rjnoe@ihlts.UUCP (Roger Noe @ N41:48.5, W88:07.2) (02/15/84)
To begin with, one must define elementary functions for complex numbers in such a way as to be consistent with their real counterparts. Among other things, this means than ln must be the inverse of exp. Further, the same rules of logarithms and exponents must apply. For example, ln(a^b) == b*ln a, e^(x+y) == e^x * e^y, etc. Any other technique leads to inconsistencies, forcing one to treat numbers with complex components as entities entirely different from real numbers. This defeats the purpose of complex numbers. Similar problems are encountered by treating negative integers as fundamentally different, subject to different laws of mathematics than positive integers. A better idea was to treat them all as simply integers. The same tendency to make mathematics general-purpose can be seen in the integration of irrational numbers with the rationals to make the reals. It took many people a long time to accept the concept of zero, then rational numbers, then negative numbers, then irrationals, and finally complex numbers. (Let's skip extended reals for the moment.) But the wisest course is to for- mulate general rules in math to apply to all these sets of numbers. Until we get to complex numbers, we have a problem in mapping. With any of the other subsets, elementary functions can easily give us results that don't map into the same set; subtraction on positive numbers, division on integers, and both square root and logarithms on reals. But with extended complex numbers, there is no such problem. Any combination of elementary functions one can imagine will map onto this same set. What still causes people some problems is that not all of these map one-to-one. Mappings that are not bijections are really rather common, even when one is not used to dealing with anything but reals. The square root function is bivalued on the reals (except at zero, of course), and the square (x^2) function maps two-to-one. It is not surprising to find that logarithms on complex numbers are multivalued. To reject this is equivalent to rejecting the Euler formula: e^(i*t) = cos t + i * sin t Because e^[i*(t+2*PI*n)] has a single value for all integers n. Thus the exponential function is an infinite-to-one mapping. To reject this is to reject the basic concept of complex numbers. Real numbers are on a line, so complex numbers are on a plane. Such a concept is very useful, very natural for our Euclidean minds. Certainly one can come up with a self-consistent, but very different concept of complex numbers, but human beings would have a difficult time using it. I think mathematics exists independently of any intelligence, but that doesn't mean that all MODELS of it are the same. The model must be suited to our minds. Accepting the Euler formula means accepting traditional rules for a real raised to a complex power. But this also means accepting that ln i = i * (2n + .5) * PI because e^[i*(2n+.5)*PI] = i for all integers n. It is fallacious to presume that just because certain mappings are one-to-one in the real domain that the same must hold true in the complex domain. Keeping the utility of mathematics, we must say that ln(i^i) = i * ln i. Similar rules for multiplication of complex numbers require that i * i = -1, a pure real. The fact that i^i is infinitely multivalued is inescapable from the accepted model (or representation) of mathematics. One can come up with a different repre- sentation, but that does not make it useful or even consistent. The best approach, for those who have trouble with i^i, is to use the principal value, where n=0. Then the modulus of all complex numbers must be within the interval (-PI, PI]. Of course, any single half-open real interval with a 2*PI width works just fine. Roger Noe ihnp4!ihlts!rjnoe AT&T Bell Laboratories
akt@mcnc.UUCP (Amit Thakur) (02/19/84)
"real numbers are on a line, complex numbers on a plane." wonder: are there any numbers in 3-D? how about in N-D? as explained by Roger Noe, complex numbers are required to have a set which is closed with respect to all operations. does this mean there are no functions whose answer(s) cannot be found in the complex plane? if we created (just for fun) numbers in N-D, would we have any use for them? akt at ...decvax!mcnc!akt
amigo2@ihuxq.UUCP (John Hobson) (02/20/84)
Yes, there are numbers beyond complex numbers, they are called quaternions, that have the properties Q = a + bi + cj + dk where a, b, c, and d are any real numbers; i^2 = -1, j^2 = -1, k^2 = -1; i != j != k; and i, j, and k have relationships i * j = k, etc. The proper mapping of quaternions is to a four dimensional space. John Hobson AT&T Bell Labs Naperville, IL (312) 979-0193 ihnp4!ihuxq!amigo2
mam@charm.UUCP (Matthew Marcus) (02/21/84)
The best mapping for quaternions is to Pauli spin matrices which have
the same algebra. The reason you don't hear too much about generalizations
of complex numbers is that all such are equivalent to representations of
Lie groups, usually the rotation group. For example, Pauli matrices are
the bases of the 2x2 matrix rep. of the rotation group in 3D. See any
good quantum mech. text for more details.
{BTL}!charm!mamjoseph@orstcs.UUCP (04/03/84)
You didn't seem to notice that this is SUPPOSED to be
philosophy.