"Dan in NY" <Dan@[EMAIL PROTECTED]
> wrote in message
news:4808d135$0$7714$4c368faf@[EMAIL PROTECTED]
> "~greg" <g_m@[EMAIL PROTECTED]
> wrote in message
> news:cPKdne4XTf5bE5XVnZ2dnUVZ_g6dnZ2d@[EMAIL PROTECTED]
>>
>> "William Elliot" <marsh@[EMAIL PROTECTED]
> wrote in message
>> news:Pine.BSI.4.58.0804172334180.10208@[EMAIL PROTECTED]
>>> On Thu, 17 Apr 2008, ~greg wrote:
>>>
>>>> Consider the lattice of integral points (i,j) in (Z,Z) in (R,R),
>>>> and probe lines on the origin, shooting through the lattice.
>>>>
>>> What is a probe line on the origin?
>>> Do you mean a line through the origin?
>> ~~~~~~~~~~~~~~~~~~~~~~
>>
>> I should not have used the word "probe"
>> if was going to throw you off that much.
>>
>> However, the problem didn't originate in mathematics.
>> Moreover, in origin, it was probably closer in spirit to projective
>> geometry
>> (where the usual expression is "line on a point")
>> than to high school analytical geometry
>> (where I suppose they do speak of "line through a point").
>>
>> It's been decades since I was in school,
>> and I'm not used to picayune criticism.
>
> [... snip]
>
> Greetings Greg and other News:alt.math readers,
>
> I have been working on something similar to your comments but I couldn't
be
> sure because I didn't understand some of your fine points. The comments
by
> William Elliot and your answers helped quite a bit. I would like to see
a
> restatement of your first statement in this thread when corrections are
made
> that reflect your answers. Would you like to post it -- or send it to
me
> via email?
>
> I think my ideas are a special case of yours with an added twist. In my
> work, the irrational slope is a trigonometric function and I distinguish
> whether it is trancendental or not when the angle (in degrees) is
rational.
>
> In particular, I want to represent the sine of an angle as a rational
number
> or as an irrational number that can be written as a (finite) expression
> using the square roots of integers or the square roots of expressions
that
> contain the square roots of integers. What can be said about the angles
> that have such a sine? To generalize, I want to represent cosines and
> tangents in the same way.
>
> I have a list of expressions for various angles and I would like to
extend
> that list.
> --
> Dan in NY
~~~~~~~~~~~~~~~~~~~~~~~
It's been a very long time, Dan in NY,
but you are definitely talking about "field theory" and "Galois theory",
....and beyond.
There're a lot of books about these subjects, but almost all of them
beat around the bush, probably more than necessary. The older ones
beat around older bushes, and the newer ones beat around newer bushes.
As soon as I straighten out this apartment, I'll have a recommendation
for you. It's a very little book. But it's the most direct route I know
of,
- going through radicals and field extensions rather than too much
old fa****oned manipulation-algebra, or too much newer fa****oned
abstract-algebra.
"What can be said about ..." ?
-- They are field extensions.
More than that, I'm not qualified to say.
~greg
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