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.)

There have been many answers already, but I'd thought maybe I could try to provide an easier explanation of how cardinality works.
Consider the three infinite sets $A = \{1,2,3,...\}$, and $B = \{1,2,3,...\}$ and $C = \{1,2,3,...\}$, that is to say, each of them are just all natural numbers. Surely you agree that they have the same size, regardless of how we calculate it.
We now change $B$ a bit: We add 5 to every number in it. So now $B = \{6,7,8,...\}$. Adding 5 to every number surely does not change the number of elements in $B$. So we can say that the size of $A$ and $B$ are still the same.
Now let's do something different: Let's multiply all numbers in $A$ by 2 and all numbers in $C$ by 3. This surely does not change the number of elements in them - we only shuffled around the values a bit.
But now we also see that $A = \{2,4,6,...\}$ and $C = \{3,6,9,...\}$ - the sets we initially looked for! This means that, strangely enough, they have the same number of elements in them. That's why we say both sets are of the same size.
So why did we initially think that $\{2,4,6,...\}$ has more elements than {3,6,9,...}? That's because we implicitly used the natural numbers as a scale to measure how far we went in the sequence - we compared the elements by their numeric value, not by their index in the sequence.
And that's absolutely not wrong in itself, but we must remind ourselves that this only shows us how quickly the values in the sets grow compared to the natural numbers. That is just something different from the number of elements they contain (in relation to each other).
In math speak, our initial idea refers not to the cardinality, but the density, and the answer of @Henning Makholm covered it very nicely.
Another example
It may also be helpful if you consider this example: You know fractions, right? They are made of a numerator and a denominator like $\frac{a}{b}$. Let's look at the two sequences $\frac{1}{1}, \frac{1}{2}, \frac{1}{3}, ...$ and $\frac{1}{1}, \frac{2}{1}, \frac{3}{1}, ...$. From their construction, it's intuitive to say they have the same number of elements - we always added 1 to either the numerator or denominator in each element.
But at the same time, the former sequence only occupies the space between 0 and 1, while the latter is equal to all natural numbers (which looks a lot larger). So the absolute values involved in a sequence are no indicator of how many elements it contains (again: different concepts!).
