Why are root position, first, second and third inversion labeled by V7, V65,V43,and V42 repectively? Are there reasons for the specific numbers chosen? If not how can I memorize them? Thanks.
At one point in time (mainly the Baroque period,) a common way to notate a keyboard part was to simply write the bass part and then notate with numbers what intervals above the bass note were needed to complete the chord. This is called figured bass.
So, for instance, if you wanted to indicated a root position 7th chord, you would write 3,5,7 below the bass not. The image below shows how each inversion would be notated:
Over time, the simplified shorthand developed, which is what we use today to indicate inversions. It all comes from the tradition of figured bass.
Edit: See Kyle Strand's explanation in the comments for why these shorthands in particular were chosen. It's a good way to help remember which is which.
It's figured bass. The numbers correspond to the interval between the bass note and the notes above the bass.
In root position you have the intervals 3, 5, and 7 above the bass. 5 and 3 are just standard triad intervals so the it is simplified to V7
In first inversion you have the intervals 3, 5, and 6 above the bass. We have just a 6 to denote a triad in first inversion, so we need to notate more than 1 note so we simplify it to V65.
In second inversion you have the intervals 3, 4, and 6 above the bass. We have a 64 to denote a triad in second inversion, so instead we need to notate more than the other two notes so we simplify it to V43.
In third inversion you have the intervals 2, 4, and 6 above the bass. It's very similar to the last inversion we just replace the 3 with a 2 and we get V42.