Astronomical year numbering
Cassini gave the following reasons for using a year 0:
Fred Espanak of NASA lists 50 phases of the moon within year 0, showing that it is a full year, not an instant in time. Jean Meeus gives the following explanation:
Although he used the usual French terms "avant J.-C." (before Jesus Christ) and "après J.-C." (after Jesus Christ) to label years elsewhere in his book, the Byzantine historian Venance Grumel used negative years (identified by a minus sign, −) to label BC years and unsigned positive years to label AD years in a table. He did so possibly to save space and put no year 0 between them.
Anno Domini
This calendar era is based on the traditionally reckoned year of the conception or birth of Jesus of Nazareth, with "AD" counting years from the start of this epoch, and "BC" denoting years before the start of the era. There is no year zero in this scheme, so the year AD 1 immediately follows the year 1 BC. This dating system was devised in 525 by Dionysius Exiguus of Scythia Minor, but was not widely used until after 800.