Are we allowed to compare infinities? I'm in middle school and had a question (my dad is helping me with formatting).
We're learning about infinity in math class and there are a lot of problems like how it's not a number and how if you add one to infinity it doesn't change value.
But can you have one infinity be more than another? There are an infinite amount of odd numbers and an infinite amount of even numbers, so are there the same number of odd and even numbers?
I think so, because for every odd number $n$ there is an even number $n+1$. So the odd numbers are $1,3,5,7,\ldots$ while the even numbers are $2,4,6,8,\ldots$, and as long as you stop counting at an even number the two lists will have the same number of numbers.
But there are also an infinite amount of multiples  of $2$ and an infinite amount of multiples of $3$, but I don't think there are the same amount of both. The multiples of $2$ are $2,4,6,8,\ldots$ while the multiples of $3$ are $3,6,9,12,\ldots$ So, no matter which number you stop at, the multiples of $2$ will have more numbers.

(Side question (this is dad speaking, now): is there an easy way to explain why we need to put dollar signs around mathematical expressions to make them look prettier? My daughter doesn't know what $\LaTeX$ is, but I want to give her an explanation that isn't horribly hand-wavy.)


So, no matter which number you stop at, the multiples of 2 will have more numbers.

The problem of thinking in infinities is that you must force yourself to not think about stopping. If you stop counting, you have a finite sequence and in that case, the number of odd and even numbers will depend on where you began and where you stopped. In infinity it will always be infinity. 
