Re: [crazy_music] Re: crazy, man... crazy

Hi Jon,

<continued discussion from crazy_music>

For anyone joining at this point, it is to do with discussion
of whether it is okay for crazy_music to be called the
"Irrational Microtonal Music Workshop". We have decided
it is off topic for the group, and belongs to metatuning.

> OK yes I really should have said 'irrationals *include*
> transcendental numbers'.

Yes.

> And imaginary and complex numbers can be irrational or
> rational I suppose: 3i vs 2i/3 + 1/4.

Yes they can.

In fact Galois theory (is all
to do with starting with the rational complex numbers,
then adding in irrational numbers such as square roots
one at a time, each time adding in at the same time
all multiples of the new number by the numbers already there
inverses of it, negative vrsion of it and so on, to complete
the resulting "field" under all the arithmetical operations.

2i/3 + 1/4. is same as (8i + 3)/12, so if one thinks
of 8i + 3 as a kind of complex integer, this is a complex
rational.

You can also do things like find (3 + 8i)/(11 - 5i)

Multiply top and bottom by (11 + 5i) and you will get rid
of the imaginary part in the quotient:
(11 + 5i) (11 - 5i) = 121 -25 * (i)^2
= 146, so after multiplying out the top as well, using i^2= -1
you will end up with a complex number with rational coefficients
again.

So, complex numbers with rational coefficients are "closed"
under all the usual arithmetic operations - when you apply
any of them, result will again be a complex number with rational
coefficients.

Galois theory is used to show that certain things can't be
done with ruler and compass. Basic idea is, start with
the rational complex numbers. Considering them as coordinates
they give you all geometrical points with rational coordinates
= all the points one can construct using a ruler alone using
traditional Euclidean methods, and two points.

Now add in constructions involving circles, and one finds
this is equivalent to being able to find the square root
of any of the complex numbers constructed so far.

Now take one of thc classic ruler and compass problems from
antiquitey, e.g. to trisect 60 degrees using ruler and
compass.

With a lot of rather advanced algebra,
one can prove that this point isn't any of the ones
you get by using rational coordinates + adding in the
square root construction. So it can't be constructed
using the standard Euclidean methods of ruler and
compass construction.

It can be constructed using other methods of deploying
the rulers and compasses, and some people think they have
found a counter example by doing this, but these extended
geometrical ways of making the points were already known
to the Greeks, so it is nothing new.

Same method applies for the other classic problem of
doubling the volume of a cube, and what was the third
one, I forget now.

> Not sure if imaginary numbers are or are not
> transcendental. Graham says "A transcendental number
> can't be the solution of a polynomial equation of
> integers." which if that is a sort of definition
> would separate the two since complex numbers obviously can
> be soch a solution.

To state it more completely, a transcendental number
can't be the solution of a polynomial equation with rational
coefficients (or one could also say, a polynonial equation
with integer coefficients here, as one can multiply through
by the least common multiple of all the quotients to get
integer coefficients).

Some complex numbers are solutions of polynomial equations
with rational coefficients. Others aren't. So you get
rational, irrational, and transcendental complex numbers,
just as you do for real numbers. (Here, "real" = math
jargon for number with no imaginary part to it, i.e.
ordinary non complex number).

Rest was about things Graham was explaining, so leave it to
him!

Robert