Comparing Infinities Can Give You Counterintuitive Results

Brilliance | Dec. 10, 2017

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.

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.

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