## Archive for the ‘TeX’ Category

### allegedly “classic”

here it is in code.

$latex ii = \begin{pmatrix} \begin{pmatrix} 0 & 2 \\ 1 & 2\end{pmatrix}& \begin{pmatrix} 1 & 1 \\ 0 & 1\end{pmatrix}& \\ \begin{pmatrix} 1 & 0 \\ 2 & 1\end{pmatrix}& \begin{pmatrix} 2 & 2 \\ 1 & 0\end{pmatrix} \end{pmatrix}$
$latex i = \begin{pmatrix} \begin{pmatrix} 1 & 2 \\ 1 & 0\end{pmatrix}& \begin{pmatrix} 2 & 1 \\ 0 & 2\end{pmatrix}& \\ \begin{pmatrix} 2 & 0 \\ 2 & 2\end{pmatrix}& \begin{pmatrix} 0 & 2 \\ 1 & 1\end{pmatrix} \end{pmatrix}$

$latex iii = \begin{pmatrix} \begin{pmatrix} 0 & 1 \\ 2 & 2\end{pmatrix}& \begin{pmatrix} 1 & 0 \\ 1 & 1\end{pmatrix}& \\ \begin{pmatrix} 1 & 2 \\ 0 & 1\end{pmatrix}& \begin{pmatrix} 2 & 1 \\ 2 & 0\end{pmatrix} \end{pmatrix}$
$latex iv = \begin{pmatrix} \begin{pmatrix} 1 & 1 \\ 2 & 0\end{pmatrix}& \begin{pmatrix} 2 & 0 \\ 1 & 2\end{pmatrix}& \\ \begin{pmatrix} 2 & 2 \\ 0 & 2\end{pmatrix}& \begin{pmatrix} 0 & 1 \\ 2 & 1\end{pmatrix} \end{pmatrix}$

$latex II= \begin{pmatrix} (mgy) (nrp) & (ybr) (pog) & \\ (mro) (ngb) & (mpb) (nyo) \end{pmatrix}$
$latex I= \begin{pmatrix} (bo) (mpgnyr) & (mn) (ygbpro) & \\ (yp) (mbrnog)& (rg) (mopnby) \end{pmatrix}$
$latex III = \begin{pmatrix} (mbp) (nor) & (mor) (nbg) & \\ (yrb) (pgo) & (myg) (npr) \end{pmatrix}$
$latex IV = \begin{pmatrix} (rg)(mybnpo) & (yp)(mgonrb) & \\ (mn)(yorpbg) & (bo) (mryngp) \end{pmatrix}$

$latex jj= \begin{pmatrix} hk & g & \\ gk & hi \end{pmatrix}$
$latex j= \begin{pmatrix} -gi & h & \\ -hj & -gj \end{pmatrix}$
$latex jjj = \begin{pmatrix} gj & hj & \\ -h & gi \end{pmatrix}$
$latex jw = \begin{pmatrix} -hi & -gh & \\ -g & -hk \end{pmatrix}$

### the abyss looks back into you

here’s looking, not at euclid, but rather at this
nameless-by-me-i-think mac book. behind me a few
of the many shelves here at the livingston libe.
madeline’s encyclopedia was here before my arrival
(all those years ago). most of the rest was brought
by me or acquired since. the hugely-cool coffee-table
book about public libraries (lavish production;
photos of libes) is next to the top row
of those _britannica_s already mentioned…
under that, typography & dictionaries…
this is way too slow… the “connect four”
game… fast forwarding…

there’s a whole martin gardner *section* for
hecksake including two issues of th’ _notices_;
under his _last_recreations_ is a bunch of stuff
by the great knuth (that made up $\TeX$—thanks, don!).
the jacket for th’ _annotated_alice_ (the caroll section is
in my sight from here but not in this shot) also
deserves mention. i loved that book even before
i knew who gardner *was*.

other side of my head. “children’s book” about sacagawea
and a few of my many dollar-coins. there’s a map
more or less of course—part of a streets-of-columbus
folded-up-for-gloveboxen thingum in this case
(one of many… want one?).

the enormous _handbook_of_physics_ at lower right
was a cursemas gift from public-school days.
too awkward to be good for much actual *reading*
but nonetheless known by me to have an impressive
survey-of-advanced-math right up front.

okay. enough stalling around.

announcing _the_ten_page_ten_yearly_ (2020).
this i vow.
if it comes out at all
(i) it’ll be ten pages
(
digest-size front and back
in glorious [and expensive]
full color; 8 pages inside
in glorious cheap B&W
), and
(ii)
i won’t do it again for ten years.

watch this space.

### teach your children

here it is in code.

$ii = \begin{pmatrix} \begin{pmatrix} 0 & 2 \\ 1 & 2\end{pmatrix}& \begin{pmatrix} 1 & 1 \\ 0 & 1\end{pmatrix}& \\ \begin{pmatrix} 1 & 0 \\ 2 & 1\end{pmatrix}& \begin{pmatrix} 2 & 2 \\ 1 & 0\end{pmatrix} \end{pmatrix}$
$i = \begin{pmatrix} \begin{pmatrix} 1 & 2 \\ 1 & 0\end{pmatrix}& \begin{pmatrix} 2 & 1 \\ 0 & 2\end{pmatrix}& \\ \begin{pmatrix} 2 & 0 \\ 2 & 2\end{pmatrix}& \begin{pmatrix} 0 & 2 \\ 1 & 1\end{pmatrix} \end{pmatrix}$

$iii = \begin{pmatrix} \begin{pmatrix} 0 & 1 \\ 2 & 2\end{pmatrix}& \begin{pmatrix} 1 & 0 \\ 1 & 1\end{pmatrix}& \\ \begin{pmatrix} 1 & 2 \\ 0 & 1\end{pmatrix}& \begin{pmatrix} 2 & 1 \\ 2 & 0\end{pmatrix} \end{pmatrix}$
$iv = \begin{pmatrix} \begin{pmatrix} 1 & 1 \\ 2 & 0\end{pmatrix}& \begin{pmatrix} 2 & 0 \\ 1 & 2\end{pmatrix}& \\ \begin{pmatrix} 2 & 2 \\ 0 & 2\end{pmatrix}& \begin{pmatrix} 0 & 1 \\ 2 & 1\end{pmatrix} \end{pmatrix}$

$II= \begin{pmatrix} (mgy) (nrp) & (ybr) (pog) & \\ (mro) (ngb) & (mpb) (nyo) \end{pmatrix}$
$I= \begin{pmatrix} (bo) (mpgnyr) & (mn) (ygbpro) & \\ (yp) (mbrnog)& (rg) (mopnby) \end{pmatrix}$
$III = \begin{pmatrix} (mbp) (nor) & (mor) (nbg) & \\ (yrb) (pgo) & (myg) (npr) \end{pmatrix}$
$IV = \begin{pmatrix} (rg)(mybnpo) & (yp)(mgonrb) & \\ (mn)(yorpbg) & (bo) (mryngp) \end{pmatrix}$

$jj= \begin{pmatrix} hk & g & \\ gk & hi \end{pmatrix}$
$j= \begin{pmatrix} -gi & h & \\ -hj & -gj \end{pmatrix}$
$jjj = \begin{pmatrix} gj & hj & \\ -h & gi \end{pmatrix}$
$jw = \begin{pmatrix} -hi & -gh & \\ -g & -hk \end{pmatrix}$

### 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.

### begin{pmatrix}

$M = \begin{pmatrix} \begin{pmatrix} 1 & 0 \\ 0 & 1\end{pmatrix}& \begin{pmatrix} 0 & 0 \\ 0 & 0\end{pmatrix}& \\ \begin{pmatrix} 0 & 0 \\ 0 & 0\end{pmatrix}& \begin{pmatrix} 1 & 0 \\ 0 & 1\end{pmatrix} \end{pmatrix}$

$B =\begin{pmatrix} \begin{pmatrix} \begin{pmatrix} 1 & 0 \\ 0 & 1\end{pmatrix}& \begin{pmatrix} 0 & 0 \\ 0 & 0\end{pmatrix}& \\ \begin{pmatrix} 0 & 0 \\ 0 & 0\end{pmatrix}& \begin{pmatrix} 1 & 0 \\ 0 & 1\end{pmatrix} \end{pmatrix} & \begin{pmatrix} \begin{pmatrix} 1 & 0 \\ 0 & 1\end{pmatrix}& \begin{pmatrix} 0 & 0 \\ 0 & 0\end{pmatrix}& \\ \begin{pmatrix} 0 & 0 \\ 0 & 0\end{pmatrix}& \begin{pmatrix} 1 & 0 \\ 0 & 1\end{pmatrix} \end{pmatrix} \\ \begin{pmatrix} \begin{pmatrix} 1 & 0 \\ 0 & 1\end{pmatrix}& \begin{pmatrix} 0 & 0 \\ 0 & 0\end{pmatrix}& \\ \begin{pmatrix} 0 & 0 \\ 0 & 0\end{pmatrix}& \begin{pmatrix} 1 & 0 \\ 0 & 1\end{pmatrix} \end{pmatrix} & \begin{pmatrix} \begin{pmatrix} 1 & 0 \\ 0 & 1\end{pmatrix}& \begin{pmatrix} 0 & 0 \\ 0 & 0\end{pmatrix}& \\ \begin{pmatrix} 0 & 0 \\ 0 & 0\end{pmatrix}& \begin{pmatrix} 1 & 0 \\ 0 & 1\end{pmatrix} \end{pmatrix} \end{pmatrix}$

### more numb’-th’y exercisen

let p be an odd prime &
let g & g’ be primitive
roots (mod p).

any Primitive Root, h, satisfies
$(h^{{p-1}\over2})^2 = 1$ (mod $p$);
since $h^{{p-1}\over2}\not= 1$ (mod $p$)
[this uses “h is primitive”],
we can conclude that
$h^{{p-1}\over2} ~ -1$ (mod $p$).

etcetera. forget it.
the handwritten stuff
is beautiful though.
$\TeX$ 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).

### zine fodder

Symbols Used

$\Bbb N$ the set of natural numbers, “en”

$\Bbb Z$ the set of integers, “zee”

$\Bbb Z^+$ the positive integers, “zee-plus”

$\in$ “is an element of”; e.g. $-3\in \Bbb Z$, $-3\not\in \Bbb N$

# the number of elements for a (finite) set.
# (more generally) the “cardinality” of a set (finite or not)

$X \cup Y$ the union of (sets) X and Y
$X \cup Y$ the collection (set) of objects belonging X alone, or to Y alone, or to both.

W := C the symbol “W” stands for the code “C”
W := C “W equals C by definition

{…} the set containing “…” (typically a list of “elements”)

$\{P(x)\}_{x\in S}$ the collection of all (objects) “x”, taken from the set “S”, such that the proposition “P(x)” is true

$S^{\rightarrow}$ ess-arrow
$S^{\rightarrow}$ the “successor” of S

$\infty$ is well-known;
i’m lazy to look up aleph-null.

*********************************************************
the set of natural numbers, “en”
the set of integers, “zee”
the positive integers, “zee-plus”
“is an element of”; e.g. ,
# the number of elements for a (finite) set. # (more generally) the “cardinality” of a set (finite or not)
the union of (sets) X and Y;
the collection (set) of objects belonging
to X alone, or to Y alone, or to both.
W := C the symbol “W” stands for the code “C”
(“W equals C by definition“)
{…} the set containing “…” (typically a list of “elements”)
the collection of all (objects) “x”,
taken from the set “S”, such that
the proposition “P(x)” is true
the “successor” of S (“ess-arrow”)
is well-known; i’m lazy to look up aleph-null.

munging code is fun and easy.

### Cut The End Off The Roast

$\bullet$A New Arrangement For Calculating Slope (M. P. Goldenberg).
$\bullet$A very useful list of common LaTeX expressions. All knowlege is found in blogs.

### A Poor Workman Blames His Tools

The “Formula does not parse” message in Friday’s post isn’t (entirely) my fault. The “editor” (computer interface where I enter my copy) changes the code around at whim. It can get pretty infuriating. In this case, I’m inclined to be rather more forgiving than usual: the “less than” sign $<$ happens to be a Special Character in HTML (the “markup language” that Web pages are coded in), so its appearance in a formula (i.e., in the TeX code—our host, WordPress.com, uses the same math markup language as the hardcopy journals—kind of [at the level of the formula rather than the page or the chapter, alas])—the appearance of a special HTML character in the TeX, I say, probably oughta be expected to cause problems. I’ll soon work around it almost unconsciously. The need for random “null” characters here and there to make things work is another story.

Enough about production. Sometimes it seems like that’s all anybody wants to talk about. It’s interesting enough I suppose and hugely important to a publishing junky like me, but if you want documentation of computer tools, I expect you know where to get it. There’s a blue million technophiles out there flogging every conceivable application. What was I saying?

No, wait. Nobody took me up on my challenge to produce TeX code in a comment. DOLLARSIGNlatex [write your formula here] DOLLARSIGN gets it; mouse over an existing formula to see some code. If you’re embarrassed at having your every effort published with no chance to clean it up, start your own blog (free and easy … I know, everybody always says that …).

Now. $\null [x_1 < x_2] \Rightarrow [f(x_1) < f(x_2)] (\forall x_1, x_2 \in S)$ sez “f is increasing on S” (recall that $f:{\Bbb R} \rightarrow {\Bbb R}$ and $S \subset {\Bbb R}$f is a real-valued function of a real variable and S is a set of real numbers). In words, “bigger x‘s get bigger y‘s” (but this is slangy—notice that y doesn’t occur at all and that the x‘s that do appear are so-called “dummy variables” [that could be replaced with, say, a and b without changing the meaning of the code]).

I actually put some code like this into yesterday’s “sample quiz” (in the 2-hour afternoon class; the daily classes get it in an hour or two). I probably won’t have the nerve to put it on the quiz itself but it’s already been a pretty good conversation starter. One more or less constant refrain with material like this: “we’ll begin by using the pictures and the verbal descriptions to understand the code … but if all goes well, we’ll eventually be able to use the code to understand the diagrams better …”.

• ## (Partial) Contents Page

Vlorbik On Math Ed ('07—'09)
(a good place to start!)