On Jun 20, 2007, at 2:36 PM, Mark D. Niemiec wrote:
Granted, this value lies right along a line separating two regions,
and
floating-point inaccuracies would explain why a number might stray
slightly
to one side or other of this line, but tolerant comparisons should
remove
such artifacts. The anomaly looks particularly heinous in the
second case,
since it interrupts the linear progression of values.
The numbers in a computer do not contain all the reals, much less all
the complex numbers.
My papers assume a mathematical treatment of numbers.
Given that:
The complex plane is completely tessellated by slanted bricks, with
the upper left and lower left boundaries within the brick, and the
other sides of the brick not in the brick.
If a Gaussian or complex number arises it must be on one of the
included sides or not. The plane is completely defined by these L-
shaped sides.
Given this brick, then ceiling and residue follow -- defined
algebraically.
Given this definition, the division theorem follows, and one of its
strengths is that it is the identical division theorem obtaining
among the reals, and that residues will always be strictly less than
the divisors.
The reason I undertook this study was to answer Larry Breed's
question: how is complex residue defined?
This subject is treated more fully in the my article "Complex Floor"
in the Proceedings of APL Congress 73, Copenhagen, Denmark, edited by
Per Gjerlov, H.G.Helms and John Nielsen, and even more fully in my
"Integer Functions on Complex Numbers, with Applications", IBM
Philadelphia Scientific Center, Tech. Report No. 320-3015, February,
1973.
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