## Archive for the ‘Lectures Without Words’ Category

ten poles, ten polars,

and ten pairs-of-triangles:

ten ways to use one drawing

(from MathEdZine, of course)

to illustrate one theorem.

(desargues’ theorem at w’edia.)

performing calculations using

actual blobs of color rather than

alphabetical symbols *standing*

for colors is time-consuming

(and demanding of special tools)…

but one literally “sees” certain things

*much more readily* than with

symbol manipulation.

hey, i’m a visual learner.

at long last. this has been sitting

on the paperpile very-nearly-finished

for quite a while.

i started the “lectures without words”

series early on with 0.1: .

whose cover more or less announced

implicitly that it was one of a series

called . and that

was, like, five quarters ago.

and they’re only 8 micro-size pages.

a couple days ago i inked the graphs

and the corresponding code (the stuff

under the dotted line had *been* inked

and the whole rest of the issue was

entirely assembled). and zapped it off.

and yesterday i passed ’em around

at the end of class (to surprisingly few

students given that i’ve got freshly-

-graded exams). it went okay.

i did right away

(if i recall correctly), and in

high-art style, too (i used a

brush instead of a sharpie).

wasn’t much later.

i have plenty of notes for ,

too, and could knock out a version

on any day here at the studio

(given a couple hours and some

peace of mind) that’d fit right in.

anyhow, what we have here are,

first of all, obviously, a couple graphs

and a bunch of code. here, at risk

of verbosity, is some line-by-line

commentary.

in the upper left is

part of the graph of

the linear equation

y=(x+1)/2…

namely the part whose x’s

(x co-ordinates) are between

-1 and 1.

and my students (like all

deserving pre-calculus graduates)

are familiar with *most* of the

notations… and *all* the ideas…

in this first line.

that funky *arrow*, though.

well, i can’t easily put it in here

(my wordpress skills are but weak)

but i’m talking about the one

looking otherwise like

.

and in the actual *zine*, it’s

a Bijection Arrow.

something like ” >—->>”.

as explained (or, OK, “explained”)

*below* the dotted line.

where *three* set-mapping “arrows”

are defined (one in each line;

the in each line

denotes “is equivalent-by-definition to”).

the Injective Arrow >—> denotes that

f:D—->R is “one-to-one” (as such

functions are generally known

in college maths [and also in

the pros for that matter; “injective”

and its relatives aren’t *rare*,

but their plain-language versions

still get used oftener]).

“one-to-one”, defined informally,

means “different x’s always get

different y’s”. coding this up

(“formally”), with D for the “domain”

and R for the “range” (though i

prefer “target” in this context

when i’m actually present to

*explain* myself) means that

when d_1 and d_2 are in D,

and d_1 \not= d_2

(“different x’s”), one has

f(d_1) \not= f(d_2)

(“different y’s”).

likewise the Surjective Arrow —>>

denotes what is ordinarily called

an “onto” function:

every range element

(object in R)

“gets hit by” some domain element.

and of course the Bijective Arrow >—>>

denotes “one-to-one *and* onto”.

there’s quite a bit of symbolism

here that’s *not* familiar to

typical college freshmen.

the Arrows themselves.

“for all”

“there exists”

logical “and”

and the seldom-seen-even-by-me

“such that” symbol that, again,

i’m unable to reproduce here

in type.

still, i hope i’m making a point

worth making by writing out

these “definitions without words”.

anyhow… worth doing or not…

it’s out of the way and i can return

to the main line of exposition:

the mapping from (-1, 1) to (0,1)

in the top line of my photo here

is a “bijection”, meaning that it’s

a “one-to-one and onto” function.

for *finite* sets A and B,

a bijection f: A —-> B

exists if and only if

# A = # B…

another unfamiliar notation

i suppose but readily understood…

A and B have

*the same number of elements*.

we extend this concept to *infinite*

sets… when A and B are *any*

sets admitting a bijection

f:A—>B, we again write

#A = #B

but now say that

A and B have the same **cardinality**

(rather than “number”;

in the general case, careful

users will pronounce #A

as “the cardinality of A”).

i’m *almost* done with the top line.

i think. but there’s one more notation

left to explain (or “explain”).

the “f:D—>R” convention i’ve been

using throughout this discussion

is in woefully scant use in textbooks.

but it *is* standard and (as i guess)

often pretty easily made out even

by beginners when introduced;

one has been *working* with

“functions” having “domains”

and “ranges”, so fixing the notation

in this way should seem pretty natural.

but replacing the “variable function”

symbol “f” by *the actual name

of the function* being defined?

this is *very rare* even in the pros.

alas.

the rest is left as an exercise.

a recent post in *KTM* asked why

.

i know that one.

the “graph” in the upper right is K_4…

the “complete graph on 4 vertices”…

and has *6* (so-called) edges.

its “complement” (upper left) has *none*

of the edges. obviously there’s one

(so-called) *complete* graph having

all six possible edges and one “empty”

graph having *none* of the edges.

move down to the next couple lines:

the graphs having exactly *one* edge

match up in one-to-one fashion

with the graphs having *five* edges

(because “including five (edges)”, in this context,

is the same as “leaving out *one* (edge) out”.

so on for two-edge graphs.

each is the “complement”

(via the inclusion-exclusion principle

as hinted at a moment ago)

of a *four*-edge graph.

posting; gotta go to class.

above we have *pp* 6, 7, 8, 1;

below are *pp* 2, 3, 4, 5. you fold

it up into a little booklet (a “micro-zine”,

i sometimes say… a “mini” [also known

as a “quarto”] is four times this size

[twice as large in each direction]).

here’s section 1.1

of my textbook (and much the most-hit

post of this blog): “the set of natural numbers”.

and here’s a shot of

this and some other MEdZes

in their “folded” state.

stand by for some more links.

who knows maybe some direct

explanation even.

all these MathEdZines…

## 0.1, 0.1.1, 0.2, 0.3,

0.6, 0.7, and 1

(together with the un-

numbered nanozine version

of )…

are going with us to SPACE

a few hours from now.

where, with any luck, i'll trade 'em

all away for some really cool comics.

z:= x+yi

w:= a+bi

zw = (xa-yb) + (xb+ya)i

u:= xa-yb

v:= xb+ya

f(z)+f(w) = f(z+w) trivially.

PS

s:=

[1^2, 2^2, 3^2, 4^2, …]=

[1,4,9,16,…]=

[n^2]_(n \in \Bbb N).

=: “\Bbb N”

NB: 0 \in \Bbb N.

******************************************

S := x + 4x^2 + 9x^3 + …

()

******************************************

(1-x)*S = x + 3x^2 + 5x^3 + 7x^4 + …

(1-x)^2*S = x + 2x^2 + 2x^3 + 2x^4 + …

(1-x)^3*S = x + x^2

S = (x+x^2)/(1-x)^3

******************************************

(

x = 1/10

S = .1 + 4*.01 + 9*.001 + 16*.0001 + …

S ~ .150892

S = (.1+.01)/(.9)^3

S ~ .150892

OK

)