Re: Residue Classes

Virgil wrote:
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

I have not checked into all the references given on that page
109, may be you do if you are really interested.

It is just a wording not used today (and not sure what's due
to translation to English).

What they do is constructing a group from a half group (or so),
IN x IN /~ = Z, the integers.

What they call "residue class w.r.t. equivalence relation" is
a representant, but no clear definition is given (as over the
whole text - Math often was written that way in older times).

Thinking about the according footnote I think they have the
following in mind:

The construction modulo a relation is like ring modulo ideal
or classical Z / mZ. A German word is "Restklassen-Ring" and
an element is called "Restklasse", which at that time is not
the same as residue (not sure how it was used before).

The reason is: given some number k you dived by n. Which may
not give a natural number, something reminds - a "Rest", at
school children/students learn(ed?) that as "Teilen mit Rest".

Probably you want to check the German original text.

.

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