).

they should give medals for this kind

of copyediting. this is *hard work*.

not that it does anyone any *good*,

mind you…

let p be an odd prime &

let g & g’ be primitive

roots (mod p).

any Primitive Root, h, satisfies

(mod );

since (mod )

[this uses "h is primitive"],

we can conclude that

(mod ).

etcetera. forget it.

the handwritten stuff

is beautiful though.

is too hard.

it’s not so bad in the real editor,

of course.

oh, ps. (gg’)^[(p-1)/2]

is now seen to be congruent

to 1… and so is not a

primitive root (which, on

the day, was to’ve been shown).

ladies and gentlemen, PSL(2,7).

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.