Questions about infinity and complex numbers

Question

a) If $\infty + 1 = \infty$ then is $\infty+i$ equal to $\infty$?

b) More generally, is $\infty + ai$ equal to $\infty$, for any $a\in \Re$?

c) What is $\infty i$? I found this in a math paper.

Thanks in advance.

Answer 1

First of all, $\infty$ is not a number. It is better to think of $\infty$ as a limit and of operations like $\infty+1$ as an operation on limits.

In the case of $\mathbb R$ there are two different types of infinity: $+\infty$ and $-\infty$.

In the case of $\mathbb C$ there are infinitely many different types of infinity as there are infinitely many directions in the plane. So every number on the unit circle $e^{i \phi}$ gives rise to a type of infinity denoted $e^{i \phi} \cdot \infty$.
In particular, $i \infty$ is such a type of infinity.

If $z=x +i y \in \mathbb C$ is some complex number, then still for any sequence $w_n$ converging to $1\cdot \infty$, we have $\lim_n z+w_n=\lim_n(x+\text{Re } w_n)+\lim_n i(y +\text{Im }w_n) = \infty$, because $y+\text{Im } w_n$ is bounded, so "much smaller" than infinity.
Multiplying this equality with $e^{i \phi}$ generalizes it to all types of infinity $e^{i \phi} \cdot \infty$.

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A general remark about infinity:

Often, we are only concerned with if something converges to infinity, but not in which direction. Identifying all points of infinity gives rise to a new topological space.
In the case of $\mathbb R$ identifying $+\infty$ and $-\infty$ (gluing them together") corresponds to a circle, called $\mathbb RP^1$.
In the case of $\mathbb C$, gluing together the circle at infinity gives the a sphere, called the Riemann sphere or $\mathbb CP^1$.
We can also look at the plane $\mathbb R^2$ and only glue together the infinities at opposite directions. The resulting space is called $\mathbb RP^2$ and looks much more complicated than the ones above. In particular it is not orientable.

In this spaces, $\infty$ corresponds to a well-defined object, it is just a point in this space, just like $0$ corresponds to another point in the space.
For example on the Riemann sphere, $\infty$ is the north pole and $0$ is the south pole. But for symmetry reasons they need to behave quite similar.

Answer 2

In the past, people have extended number sets and constructed "bigger" sets (take $\mathbb N \subset \mathbb Z \subset \mathbb Q \subset \mathbb R \subset \mathbb C$ for example). (Someone would say "discovered" instead of "constructed", let's leave the distinction to philosophers.) In all those extensions, the main point is to also extend the arithmetics (i.e. define operations such as +, $\cdot$ etc.) so that they satisfy the same laws (or close to it) as before (commutativity, associativity, existence of inverse etc.)

Now, the problem here is that there is no standard, agreed way of extending the above number sets with infinity - rather there are many ways, different to each other, all useful for something, rather than one canonical, agreed upon, way.

For example:

• You can extend $\mathbb R$ with either one infinity $\infty $ (on "both sides") or with two infinities ($+\infty$ and $-\infty$, one at each "side"). Addition and multiplication makes sense, but there is no additive inverse anymore, and only the first case has a multiplicative inverse, but only in the second case you can extend the natural order (relation $\le$).

• The former one extends to the 2D real plane in two different ways: either there is just one infinity (this is also useful in complex analysis) or there is one infinite point associated with every direction - this is useful in projective geometry.

• In set theory, we speak of cardinal numbers and ordinal numbers. Those can be infinite and can have different sizes (some are bigger than others). They admit some arithmetic (you can define addition and multiplication on both).

There is much more to it, I am just putting some of the standard undergraduate knowledge here. You can also try to extend the standard number sets with infinity/ies and define arithmetic operations in your own way; many people have tried that, so chances that you will discover a new and useful way of doing that are slim - but you never know...

The result is that just saying "infinity" without providing context is ambiguous. If you specify which theory the "infinity" is coming from, it will be possible to answer your questions. Without a context, those questions are just not well defined.

Edited to add: As your question is about complex analysis, I guess your "infinity" is the one, single element $\infty $ added to $\mathbb C $. There is a standard way of adding it in complex analysis. In that context:

• You set $\mathbb C^*=\mathbb C \cup \{\infty\} $,
• You define topology on $\mathbb C^*$ by defining the neighbourhoods of $\infty$ (sets containing all complex numbers wih "big enough" modules),
• That gives you the notion of convergence, which then lets you define $\lim_{z\to\infty}f (z) $ for complex functions $f $ defined on some neighbourhood of $\infty $, and
• Only then you can start extending functions on $\mathbb C $ to $\mathbb C^*$ by defining $f (\infty)=\lim_{z\to\infty}f (z) $.

Once you've done all that work, you will start taking functions such as $f (z)=z+1$, $f (z)=z+ai $, $f (z)=iz $, and that will let you conclude that, in that context, it is true that $\infty+1=\infty $, $\infty+ai=\infty $, $i\infty=\infty$ etc.