even/odd analogy for ternary?
Jeremy Targett
2 mod 3? I need to refer to these properties and would rather use
established names than invent my own. (If I do have to invent names, what
do you suggest? First thing off the top of my head is "even", "surodd" and
"subodd". Maybe if the use of "even" was judged confusing, "threeven"
would be a cute coinage)
Robert Israel
Standard terminology is "divisible by 3" (or "congruent to 0 mod 3"),
"congruent to 1 mod 3" and "congruent to 2 mod 3". Your coinages may
sound cute to you, but they would be bizarre to most mathematicians.
Robert Israel isr...@math.ubc.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada
Jeremy Targett
> In article <eajc7h$i3c$1...@us23.unix.fas.harvard.edu>,
> Jeremy Targett <ot...@elsewhere.example.com> wrote:
>>Are there terms in use anywhere for numbers equal to 0 mod 3, 1 mod 3 and
>>2 mod 3? I need to refer to these properties and would rather use
>>established names than invent my own. (If I do have to invent names, what
>>do you suggest? First thing off the top of my head is "even", "surodd" and
>>"subodd". Maybe if the use of "even" was judged confusing, "threeven"
>>would be a cute coinage)
> Standard terminology is "divisible by 3" (or "congruent to 0 mod 3"),
> "congruent to 1 mod 3" and "congruent to 2 mod 3". Your coinages may
> sound cute to you, but they would be bizarre to most mathematicians.
It's not that I'm trying to sound cute, I just want to avoid repeating a
phrase like "congruent to 1 mod 3" over and over in a text that attempts to
be conversational, and requires this notion. For example, a fact stated
like "an odd plus an even yields an odd" is easier to take in than "a
number congruent to 1 mod 3 plus a number congruent to 2 mod 3 equals a
number congruent to 0 mod 3". The payoff for a short name is even higher
when less obvious properties are to be stated.
In any case I would of course define my terms before using them.
Thanks - JT
Gene Ward Smith
"Even" won't work, as it's in use. Just give them names like threeven,
throd, and thud or something.
Bill Taylor
> Are there terms in use anywhere for numbers equal to 0 mod 3, 1 mod 3 and
> 2 mod 3?
Sadly, no. It would indeed be nice to have a single adjective
and avoid clumsy circumlocution.
I suggest treble, excessive and deficient,
for 3n, 3n+1, 3n-1 respectively.
Intriguingly, there WAS (is?) a term for 4n and 4n+2,
namely, "evenly even" and "oddly even".
There was never anything for 4n+1 and 4n-1; though
excessive & deficient would do there too. As they would
for 6n+/-1, used in prime number tasks at times.
---------------------------------------------------------------------
Bill Taylor W.Ta...@math.canterbury.ac.nz
---------------------------------------------------------------------
Q: How do you call a headache after pondering about the 3n+1 problem?
A: Collatzeral damage.
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Robert Israel
Jeremy Targett <ot...@elsewhere.example.com> wrote:
>Robert Israel <isr...@math.ubc.ca> wrote:
>> In article <eajc7h$i3c$1...@us23.unix.fas.harvard.edu>,
>> Jeremy Targett <ot...@elsewhere.example.com> wrote:
>>>Are there terms in use anywhere for numbers equal to 0 mod 3, 1 mod 3 and
>>>2 mod 3? I need to refer to these properties and would rather use
>>>established names than invent my own. (If I do have to invent names, what
>>>do you suggest? First thing off the top of my head is "even", "surodd" and
>>>"subodd". Maybe if the use of "even" was judged confusing, "threeven"
>>>would be a cute coinage)
>
>> Standard terminology is "divisible by 3" (or "congruent to 0 mod 3"),
>> "congruent to 1 mod 3" and "congruent to 2 mod 3". Your coinages may
>> sound cute to you, but they would be bizarre to most mathematicians.
>
>It's not that I'm trying to sound cute, I just want to avoid repeating a
>phrase like "congruent to 1 mod 3" over and over in a text that attempts to
>be conversational, and requires this notion. For example, a fact stated
>like "an odd plus an even yields an odd" is easier to take in than "a
>number congruent to 1 mod 3 plus a number congruent to 2 mod 3 equals a
>number congruent to 0 mod 3". The payoff for a short name is even higher
>when less obvious properties are to be stated.
Perhaps rather than new names you might use some notation: let
[j] = j + 3Z be the set of integers == j mod 3, and then you can
state your fact very concisely as [1] + [2] = [0].
James Dow Allen
Bill Taylor wrote:
> Jeremy Targett wrote:
>
> > Are there terms in use anywhere for numbers equal to 0 mod 3, 1 mod 3 and
> > 2 mod 3?
>
> Sadly, no. It would indeed be nice to have a single adjective
> and avoid clumsy circumlocution.
>
> I suggest treble, excessive and deficient,
> for 3n, 3n+1, 3n-1 respectively.
This relates to a question I have, espousing hexagon
grids for image processing.
The 4-neighbor topology offers obvious names, like
up, left, down, right
and the 8-neighbor topology also:
North, NE, East, SE, South, SW, West, NW
but there don't seem to be corresponding names for
6-neighbor topology. (Are there?)
I wonder if this sort of "language defect" rather than
any technical disadvantage, is a reason why hexagon
grid is not popular in image processing, despite its
superior properties.
James Dow Allen
Jeremy Targett
> I suggest treble, excessive and deficient,
> for 3n, 3n+1, 3n-1 respectively.
Thanks Bill, I like it!
--JT
Robert Israel
James Dow Allen <jdall...@yahoo.com> wrote:
>This relates to a question I have, espousing hexagon
>grids for image processing.
>
>The 4-neighbor topology offers obvious names, like
> up, left, down, right
>and the 8-neighbor topology also:
> North, NE, East, SE, South, SW, West, NW
>but there don't seem to be corresponding names for
>6-neighbor topology. (Are there?)
Why not, say, NE, East, SE, SW, West, NW?
Gene Ward Smith
Bill Taylor wrote:
> I suggest treble, excessive and deficient,
> for 3n, 3n+1, 3n-1 respectively.
Hmmm. What about natural, sharp, and flat?
Bill Dubuque
>Jeremy Targett <ot...@elsewhere.example.com> wrote:
>>
>> Are there terms in use anywhere for numbers equal to 0 mod 3, 1 mod 3 and
>> established names than invent my own. (If I do have to invent names, what
>> do you suggest? First thing off the top of my head is "even", "surodd" and
>> "subodd". Maybe if the use of "even" was judged confusing, "threeven"
>> would be a cute coinage)
>
> for 3n, 3n+1, 3n-1 respectively.
Jeremy later wrote:
>> It's not that I'm trying to sound cute, I just want to avoid repeating a
>> phrase like "congruent to 1 mod 3" over and over in a text that attempts
>> to be conversational, and requires this notion. For example, a fact stated
>> like "an odd plus an even yields an odd" is easier to take in than "a
>> number congruent to 1 mod 3 plus a number congruent to 2 mod 3 equals a
>> number congruent to 0 mod 3". The payoff for a short name is even higher
>> when less obvious properties are to be stated. In any case I would of
>> course define my terms before using them.
If you write excessive deficiency = treble
versus 1 + 3Z + -1 + 3Z = 3Z
or 1 + -1 = 0 (mod 3Z)
then most mathematicians will stop reading your work
right at that point. It smacks of mathematical crank
(or trebles worse than such excessive deficiencies)
Natural language names for congruence classes would
obfuscate the important relationships between the
arithmetic of Z and its quotient ring Z (mod mZ)
If you disagree, then please go prove all of the
theorems in Gauss' Disquisitiones Arithmeticae
using only natural language names for integers
and their congruence classes. Maybe then you'll
realize there will be no reward for your trebles.
TIP It'd be odd to study the trouble with trebles
without even first studying the trouble with twos -
see my prior posts on parity rings (no parody intended).
http://google.com/groups/search?q=group%3A*math*+dubuque+parity+ring
--Bill Dubuque
Jeremy Targett
> If you write excessive deficiency = treble
> versus 1 + 3Z + -1 + 3Z = 3Z
> or 1 + -1 = 0 (mod 3Z)
> then most mathematicians will stop reading your work
> right at that point.
I'm not writing for mathematicians. I need to use the concept in a text
aimed at lay people, and so it needs to be conversationally stated. It's as
simple a concept as evenness and oddness, so why require the reader to learn
what mathematicians mean by "congruent" and "modulo"? (Even if these terms
are also conceptually simple, new terms will often cause trouble for readers
not used to thinking like mathematicians.)
In the end, despite liking Bill Taylor's terms, I took the advice of Robert
Israel and explained in the text what I the reader should understand by
[0]-numbers, [1]-numbers and [2]-numbers.
Gene Ward Smith's suggestion of flat, natural and sharp was nice too!
David R Tribble
>> I suggest treble, excessive and deficient,
>> for 3n, 3n+1, 3n-1 respectively.
>
Gene Ward Smith wrote:
> Hmmm. What about natural, sharp, and flat?
Or, like the three colors of quarks: red, green, and blue.
And there's also up, down, and strange.
And top, bottom, and charm.
How about left, center, right?
Liberal, moderate, conservative?
Richard Henry
Chemists use prefixes like ortho, bis, para. I don't know if their
implicit meanings would interfere in understanding what you mena in a
mathematical sense.
Bill Taylor
> Why not, say, NE, East, SE, SW, West, NW?
Damn! Robert beat me to the punch, here.
But it leads to a nice extension, of even MORE use to a computer
graphicist than a mathematician. Because the fact is, that using
the natural directions enforced by the rectilinearity of typeface
and screens, there are actually TWO hexagonal grids
you might like to use.
This one...
\__/ \__/ \__/ \__/ \__/ \__/ \__/ \__/
_/ \__/ \__/ \__/ \__/ \__/ \__/ \__/ \_
\__/ \__/ \__/ \__/ \__/ \__/ \__/ \__/
_/ \__/ \__/ \__/ \__/ \__/ \__/ \__/ \_
\__/ \__/ \__/ \__/ \__/ \__/ \__/ \__/
_/ \__/ \__/ \__/ \__/ \__/ \__/ \__/ \_
\ / \ / \ / \ / \ / \ / \ / \ /
...or this one...
| | | | | | | | | | | | |
\ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \
| | | | | | | | | | | |
/ \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
| | | | | | | | | | | | |
\ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \
| | | | | | | | | | | |
/ \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
| | | | | | | | | | | | |
...and clearly the directions are N,NE,SE,S,SW,NW for the former,
and E,SE,SW,W,NW,NE for the latter!
How cute is that!?
--------------------------------------------------------------------
Bill Taylor W.Ta...@math.canterbury.ac.nz
--------------------------------------------------------------------
A Mobius stripper will never reveal her backside!
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James Dow Allen
Bill Taylor wrote:
> Robert Israel wrote:
>
> > Why not, say, NE, East, SE, SW, West, NW?
>
> Damn! Robert beat me to the punch, here.
FWIW this is the nomenclature I was already using.
It just seems very unsatisfactory. "NE", "NW",
are not as intuitive as "up", especially since the
implied 45-degree isn't even correct.
I guess gravity (which leads to natural sense
of horizontal and vertical) is why human language
supports 4- but not 6-neighbor 2D structure.
(Of course there is terminology for 3-D 6-neighbor,
add "out", "in"). Would bee language support hexagon?
James
James
cbr...@cbrownsystems.com
C.nat.nat = C.nat = C works.
On the other hand C## = D; but also E## = F#.
Likewise, Ebb = D; but also Cbb = Bb.
Maybe in a semi-tone scale where A## = Cb, it would work better...
Cheers - Chas
James Dolan
bill taylor <w.ta...@math.canterbury.ac.nz> wrote:
i'm not sure whether i might prefer simply 12 o'clock, 2 o'clock, and
so forth for the former, and 3 o'clock, 5 o'clock, and so forth for
the latter.
--