Re: Residue Classes

In article <GdE3i.30001$V75.249@edtnps89>,
"Larry Hammick" <larryhammick@xxxxxxxxx> wrote:

"Hatto von Aquitanien"
me:
I can think of two reasons why he should be using the term "equivalence
class" rather than "residue class".

What would those be?
One is that "residue class" is a long-established bit of jargon in number
theory. "Residue" is just a synonym for "remainder", on division. But
division is not involved in the current construction.
The second reason is that this formal construction is analogous to various
others around math, in which cases residues are irrelevant; examples are
-- the definition of a rational number a/b as the equivalence class of the
element (a,b) in ZxZ,

Minor nit: isn't it Zx(Z\{0}), for Z = the set of integers and 0 its
additive identity?

for the equivalence relation
wz = yx
between two elements (w,x) and (y,z).
-- the definition of a complex number as an equivalence class of polynomials
over R, modulo the polynomial x^2+1.
There are many others.
Anyhow, a residue class is a special case of an equivalence class, so that
author's jargon is not outright wrong.
LH
.

Relevant Pages

  • Re: Residue Classes
    ... class" rather than "residue class". ... The second reason is that this formal construction is analogous to various ... -- the definition of a complex number as an equivalence class of polynomials ...
    (sci.math)
  • Re: Residue Classes
    ... "Residue" is just a synonym for "remainder", ... But division is not involved in the current construction. ... -- the definition of a rational number a/b as the equivalence class of the element in ZxZ, ... The construction modulo a relation is like ring modulo ideal ...
    (sci.math)