From: Brian Harper (harper.10@osu.edu)
Date: Wed Sep 03 2003 - 15:59:25 EDT
DFS said:
One line, the one commonly
noted, goes from integers to rationals to irrationals and imaginaries.
And each set of numbers, including the modular numbers you mentioned later,
represents a real part of our universe. One cannot count the flow of water
in the same way one counts apples. Imaginary numbers seem totally ridiculous
unless you are in the other dimension which they represent, in which case
they are indispensible. Any electrical engineer uses irrational numbers
daily, in practice if not in theory. The waves of electricity, which most of
us have seen graphed, are heavily based on both irrational and imaginary
numbers. This is a case where I argue that they are based - not modeled in
an abstract sense. The rotation of a magnetic field near electrical
conductors produces electricity which is very precisely represented by these
imaginary and irrational numbers. Tell me that isn't real? I'll admit you
can't see it, smell it, taste it or hear it - but the calculations will tell
you what will happen if you try to get in its way.
Euclid described the two 'party trick' geometries. And Einstein demonstrated
how they represent reality when one gets beyond our 'normal' limits of size.
The Romans used Roman Numerals successfully for years. They had no zero.
Does zero exist - certainly! Our need for math increases with our realm of
knowledge. If we had no need for the microscopic or macroscopic - we would
have no need for these two 'non-Euclidean' geometries.
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