The difference between 0 and 1 is 1, and the difference between 0 and 2 is 2. But that's only in terms of whole integers. Let's talk infinity. There are infinite numbers between 0 and 1; think about taking 0.1 and dividing it by 10 again and again forever. The number gets infinitely small and never reaches zero. The set of numbers between 0 and 1, then, is an infinite set. Likewise, there is an infinite quantity of numbers between 0 and 2. In those simple terms, you can equate the two sets. Another way to compare those two infinite sets is by way of bijection. Watch the videos below to get it all sorted out.
Comparing Infinities Can Give You Counterintuitive Results
The concept of infinity is infinitely fascinating. But before we really get into some mind-boggling number stuff, it's important for you to first understand this: There is more than one kind of infinity. In fact, there are infinite infinities, and some infinities are bigger than others. And when it comes to comparing infinities, the results can seem rather counterintuitive. Here's an example: There can be just as many numbers between 0 and 1 as there are between 0 and 2.
How To Compare Infinities
There are different kinds, and they can be used in many different ways.
from Undefined Behavior
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How To Count Infinity
The definition of infinity is concrete, right? Not so much. It's hard enough to contemplate the basic idea of infinity. But when you really start playing with the concept, amazing things happen.
How Many Kinds Of Infinity Are There?
Betcha thought that there's only one...
from Vi Hart
Key Facts In This Video
In math, all a number needs to be infinite is to be bigger than any finite number. (0:09)
There is no set of surreal numbers because there are too many to fit in this set. (7:38)
Each unique counting number also has a unique prime factorization. (8:45)
Counting With Bijections
Bijections are all about making sure each number in a set has a buddy in another set.
from MIT OpenCourseWare