– How can some infinites be bigger than others?

Question

I have just come across the concept that there are different types of infinite numbers (whole integers, irrational numbers etc.) and the concept that for example, the amount of irrational numbers between 0 and 1 is higher than the amount of whole numbers from 1 to infinity.

I guess I just don’t understand how an infinite amount of something can be bigger/smaller than an infinite amount of something else…

Please un-f**k my brain 😀

In: 4

Answer 1

Infinity is a concept, not a number… and cannot be directly compared to them. There is no number “infinity”, it’s about how you got there.

Most commonly infinity comes up when you’re looking at long term trends and patterns. However, even if two different variables will get “infinitely large” given an unlimited amount of time, often one is still always bigger than the other. Hence, an “infinity” divided by an “infinity” could be anything, from 0 to 1 to 7 to another infinity.

If you write an equation like x^2 / 4x, then as x becomes infinitely large, you have an infinitely large top and bottom of the fraction. However you can also very clearly see that beyond x=4, the top is always bigger than the bottom and the difference just keeps growing. So here we have 2 different infinities which are also infinitely apart from each other. They’re both “infinity”, but one is very much bigger than the other.

In your “irrational numbers” example, the trick is to show that you can pair off each integer with a unique irrational number, and then showing that some irrational numbers will still be missed. Therefore, even though the number of integers is infinite, the number of irrational numbers is “more infinite” since we missed some.

Spoiler because this feels like a homework question.

>!For the sake of example, for each integer, I’m going to map it to the irrational number that’s written as “0.” followed by that integer, then that integer+1, then +2, and so on indefinitely. So irrational number 17 is 0.1718192021222324…. Clearly this will be unique for each possible integer, and the pattern should be easily understood for any possible integer, so I’ve fulfilled my obligations about building this pairing. Yet also clearly I’m missing a lot of irrational numbers, like any that would begin with `0.00`. Ergo, there are more irrational numbers between 0 and 1 than there are integers between 1 and infinity.!<