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.
Astronomical year numbering
The prefix AD and the suffixes CE, BC or BCE (Common Era, Before Christ or Before Common Era) are dropped. The year 1 BC/BCE is numbered 0, the year 2 BC is numbered −1, and in general the year "n" BC/BCE is numbered "−("n" − 1)" (a negative number equal to 1 − "n"). The numbers of AD/CE years are not changed and are written with either no sign or a positive sign; thus in general "n" AD/CE is simply "n" or +"n". For normal calculation a number zero is often needed, here most notably when calculating the number of years in a period that spans the epoch; the end years need only be subtracted from each other.