Question

How come some infinities are bigger than other infinities?
Like Eg=> There are more numbers between 0 and 1 than any real numbers!

Answer 1

Well, firstly, that's not true. Perhaps you mean that there are more real numbers between 0 and 1 than there are integers?

More importantly though, you can only use the word "more" in a very loose sense. Strictly speaking, when comparing sets of numbers (like the set of integers, or the set of real numbers), you might ask whether you can come up with some kind of one-to-one correspondence between sets, where each element in one set corresponds to exactly one element in the other and vice-versa. If you can do this, then the two sets are the same "size". If the sets are infinite, then we say that the sets have the same "cardinality".

If a set can be put in a one-to-one correspondence with the natural numbers (1,2,3, ...) then the set is classified as "countably infinite". Such sets include the natural numbers themselves, the integers, and the rational numbers.

The irrational numbers and the real numbers are two sets that *cannot* be put in one-to-one correspondence with the natural numbers. They are called "uncountably infinite", and they represent the next class of infinite sets. In this limited sense, we say they are "larger".

Answer 2

https://www.youtube.com/watch?v=A-QoutHCu4o&index=1&list=PL15B1545C780DD16B

Answer 3

check this of minutephysics @Jemurray3

Answer 4

I watched this and asked this question.
If you can relate your answer with this video, it will be helpful.
Thank you, i m little confused

Answer 5

The video states "There are more real numbers between zero and one than there are integers", which is exactly what I said above. One way that you can think about this is to ask, "is it possible to make an ordered list (which might be infinitely long) of the numbers in a set?" If you can do that for an infinite, then the set is countably infinite. If you can't, the set has greater cardinality, and is in that sense "larger".

For example, I will list all of the perfect squares:

1. 1
2. 4
3. 9
4. 16
5. 25
...

so on and so forth. As you can see, each perfect square can be partnered with exactly one natural number, so the set of perfect squares and the set of natural numbers have the same cardinality. They are both "countably infinite."

The point is that you *cannot* possibly make such a list for all real numbers between zero and one. Think about how you might go about organizing it - it can't be done. So the set of real numbers between zero and one has greater cardinality, and is called "uncountably infinite."