37 pm, John Larkin
[....]
I still can't get my head around the fundamental reason why there's
only one kind of positive resistor but two kinds of negative resistor.
I think I can add to the confusion:
We are thinking "number line like this:
-3, -2, -1, 0, 1, 2, 3, ..... really big, infinite, -really big ...
There are two ways to get to a negative number. One takes you past
the "infinite" value. In truth though. we have a 2D world we can
avoid hitting the singular value by going a little reactive.
.................................
.................----------......
..-...0...-------....*.....------
...-------.......................
.................................
As often happens in math, the pole is the center pole of a spiral
staircase. You have to go around the pole twice to get back where you
started.
Don't you remember how much grief certain parties gave me for using
the term "infinite"?
(I don't know why I call them parties, when they never seem to have
any fun.)
Some people are concerned on theological grounds any time the subject
goes to that of the infinite. They have near fits when you start
comparing the sizes of infinities and their heads explode if the
question of whether there is an infinite number of infinitudes. In
this case we merely have a 1/0 sort of infinity or a simple pole.
This is not to suggest that all poles are simple. I work with one who
does quantum physics, but that is way off the topic of this question.
What is the difference in size is the set of irrational numbers versus
the set of rational numbers? Explain your answer.
The rational numbers can be put onto a list that just runs down the
page. Therefor they are only one dimensionally infinite. The
irrationals require at least two dimensions of infinite listing.
For any who don't get what we are joking about here:
Take any rational number in the form of:
ABCD..../UVWX....
Prepend on either number enough zeros to pad the number of digits to
be equal.
Prepend a "1" on each number.
Interleave the digits of this number like this:
AUBVCW....
You now have an integer value that will be different for every
rational you started with.
This integer is the index for where to write down this rational number
on the list.
Because of the prepending of "1", we know that the first two digits of
this index will always be "11". This means that we will have room
left over for 98 times as many more numbers.
****************************
Assume that you have a list of the irrationals.
Assume that they are padded on the right with many zeros below the
decimal or consider white spaces as zeros.
From the first number, take the first digit to the right of the
decimal. From the second take the second. Continue down the list
creating a string of digits.
For each of these selected digits add one with no carry. 0 becomes 1,
1 becomes 2 .... 9 becomes 0
Place a decimal point in front of this string of digits.
You now have a number between 0 and 1 that differs from the first in
the first digit and differs from the second on the second digit and so
on. Therefor this number must not be on our proposed list. This
proves that in a list that is infinite in only one direction the
irrationals can't be listed.
It is left to the reader to prove whether or not there exists a rule
that would allow all the irrationals to be placed in two dimensions.