Re: general discussion

On Sat, 6 May 2006 14:19:10 +0100, Jonathan L Cunningham
<spam@xxxxxxxxxxxxxxxxxxxx> wrote in
<news:1hex30n.146ghvoa0q1scN%spam@xxxxxxxxxxxxxxxxxxxx> in
rec.arts.sf.composition:

Gerry Quinn <gerryq@xxxxxxxxxxxxxxxxxxx> wrote:

In article <1hetr34.1fk2xbuaf3zmoN%spam@xxxxxxxxxxxxxxxxxxxx>,
spam@xxxxxxxxxxxxxxxxxxxx says...

I suspect it is some mystical confusion caused by the
use of a definite article in the *usual* interpretation
of the symbol "i". Saying "i" is *the* square root of
minus one is making *two* implied ontological
assertions, which you seem unaware of. The first is that
there *is* a square root of minus one, the second that
it can be referred to by the definite article, i.e. that
it is unique (or the plus/minus pair is unique).

The problem as I see it is that the use of i is intrinsically
misleading because the concept of i as the square-root of -1 does in
fact involve a lot of associated concepts, in particular a number
system that maps onto a plane rather than a line. Your modular version
doesn't seem to have anything corresponding to that mapping. (There
may be overtly similar systems that do have some such correspondence,
whole or partial, and in any such case I would consider the use of 'i'
justifiable.)

Thanks.

I can't help wondering if that would happen so strongly if our
language lacked the definite article.

Don't really think so: the problem is with the context and
remains even if one speaks carefully and says that i is an
arbitrarily chosen root of x^2 + 1 = 0 in the algebraic
closure of the reals. It's 'the' square root of -1 in the
same useful sense that C is 'the' algebraic closure of R: up
to isomorphism.

But also it's a hobby-area of mine. For example you could take
s = sqrt(2), and have two-component numbers (x,y) = x + s*y, which
have some (different) similarities to (conventional) real and
imaginary numbers.

You can see the resemblance: if you stick to integer values for
x and y, then when you multiply two y components you get an x
component. The correct (or standard) name for such a
thing escapes me for the moment, but you can find it in Wikipedia
(I'm sure Brian knows it).

That would be the algebraic integers in the extension field
Q(sqrt(2)), analogous to the Gaussian integers in
Q(sqrt(-1)); it's an integral domain. In general if D is a
square-free integer (except 1) that's congruent to 2 or 3
mod 4, {a + b*sqrt(D) : a, b in Z} is the ring of algebraic
integers in Q(sqrt(D)); when D == 1 (mod 4) it's a little
more complicated, as I recall.

I don't know of a single generic term for these critters,
though. And some of them are nasty, because they don't have
unique factorisation into irreducible elements (i.e., the
analogues of prime numbers: elements that can be written as
a product only if one of the factors is a unit [= invertible
element] and hence analogous to 1 or -1 in the integers).
Let me see if I can find an example with D > 0. Right, here
we go: take D = 10, and let s = sqrt(10). Then
(4 + s) * (4 - s) = 6 = 2 * 3, and it's possible to show
that 4 + s, 4 - s, 2, and 3 are all irreducible in Z(s).

Quick sketch, which you may have seen before: For integers
a and b define N(a + bs) = (a + bs) * (a - bs) = a^2 - 0b^2.
Show that N((a + bs) * (c + ds)) = N(a + bs) * N(c + ds).
Show that a + bs is a unit iff N(a + bs) is 1 or -1.
N(4 + s) = N(4 - s) = 6, N(2) = 4, and N(3) = 9, so if any
of these were reducible, it would have to have a factor
a + bs with N(a + bs) = 2 or 3; consideration of N(a + bs) =
a^2 - 10b^2 modulo 5 shows that this is impossible.

[...]

So the idea that there is such a thing as *the* complex
plane is, perhaps, slightly weaker for me.

It's simply a matter of definition. The term 'complex
numbers' refers specifically to the algebraic closure of the
real numbers, a field that is provably unique up to
isomorphism; the only wiggle room is in the choice of which
isomorphic version to use. (And the term 'complex plane'
probably tends to suggest the representations of the form
(x, y), x + iy, and r*exp(it), where x, y, r, and t are real
numbers and in each case the operations are defined in the
familiar way, rather than purely abstract algebraic
constructions like R[x]/(x^2 + 1).)

Of course there are other mathematical objects with
constructions similar to one or another construction of the
complex numbers, by virtue of which they may share some
properties with C, but they aren't '(the) complex numbers'
or '(the) complex plane': those names are already taken.

[...]

Brian
.

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