Wikipedia:Reference desk/Archives/Mathematics/2014 January 16

= January 16 =

Defining infinity plus a complex number
We know that:

$$ \infty + r = \infty $$ for any real number

$$ \infty + \infty = \infty $$ (this can be simplified to $$ 2 \times \infty = \infty $$, but we're focusing on adding numbers to infinity, not multiplying numbers by infinity.)

$$ \infty + -\infty $$ is an indeterminate form

But has anyone ever thought of defining infinity added to a complex number?? This does produce new numbers that are not in finite positions on the complex number plane, but that can be defined as the sum of infinity or negative infinity and a (finite) complex number. I notice that a number of this kind occurs as follows:

Let's define tetration $$a {\uparrow\uparrow} b$$ for negative values of b. We get:

$$2 {\uparrow\uparrow} 0 = 1$$

$$2 {\uparrow\uparrow} -1 = 0$$

$$2 {\uparrow\uparrow} -2 = -\infty$$

No new kind of number appears yet. But in defining $$2 {\uparrow\uparrow} -3$$, we get a new number; this is $$ \infty + ((\pi \times i)/(\ln(2))) $$. (Pi*i is the numerator; ln(2) is the denominator.)

Thus, we have a value of infinity plus a complex number. Is this kind of number mentioned anywhere in Wikipedia?? (It doesn't matter exactly what number other than that it meets this criterion.) Georgia guy (talk) 21:36, 16 January 2014 (UTC)
 * The most usual sort of infinity used in conjunction with the complex numbers is the infinity of the Riemann sphere. In that context, &infin; plus any complex number is again &infin; (but &infin; + &infin; is undefined).  I have never heard of anyone trying to define tetration on the Riemann sphere. --Trovatore (talk) 21:43, 16 January 2014 (UTC)
 * Not only that those (non)numbers exist and have been known, but you can even see definite integrals with limits of the form $$\int_{a+bi}^{\infty+bi},$$ which basically means integration alongside a straight line parallel to the real axis. Mathematica uses directed infinity for such points in the extended complex plane other than $$\pm\infty$$ and $$\pm i\infty$$. — 79.113.226.240 (talk) 18:43, 17 January 2014 (UTC)

Defining infinities on the Riemann sphere
Go to Riemann sphere. It considers the complex number plane extended as having only one infinity for all values. Why do we have to define it this way?? Why can't we just define groups of infinities; the simplest of these are:

$$ \infty $$

$$ -\infty $$ (this one is useful in defining tetration because it is $$2 {\uparrow\uparrow} -2$$

$$ \infty \times i $$

$$ -\infty \times i $$

All of the above multiplied by $$ (1+i)/(sqrt(2)) $$

Then there's an infinity for every direction; the 8 above are those multiplied by the eighth roots of 1. Georgia guy (talk) 23:34, 16 January 2014 (UTC)
 * Well, you can define anything you want, if it's just for the fun of defining stuff, but it doesn't necessarily connect with any interesting or useful theory. In this case, note that you lose the ability to divide by zero, at least if you want to preserve the property that (az)/(aw)=z/w for nonzero a.  And being able to divide by zero is a big part of the point of the Riemann sphere, as that lets you develop a nice theory of Möbius transformations. --Trovatore (talk) 23:45, 16 January 2014 (UTC)
 * Your "directional infinities" are somewhat similar to the real projective plane, in which "infinity in the direction of +x" is the same as "infinity in the direction of -x" but distinct from "infinity in the direction of +y". See Real projective plane and line at infinity. The Riemann sphere is instead the complex projective line. Different definitions yield different notions of infinity. « Aaron Rotenberg « Talk « 00:29, 17 January 2014 (UTC)


 * Not directly related to your question, but you might like to read about the surreal numbers. 50.0.121.102 (talk) 08:03, 18 January 2014 (UTC)