Math determined that dropping the bombs saved more lives than invasion through conventional means.

I'd be hesitant to trust the math too far, especially since it's based on many assumptions as to how the war would have played out despite one side being severely outgunned.

Call me an anti-military cynic, but I'm sure a bit of it was also the idea that they now had a nuke, and wanted to see what kind of damage it could do. Perhaps the numbers there were better than if it had been dropped on any other viable target (especially a European one): that math, at least, I can believe.