I understand that 4/4 4, in this case means quarter notes. So would I be wrong
in saying that 4/8 is just 4/4 at a faster tempo? It just sounds like I'm increasing the tempo. So if I had 4/9, would I be able to the same thing? The tempo is just a quarter note faster?
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The bottom number indicates the rhythmic division that represents one beat. The top number indicates how many of those rhythmic divisions are in a bar.

4/4 means four quarter notes.
4/8 means four eighth notes.
4/9 doesn't exist (for the most part) because 9th notes don't exist.

Time signatures don't say anything about tempo.
The performer should be able to read the tempo from the marking at the start of the piece. If you notate the time signature "4/9" he would have to say "okay, 120bpm at 4/4, lowest common denominator is 36... multiply by uhhh..."
The lowest number in a time signature is nearly always a power of 2. The difference between 4/4 and 4/8 is the subtle difference between a quarter note and an eighth note. Mathematically there is no difference but tradition has assigned them different roles that are understood by a performer.
Bearing in mind that the performer is the only one who is going to read your time signature, there's not much point in being overly smart about time signatures. Stick to the traditionally accepted route and you will communicate yourself much easier.