BC or BCE is Not Negative

I have seen some math textbooks that in introducing negative numbers use AD/BC as an example of signed numbers. (The terms CE and BCE are simply substitutions for AD and BC, so the same analysis applies.) However, in actual practice the AD/BC system does not represent an example of signed numbers because there is no zero. An event that occurred on March 15, AD 14 (note on positioning of AD and BC) did occur 4 years later than one that occurred on March 15, AD 10, which is (+14)-(+10)=+4. And an event that occurred on March 15, 10 BC did occur 4 years later than one that occurred on March 15, 14 BC, which is (-10)-(-14)=+4. However, an event that occurred on March 15, AD 2 did not occur 4 years later than one that occurred on March 15, 2 BC, despite the fact that (+2)-(-2)=+4; 4 years later than March 15, 2 BC is March 15, AD 3, despite the fact that (-2)+(+4)=(+2) and not +3. This problem is caused by the fact that in the year numbering used by historians 1 BC is immediately followed by AD 1 with no zero year in between. One can see the results of this on the following table:

This off-by-one problem will occur whenever one crosses the AD/BC boundary in calculations. Since this additional complication has nothing to do with using signed numbers in mathematics, AD and BC should not be used in exercises on signed numbers. No, one does not avoid the off-by-one problem by using CE and BCE. (There is the non-standard astronomical year numbering scheme which is used to avoid this off-by-one problem by inserting a year zero; however, while this may be standard in some fields, such as astronomy; it is not the common standard used by history textbooks, which is what the common everyday standard is derived from.)

Footnotes

note on positioning of AD and BC: English teachers will tell you that AD goes before the year number and BC goes after; e.g., it's 44 BC, but AD 44. I have followed this convention, even though it seems most likely that in actual practice that most people go ahead and put the AD after as well. Note that both CE and BCE are placed after the year number.

note on CE and BCE: CE stands for Common Era, and BCE stands for Before Common Era; but otherwise are the same as AD and BC -- i.e., they do not introduce a year zero either --, so the off-by-one errors described in this page also occur for them as well.