No, a fractional patent is not what you have left over after the Patent Trial and Appeal Board gets a hold of your patent. Fractional patents are actually patents with fractional numbers.
The practice started back when the United States Patent and Trademark Office was trying to reassemble the collection of patents issued before the Office numbered patents. These pre-number era patents are known as the “X” patents, and, as the Office was assigning these patents numbers, they would occasionally come across patents whose issue dates were between already numbered patents.
In order to keep the chronological sequence, the USPTO needed to issue fractional numbers.
The USPTO even created a letter code system to identify the fractions:
The surprising thing is that the fractional number scheme continued from time to time even after the USPTO began numbering patents. Patents 126½, 1400½, RE1217½, RE1242½, D1093½, and D90793½ are all examples of fractional patents, as is fractional patent 2712152½, seen here:
And fractional patent 3262124½, seen here: