Re: Calculus

[Les Schaffer <schaffer@xxxxxxxxxxxxxxxxx>]

> Why is it that one cannot divide by zero ? Seems like special
> pleading.

well, whats-it-mean( divide-by-zero )? add that meaning to the laws,
and it'd be legal.

here is something almost as difficult:

  whats-it-mean( divide-something-by-a-half )?

well, dividing by a number means (is defined as) multiplying by its
reciprocal, and what is a reciprocal?

well, if i have 2 apples, is there an (arithmetical) operation that
returns one of them? yeah, i take one, you take one. we call that
'half', and represent that 'half' in shorthand: 1/2. a less confusing
shorthand would be recip(2):  2 * recip(2) = 1. in other words, don't
call it "1 divided by 2" so fast.

so now we have natural numbers and their reciprocals.

what is recip(2) * 4 = ?

no sweat: recip(2) * 2 * 2 = 1 * 2 = 2

which can be thought of as saying take the 4 apples, in pairs, and
then divide by 2 as before.

what is recip(2) * 1 = ?

this takes an operation similar to what we did with the two apples. we
assume we have a good paring knife, and we agree on whats the middle
of the apple, and slice it. we now have two piece, which we call
'half's'. so now, we have 2 half's and multiply by recip(2) and we get
1. 1 what? 1 'half'. (we'll need a good sharp knife later on in any
case, to construct the real number system using 'Dedekind cuts', named
after Dedekind, who first sharpened this particular knife.)

what is recip(3) * 4 = ?

no sweat, take each apple, and divide it into 3 parts, we do this 4
times, so we get 12 parts. we call the parts 'thirds', or in the
Bronx, "toids".

now recip(3) * 4 = recip(3) * 3 * 4 (thirds) = 4 (thirds). we can
write that in shorthand as "4/3".

one can continue on to define all the rational numbers. and if we are
careful enough, we can give a meaning to that '/' symbol.

but lets head for zero now.

consider fourths, fifths, sixths, ..., or in legal shorthand:

1/4 1/5 1/6 1/7 ... 1/1000000 etc

well from the earlier culinary discussions, it means to progressively
divide the apple into smaller and smaller equal pieces. 7 pieces, 8
pieces, etc etc.

now its not hard to see where this is going: we have a rather dull
knife by the time we get to 20 or 30. and pretty soon we're into
slicing apple skins down through the thickness of the skin, with a
knife thats fatter than the skins. so we switch operations from apple
rationing with a knife to using one of those things they use for
taking samples of cell tissue (micro something, oh yeah, microtomes

http://www.ebsciences.com/leica/microtome.htm
http://www.sci.sdsu.edu/emfacility/microtome.html

but then we have to go back and start all over, slicing the pieces
from the knife into smaller sections. but even this goes so far, and
we switch operations to, ...

well, now we have two choices, we stick in academia, or we get down
and dirty. the fact is, if you had one apple, and you had to ration it
amongst 20 hungry people, they are not as well fed as if you gave each
of two people a half. i am sure adam and eve figured this out long
before dedekind, once they had children.

and the plain and simple truth is, if i take an apple and divide it up
into 10-ths or finer, and i get one of those pieces, i have in plain
english, "nuttin", or in yiddish we say "bupkiss". i might as well
have zero apples, in other words forget the damn apple, it cant do 30
of us no good.

so, whats 1/0, 1 divided by zero?  its what the capitalist would have
us sitting around arguing about while he takes the other 29 apples to
sell at the apple market. in retaliation, we create a culture around
this 1/0, making it fit in continuously with its neighbors like divide
by 1/1000, divide by 1000000. we collectively agree to call it bupkiss
as well -- the hell with that damn apple, we need to recover the other
29.

les schaffer

p.s. in the hyperreal number system, where the infinitesmail and _its_
reciprocal, infinite, is well defined, we still don't get a proper
defintion of 1/0

p.p.s homework: what is 1 divided by -100?