Does infinity have a 0-like characteristic

Question

If you pick a number between 0,1, there are infinitely different possibilities, then if you were to pick a number between 0 and 0.25, infinite possibilities, BUT wasn't there infinite possibilities between 0,1 so it could be infinite/4, meaning that infinite=infinite/4, so infinity has a 0-like characteristic in that way, also, if you divide a number by infinity, you would get undefined, because infinity is undefined itself, but 0 is defined, also if you divide a number by 0, it's undefined, and if the dividend is 0, the quotient would be 0. So isn't infinity like 0 in this way. At the same time, 1/x, when x gets bigger, the number gets smaller, but it never gets to 0, I guess if you don't know that X is, 1/x would be impossible to solve, same way to infinity, because we don't know what that value is, so perhaps my last argument about 1/0 = 1/infinity = undefined, isn't true.

I'm a 7/8 student, so my knowledge is very limited.

Answer 1

To do mathematics, we have to have precise definitions of all the terms we use. So, the question we have to ask before anything else here is, what do you mean by "infinity"? There is an established definition of "infinite", but it's a property of sets (that is, of collections of things): a set is finite if you can label the things in the set $1, 2, \dots, n$ for some natural number $n$, and the set is infinite otherwise. So, for example, it's indeed true that "there are infinitely many real numbers between 0 and 1".

Actually, we can also say what it means for two sets to have the "same size" (the technical term is cardinality): two sets $S$ and $T$ have the same cardinality if there is a one-to-one correspondence between the two sets, that is, if we can pair up elements of the sets so that each element of $S$ is paired with exactly one element of $T$ and vice versa. So, for example, we can say that the interval $[0, 1]$ (the set of all real numbers between $0$ and $1$, including the endpoints) has the same cardinality as the interval $[0, 0.25]$, because we can pair up each $x$ in $[0, 1]$ with $x/4$ in $[0, 0.25]$. (This isn't the only notion of "size"—for example, if we're talking about intervals, length is a different notion of size, and of course the length of the interval $[0, 1]$ is four times the length of $[0, 0.25]$. But cardinality is a notion of "size" that makes sense for all sets, regardless of whether they consist of numbers or other objects.)

By the way, to connect these two notions, notice that a set being finite means that it has the same cardinality as the set $\{k \in \mathbb{N} : 1 \leq k \leq n\} = \{1, 2, \dots, n\}$ for some natural number $n$. (To cover the edge case of the empty set, I'm adopting the convention here that $0$ is a natural number, so when $n = 0$ this is just the empty set. By the way, if you haven't seen it before, the symbol $\mathbb{N}$ just means the infinite set of all natural numbers, that is, $\{0, 1, 2, 3, \dots\}$.)

However, neither of these tells us what's meant by "infinity". The word suggests that it's a thing in itself that's somehow like a number, but also has something to do with the property of "being infinite". It turns out there are quite a few different ways to mathematically formalize this, and these have different implications for how arithmetic with "infinity" or "infinities" works. To name a few:

• The extended real number line is probably the closest to what you're thinking of. This is a system of arithmetic that consists of the usual real number line, plus two extra symbols, $\infty$ and $-\infty$, whose arithmetic matches how "limits approaching infinity" work in calculus—that is, intuitively, they encode what happens when a quantity "gets larger and larger without bound" (or, in the case of $-\infty$, more negative rather than larger). In this number system, it's indeed true that $\infty = \infty/4$. But also, in the extended real number line, we do actually have $1/\infty = 0$; adding these extra "infinite" elements to the system lets us meaningfully define this division. This is because, in terms of limits, $1/x$ approaches zero as $x$ "approaches infinity" (that is, grows without bound). On the other hand, some arithmetic operations still can't be defined: division by zero, $\infty - \infty$, $\infty/\infty$, and $0 \cdot \infty$ are all still undefined.
• A slight variant on this is the projective real number line. This is the same as the extended real line except that $\infty$ is unsigned, neither positive nor negative, so in this system $-\infty = \infty$, while $\infty + \infty$ is also left undefined. We can also define division by zero in this system: $1/0 = \infty$, because as $x$ approaches zero, $1/x$ gets larger and larger in absolute value, and there's no problem with the sign since we don't have a distinction between $+\infty$ and $-\infty$ here. (However, $0/0$ remains undefined.)
• What if we want numbers to refer to sizes of infinite sets? It turns out there are many "different sizes of infinity", as Cantor's theorem famously shows. Cardinal numbers are a number system that encodes cardinalities of sets, and we can indeed add, multiply, and exponentiate cardinal numbers, though subtraction and division aren't defined.
• Similarly, if we want to encode not just abstract sets but a well-ordering (intuitively, like a counting process, except possibly extended infinitely), there's the related system of ordinal numbers.
• And if you want to get really fancy, there are number systems called the hyperreal and surreal numbers, which are rather complicated to describe but include a whole array of "infinite" and "infinitesimal" elements.

An important takeaway here is that there isn't just one notion of "infinity", there are a whole bunch of them, and which one makes sense in a given context really depends on what you're trying to study. Going back to your question of how zero and infinity are similar, here's one thing we can say: in the projective real number line, $0$ and $\infty$ are "reciprocals" in the sense that $1/0 = \infty$ and $1/\infty = 0$.

Answer 2

Yeah, you're somehow correct, their properties are very well comparable.

In fact, there are a lot of spaces and topics in mathematics where $0$ and $\infty$ are corresponding to each other.

Two examples: First, $\mathbb{S}^1 = \{(x,y)\in\mathbb{R}^2 | \sqrt{x^2+y^2} = 1\}$, the unit circle. How can you create this circle? You can take two sets of $\mathbb{R}^1$ and glue them together by the function $x \to \frac{1}{x}$. What does that mean?

You take the first $X=\mathbb{R}$ and then you take the second $Y=\mathbb{R}$ and bend/curve it so that the $0$ of $X$ will be directly opposite of the $\infty$ of $Y$ and the $2$ of $X$ will be directly opposite of the $\frac{1}{2}$ of $Y$. Now you glue these sets together, so that $X$ and $Y$ cannot be differentiated anymore. See the following image: This is an example, where $0$ of the one set and $\infty$ of the other are the same.

Another example: Projective spaces, e.g. $\mathbb{P}^1$, the set of all lines in $\mathbb{R}^2$ passing through $0$. There $\infty$ and $0$ are also two similar points with similar properties - both are special points that can cause the same problems. These numbers are defined on $\mathbb{P}^1$ as $0 = [0:1]$ and $\infty=[1:0]$, also looking pretty similar.

This is just touching the surface of those two examples, to show you what $0$ and $\infty$ have in common.

I hope you can understand my writing, I tried to use as little advanced mathematics as possible. Just ask if anything is unclear.

If you want to know more about projective spaces, you can look on the internet or just ask, I'm happy to pass on my knowledge.