One interesting question

- Joe sworn in as VP of USA at 11:something a.m.

- George's presidency ended at 12 pm on Jan 20th

- Barack is sworn in as Prez at 12:05 p.m. on Jan 20th

so my question is, for those in between 5 minutes from 12:00 to 12:05 was Joe the Prez? So, technically is O the 45th Prez? That would mean the same for all the other previous Prez's too?

If you know the technicalities please enlighten me!

P.S. : Just to avoid search traffic, I've got the first names and some abbr.