*ignore what I wrote below. I had pythag in my head with the
diagonal line thing. I suppose you could substitute counting
squares down off the paper onto the table instead of cutting
across imaginary diagonals... but I'm tired and don't feel like
rewriting this brain fart. Read ahead with caution to a tired
brain [edit] ORIGINAL answer.. which is unbelievable wrong.:
Here's the blame: The whole "i" thing has a lot of spook and awe
around it. But look: What's a square root? You're making a
square of a piece of paper let's say. You want to see the
distance from corner to corner. But there's a problem: You can't
use square units to make a diagonal. Big problem. Ah, but look:
Square root. Problem solved. You fold the paper down the
diagonal and you measure it. Done. But you're left with an
annoying bit. That diagonal doesn't exist in nice squares. Thus
begins the issues. This issue shows up in analog to digital
_all_ of the time. Want square pixels to make a diagonal line?
You can't. You can FAKE IT but at some point, you gotta face the
gaps. So ok. negative numbers. They have some clear rules in
arith. Add, subtract, multiple, divide. Simple enough anyway.
But remember, when you're dealing with numbers, they're on a one
dimensional line. You want two dimensions, you go to a 90 degree
angle from zero and go up and down. Same number line, running 90
degrees up. You got lots of tricks you can do there, but they're
always running at 90 degrees of each other, and you're playing
with the spaces inbetween. But when you want to "cut across"
that nice grid, you gotta play with geometry tricks. These start
getting weird quickly. Want that perfect circle? Eh, you ain't
gonna really get it. You can't entirely turn circles into
squares, just get close-ish. But, generally good enough. Like
the square root, folding those squares across the diagonal and
you're generally fine. But then you get into real issues. You
start leaving the paper! -1. Nice -1 sits there on the one
dimensional number line. You can slide it back and forth. You
can move it higher and lower. All sorts of neat things with the
-1. Then you say, "Hey, I want to make a diagonal connecting -1
to -1 on a grid. GREAT! Just ... wait. hang on.... -1 is off the
paper. It's on the table somewhere. Whatever. You got a negative
1 sitting on the left side of the paper on the table. You got
another negative 1 sitting just below the paper on the same
table. So ok, let's FOLD THE PAPER across the diagonal in a
place where there IS NO PAPER. Shit. Damn it. Well, maybe we can
do some stuff with circles and get PART of that -1 to -1
diagonal (the square root) back ONTO the paper. Oh good! We can
do that. Some of it. Some nice geometry tricks gets some lines
going back onto the paper. Just not all of it. But enough to get
SOME real number action there. An arc of a circle here and
there. Ok. Good ol' Pi to the rescue. Now you're like, "Hm, this
imaginary spot off the paper that i can get partly onto the
paper but not entirely... I'm going to take the diagonal line
that I can't draw anywhere - the square root of -1... and not go
up a 90 degree angle into a square... but a dimensionality
that's the same as itself. Eeerrrr.. nothing's on the paper
here. Well, some of it is. Enough of it is to work _something_
out and one of the answers ends up on the paper. 0.2ish Am I
explaining this entirely the wrong way? ABSOLUTELY. Would this
be a terrible way to teach math? yes it would. But it's how I
happen to see numbers and how I can mentally justify i^i ending
up on the paper. I mean, you're dealing with Pi in there for
example. You're dealing with a 2. You got some real number stuff
happening here. Stuff that's on the number line. Stuff that's on
paper. That we can do some flipping to get an imaginary, not
fully measurable 'something' back onto the paper from somewhere
in an impossible to measure vicinity shouldn't be surprising.
Anyway, it's how I see math. I flip things around, abstract
numbers to the number line and things like square roots I see as
folding paper diagonally...and stuff. It's a naive view of this
stuff but it's how I reconcile it somewhat. Even if it's wrong,
I'm ok with that. === It's salvaged somewhat by keeping the
concept of tying the number line to a piece of paper. and other
ways where you try to remember what analogues these mathematical
functions have when applied to paper sitting on a tabletop.
