Consider for instance the following chord (C7) :
- C E G Bb
- E G Bb C
- G Bb C E
- Bb C E G
How can I number these inversions? I remember it was something with a '+'.
It's just basic figured bass which on the V chord in a major key would look like:
The above is attached to a Roman Numeral for demonstrations, but would be omitted if you were just doing the figured bass. Figured bass is always built off the key so all the notes used above are strictly within the key. If you need to alter notes of the key to represent it then you would need to use a "+" or a slash though the number that represents the desired pitch to raise it and a "-" in the same fashion to lower it.
The numbering stems from the way in which you count from the lowest note up. Lets take those examples you mention. It is good to understand the reasoning behind it so as it not to become a memory exercise.
F major's dominant seventh chord (ie. C7) in root position.
Now let us count from the bottom note. It is C then the next note is E. That is a third. Then we have E to G that is a third again or a fifth from the root. Lastly we have G - Bb. That is a third again and the seventh from the root.
So we have 3, 5 and 7 when you count from the bottom note (The root.) That is why the full notation for a seventh chord in root position is 7/5/3. Most of the time like it is in this case it is just plain V:7
F Major's dominant seventh chord in first inversion.
Now we have the first inversion chord so the E is in the bass. Lets do the same again. E-G is a third. E - Bb is a fifth and E - C is a sixth. Three, five and six. Hence the notation 6/5/3 or more commonly just 6/5.
F Major's dominant seventh chord in second inversion.
Now we have the second inversion chord. Now the G is in the bass. Let us count again. G-Bb is a third. G-C is a fourth and G-E is a sixth. So we have 3, 4 and 6.
Hang on to your hats now we cannot call the second inversion seventh chord a 6/4. that name is already taken by our regular triads in second inversion. So we call it V:4/3 instead.
F Major's dominant seventh chord in third inversion.
So the third inversion chord. We start on Bb-C a second. Bb-E a fourth and Bb-G a sixth. So we have 2, 4 and 6. Still we cannot call it a 6/4 chord so we instead call it a 4/2 chord.