The notes of a triad can be rearranged so that different notes are in the bass. These are called inversions.

When the first note of the triad is in the bass, like all the chords we have seen so far, the chord is said to be in root position:

This chord, if it is the key of c major, can be written I⁵₃, meaning it is the notes of the tonic chord arranged so that there is a fifth and a third from the bass. But since thirds are taken as read, it is more usually written as just I.

If the third of the triad is in the bass, we have what is called a first inversion:

Because this chord has a sixth and a third above the bass, it can be written in full as I⁶₃, but again thirds are taken for granted so it can be abbreviated to just I⁶.

Unsurprisingly, if the fifth of the chord is in the bass, we have a second inversion:

 

This chord has a sixth and a fourth above the bass and is written I⁶₄, no shorthand allowed. I guess it could be written I⁴ with the third above the fourth implied, but I’ve never seen that.

This odd convention of a stack of numbers alongside the roman numeral of the root may seem unnecessarily complicated, since it is saying “the notes of such and such a root triad rearranged so that these intervals are above the bass”. They actually come from an old style of musical notation called figured bass where the composer would write out the melody and the bass line in full, but would then write these numbers under the bass line to indicate to the keyboard player which chords they should be playing. In thatcase the numbers (without roman numerals) did indicate the intervals above the given bass note. But when that numbering was carried over into harmonic theory, the root of the chord was considered (rightly) much more important than the bass note of the chord, and so we have this rather counter-intuitive naming system that you just have to get used to.