Infinity and Imaginary NumbersDate: 06/15/2003 at 13:25:44 From: Brandon Subject: Infinity and Imaginary Numbers I've been thinking about both of these concepts and their possible connections. I would like to know if these ideas are currently established as valid or not. To start with I see _i (the imaginary number) as having the ability to be negative and positive simultaneously, thus allowing it to be negative when squared. A sort of singularity of two opposite properties of numbers. That leads me to believe _i is a property also, and obviously not a number (as it is currently understood). Just as -,+ could not exist without multiplication by 1, _i cannot exist without it. So they fall into another category, and I'll continue calling that category properties. Now infinity seems to have different sorts, like directional infinities and a sort of omni-directional infinity. The omni- directional infinity encompasses both directions and is thus positive and negative at the same time. It would then suffice to say it has the property of _i. This leads me to believe that omni-directional infinity is a directional infinity * _i, or symbolically where inf = omni-infinity and inf+- = directional (+/-), inf = inf+- * _i. It seems simple and intuitive to me but I haven't seen it formalized. It also adds some insight to infinities that I see as useful. For instance, that you can change inf to inf+- by squaring it. inf = inf+- * _i, inf^2 = (inf+-)^2 * (_i)^2, inf^2 = inf+- * -1, inf^2 = inf-. And in calculus lim x = inf, lim x = inf+- * i, lim x/i = inf+-, x->inf x->inf x->inf lim -xi = inf+-, and when the L is figured... -inf*i = inf- x->inf There's a whole lot more to it that I can think of, but I don't want to waste time on an already established concept, or even worse, one that isn't right. Brandon Date: 06/15/2003 at 19:41:20 From: Doctor Tom Subject: Re: Infinity and Imaginary Numbers Hi Brandon, If you look at complex variables, there is a reasonable way to deal with signs and infinity. It is not a good idea to think of numbers like i and -i as having signs in the usual way, but that doesn't mean the idea is meaningless. Eveny complex number can be assigned a location on the complex plane with its real and imaginary axes. Instead of a simple +/- sign, an infinite number of "signs" is assigned (although they are not called "signs," but rather "arguments." To find the argument of any number other than zero, draw a line from 0 to that number and measure the angle that line makes with the ray from 0 out the real axis. The argument of all real numbers is 0; the argument of all negative reals is 180 degrees, the argument of any number on the positive imaginary axis is 90 degrees, the argument of (1 + i) is 45 degrees, and so on. When you multiply two numbers, the resulting number will have an argument that is the sum of the arguments of the multiplicand and multiplier, if you're careful to subtract any angle of 360 degrees or more. Thus the product of two negatives has an argument of 360 degrees - oops - that's all the way around, so subtract 360 and you get zero degrees - a positive number. i times i is 90 + 90 = 180 --- a negative real, et cetera. This always works, so there are, in a sense, an infinite number of possible "signs" (called arguments) for complex numbers, and all of them, except zero, have an argument. In math books, for technical reasons you will never see these arguments expressed as degrees. They are always measured in radiams from 0 to 2 pi, but the idea is identical. In fact, the location of a complex number is determined by its "magnitude" (distance from the origin) and argument. To multiply two complex numbers, multiply the lengths of their magnitudes to get the magnitude of the product and add the arguments (subtracting multiples of 2 pi (or 360 degrees) if necessary. There's a uniform way to handle infinity, too. There is just one infinity, and the term "as z approaches infinity" simply means that z gets farther and farther away from zero in any direction, even by spiraling out or any other path you can think of. There is an easy mapping of the complex plane plus the single "point at infinity" to something called the "Riemann sphere." Imagine a sphere sitting on the complex plane at the origin, so its "south pole" touches the origin. To do the mapping, draw a straight line from the north pole to the point on the complex plane, and wherever that line punctures the sphere is the corresponding point on the sphere. As points on the complex plane get farther and farther from the origin (in any direction), that puncture point gets closer and closer to the north pole, so the north pole is the image of infinity on the Riemann sphere. This all works incredibly nicely and gives a completely uniform way to talk about all the complex numbers, including infinity. And of course, as with zero, there is no reasonable way to assign a "sign" (argument) to the point at infinity, either. - Doctor Tom, The Math Forum http://mathforum.org/dr.math/ |
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