Understanding the behaviour of infinity is one of the major accomplishments of mathematics. Sadly, the infinite is often misunderstood and could lead to various paradoxes when used or interpreted the wrong way. This FAQ attempts to explain the role of infinity in mathematics and tries to resolve a few apparent paradoxes.
Infinity is not a real number!
Very often, people try to work with infinity like they work with ordinary real numbers. They soon end up with paradoxical situations, like
It's certainly true that 2\infty=\infty. Divide both sides by infinity and we get that 2=1.
This is an absurd result.The resolution to this apparent paradox is simply that infinity is not a member of the set of real numbers. We can often adjoin infinite quantities to \mathbb{R}, but these infinite things do not behave like ordinary real numbers and not every calculation with infinite things is allowed. For example \frac{\infty}{\infty} will often not be allowed!
When asked to solve a problem in the set of real numbers, then working with infinity is very often not allowed (because again: infinity is not a real number). If you do want to work with it, then you need to mention this explicitely and you need to be very careful about the rules you use!
Why bother with infinity if infinity doesn't exist in reality?
Firstly, we don't know that. But actually it's irrelevant whether infinity exists in reality. Mathematicians work with infinity because it's easier than not working with infinity.
Say, for example, that you take a group of people and you want to measure their height. You may obtain answers like 1.70m, 1.76m, 1.84m, etc. However, you want to make a mathematical model for the experiment. It is now convenient to say that the space which contains all the lengths is just \mathbb{R}. Of course, you will never encounter a length like \sqrt{\pi} meters, but that's not important. Choosing \mathbb{R} was easy and convenient.
Furthermore, if we choose \mathbb{R}, then we can apply the mighty methods of calculus on our outcomes. For example, we may find a curve that fits our outcomes best and we can find the slope and area under that curve. If we didn't choose our space infinite, then this was impossible (or very difficult at best)!
So, what is infinity?
The first thing that we must understand is that there is no unique definition of "infinity" in mathematics. There are very different kinds and flavours of infinity. All these different interpretations of infinity have their purpose.
Sometimes, infinity is just a symbol, this happens for example
- In limits: with notations such as \lim_{x\rightarrow +\infty}{f(x)}, or \lim_{x\rightarrow a}{f(x)}=+\infty.
- The order of an element in a group.
However, while the above are simply symbols, it is often very useful to give them some kind of meaning anyway. We do this by adjoining some infinite quantities to our original set. This way, the \infty-notation in limits becomes an actual limit. Furthermore, we are often allowed to do all kinds of arithmetical operations on the infinite quantities.
- The http://en.wikipedia.org/wiki/Extended_real_number_line" is \mathbb{R}\cup \{+\infty,-\infty\}.
- The http://en.wikipedia.org/wiki/Real_projective_line" is \mathbb{R}\cup \{\infty\}.
- The http://en.wikipedia.org/wiki/Riemann_sphere" is \mathbb{C}\cup \{\infty\}.
- In http://en.wikipedia.org/wiki/Nonstandard_analysis" , there are infinite numbers and infinitesimal numbers.
And finally, infinite can be both. It can be both denoting that something is very big, and at the same time it can be worked with.
- The http://en.wikipedia.org/wiki/Cardinal_number" are used to see how big an infinite set is.
Let's pick one of each of these lists and let's explain them a bit further.
How is infinity used in limits?
In limits, infinity is just a notation. For example, the notation \lim_{x\rightarrow +\infty}{f(x)}=a means "if we take x to be really big, then f(x) will be very close to a. If we take x to be even bigger, then f(x) will be even closer to a". For example, consider the function f(x)=\frac{1}{x}. If we take x=1000, then f(x)=0.001 is very close to 0. If we take x=100000, then f(x)=0.00001 is even closer. We say that f(x) converges to 0.
What we've just done can be formalized as follows: we say that \lim_{x\rightarrow +\infty}{f(x)}=a ifwe can make the distance between f(x) and a to be as small as desired by taking x to big enough.
As you see, we haven't really done anything with infinity here. The \infty notation just meant that our numbers can be arbitrarily large (but still finite).
What is the extended real line?
The extended real line was invented because mathematicians weren't really satisfied with limits. As you saw, limits just denoted that something gets arbitrarily close to a number a if x is chosen very big. However, it kind of makes sense to say that "if x=+\infty, then f(x)=a". In the real numbers, this doesn't make sense at all, however. Because
infinity is NOT a real number. That's why we adjoin two new elements to \mathbb{R}, namely +\infty and -\infty. Now it does make sense to say that f(a)=+\infty.
We can even do arithmetic with infinity! For example:
+\infty+\infty=+\infty,~~\frac{1}{+\infty}=0,~~2<+\infty
However, not all things make sense. For example +\infty-\infty is undefined. See
http://en.wikipedia.org/wiki/Extended_real_number_line for more information.
The projective line uses the same idea. Here we adjoin an element \infty to \mathbb{R}. The idea here is that \infty encapsulates both negative and positive infinity. So the real line becomes circular: going to the right will get you to \infty, but going to the left will also get you there. With this notion of infinity, we can even define things like
\frac{1}{0}=\infty
We couldn't do this in the extended real line because the answer could be positive or negative infinity. However, \infty-\infty is still undefined, and we can't say now that 2<\infty. See
http://en.wikipedia.org/wiki/Real_projective_line for more information.
What are cardinal numbers?
Cardinal numbers are used to distinguish between different sizes of infinity. Some infinite sets are bigger than other infinite sets in some sense.
To illustrate the idea, let's first look at finite collections. Let's say we have a really smart toddler, and let's name him Greg
. You then give Greg two sets of marbles and you ask "do these sets contain an equal number of marbles"? Greg can't count the marbles (he's too young), but he can use another method to determine whether the two sets are equal. Greg can simply pick a marble from the first set and a marble from the second set and put them together. He keeps doing that same thing until all the marbles are alligned with other marbles. If there are left-over marbles that cannot be assigned to another marble, then Greg can decide that one set was bigger than another.
The same thing happens with infinite sets. We can't really count the infinite sets, but when we're given sets A and B then we can easily say when those two sets have an equal number of elements:
We say that A and B have the same cardinality if there exists a one-to-one correspondence between A and B.
This looks like a plausible definition, and if you think about it: it's about the only thing we can do for arbitrary sets! However... there's a catch. Consider the sets
A=\{0,1,2,3,4,5,...\}~~\text{and}~~B=\{0,2,4,6,...\}
You could argue that A is definitely bigger than B because B is contained in A. This is not true, however! We can find the following one-to-one correspondence
A\rightarrow B:n\rightarrow 2n
Every element in A is now assigned to an element of B and conversely. So A and B are the same size! This is a paradoxical situation, but we just got to get used to it. It's the nature of infinity.
Using some clever reasoning, we can even say that the set of natural numbers, the set of integers and the set of rational numbers all have the same number of elements. The set of real numbers is much larger though! Cantor was the first to discover these things.
