rare MEdZ-related post

April 10, 2011 in Graphics, MathEdZine

i’ve just edited in a subscript-backslash-zero
to the top line…
which now reads $S = ({\Bbb Z}\times {\Bbb Z})_{\backslash 0} =...$ …
to repair an *earlier* repair, done sloppily.

somewhere along the line i tacked on the “—{(0,0)}”
*without* adjusting the “zee-cross-zee” ( ${\Bbb Z}\times{\Bbb Z}$ ).
a beginner-like blunder, i confess. onward! *more* mistakes!
(just get down in the dirt and *calculate*, by golly.)

anyhow, owners of Math Ed Zine #0.4— $\Bbb Q$ by name—
should please to adjust the appropriate page in their issues.

which, being interpreted, means that
the set of *rational numbers* (Q) can be
represented as the collection of *lines through
the origin* (in the usual (x,y)-plane),
having *rational slope*. The slope condition,
for a given line, is equivalent to the condition
that there be an *integer* pair lying on the line
(nonzero; it gets to be something of a pain…).

the algebraic process whereby S…
nonzero-integer-pairs…
“maps onto” Q
is called “factoring by a relation”.
the relation in this case is called “tilde” (~).

tilde is defined by
” (x_1, y_1) ~ (x_2, y_2)
MEANS THE SAME THING AS
x_1 * y_2 = x_2 * y_1″

(“cross-multiplication” is in effect;
tilde is the relation we want “because”
${{y_1}\over{x_1}} = {{y_2}\over{x_2}}$
when $x_1 y_2 = x_2 y_1$ ).

oh heck. there’s that infinite-sloped line.
belongs to S/~, too. OK. modify the $\Bbb Q$ .
let’s call it ${\Bbb Q}^\infty$ , say. okay.
that’s it.
.

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2 Replies

vlorbik

April 10, 2011 at 5:37 pm

we shoulda just had x nonzero
and got it over with…

[
x, not y, because we're using "slopes"
of the form "(y_2 - y_1)/(x_2 - x_1)".
]

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vlorbik

April 17, 2011 at 10:51 pm

weirder and weirder. i recently found an
issue of the zine wherein i’d *made* the
correction to $\Bbb Q^*$ …
but *without* cutting “zero” out
of zee-cross-zee.

anyhow any and all future printings
will be correcter & very likely prettier
than any’ve been so far.

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