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When we talk numbers in music theory, we are talking about where a note is in the major scale. Lets look at the major scale in the key of C.
The first note we play in this key, the note C, we call "1". The second note we play, the note D, we call "2". The third note we play, the note E, we call "3" and etc. When we get through the entire octave, we start on C again, which we again call "1".
There are many good reasons to do this. We can describe musical distances in numbers. The distance between C ("1") and E ("3") is called a "third", or the distance 1 to 3. The distance between C ("1") and G ("5") is called a "fifth", or the distance 1 to 5. The distance from C to B is called a "seventh" as it is 1 to 7.
We can use this to describe chords as well. I the key of C, the I chord is C, the IV chord is F and the V chord is G. I prefer to use roman numerals to describe chords and alphanumeric numbers to describe chords so you know if I'm talking about single notes or chords. After all, a C note sounds very different from a C chord.
This is a brief introduction to what we mean by the number system in music. To look more into this, check out my book "Easy Music Theory for Fiddle and Mandolin" available at the "books" link above.