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March 27

Number theory

Consider a+bi where a and b are non zero. Is there any meaning attached to calling this complex number positive or negative and is there such a thing as a prime complex number? 92.7.27.62 (talk) 19:51, 27 March 2009 (UTC)[reply]

For the first, the most reasonable way would be to call a+bi positive iff a is positive, but I doubt there's much use in that. For the second, the Gaussian primes, a subset of the Gaussian integers, might be what you're looking for. —JAO • T • C 20:03, 27 March 2009 (UTC)[reply]

Each of a and b can be independantly positive or negative. That gives 4 possible combinations of polarities, and those are the 4 quadrants of the complex plane. Cuddlyable3 (talk) 20:06, 27 March 2009 (UTC)[reply]

There's no way of assigning the notions of positive and negative to complex numbers that behaves the way you would want it to under addition and multiplication. Suppose, for example, you decide i is positive, then presumably −i is negative. A positive times a negative should be negative, but i × (−i) = 1. If you're not worried about addition and multiplication then you can put various preorders on the complex numbers, like the ones mentioned above. Chenxlee (talk) 20:13, 27 March 2009 (UTC)[reply]

The most common use of "positive" for a complex number is that the number has to be a positive real number; for example see definite bilinear form. Similarly, if you see someone write "z ≥ 0", in a context where z is supposed to be a complex number, they mean that z is real and nonnegative. — Carl (CBM · talk) 20:19, 27 March 2009 (UTC)[reply]

As the others have said, there isn't much use for a notion of order on the complex numbers. However, I wanted to emphasize that the notion of primes in the Gaussian integers is very important: for example, it is used in one of the Proofs of Fermat's theorem on sums of two squares#Dedekind's two proofs using Gaussian integers (second one), and the general notion of primes in integer-like rings is central to algebraic number theory. Eric. 131.215.158.238 (talk) 23:37, 27 March 2009 (UTC)[reply]

See Ordered field. Usually, the product of two positive numbers has to be posiive, and the sum of two positive numbers has to be positive. So if we were to define three sets partitioning the complex numbers, P, N and {0}, calling P the set of "positive numbers" (requiring P to be closed under addition and multiplication), N to be the set of all negatives of elements in P, and 0 to be the additive identity, such that P and N are disjoint, we would have a suitable ordering. Such a triple cannot exist (you can prove this if you like; it fails to have a P closed under multiplication and addition). Therefore, no interesting ordering can exist on the complex numbers. Similarly, since the complex numbers form a field, the notion of a prime isn't very interesting. Perhaps it is more interesting in the Gaussian integers. In any case, these concepts are basically questions in ring theory. --PS T 06:07, 28 March 2009 (UTC)[reply]

I think it's a little unfair to say that "no interesting ordering can exist on the complex numbers." I consider a well-ordering of the complex numbers very interesting, for instance. Also in special cases words like "positive complex number" can be very convenient. I saw an interesting variant on the Frobenius problem that used exactly that terminology. Like the real Frobenius problem, it relies on addition not always being invertible. Black Carrot (talk) 23:06, 28 March 2009 (UTC)[reply]

There are perfectly sensible, natural, and interesting orderings on the complex numbers, such as putting a + ib ≥ 0 if and only if a ≥ 0 and b ≥ 0 (this makes C a lattice-ordered field). The catch is that the order is not total. — Emil J. 13:10, 30 March 2009 (UTC)[reply]

What I meant to say (or should have said) was that there are no interesting orderings on the complex numbers, in the context of ring theory. --PS T 02:47, 2 April 2009 (UTC)[reply]