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Infinity and Imaginary Numbers

Date: 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/ 
Associated Topics:
High School Imaginary/Complex Numbers

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