Year 0

Date: 10/19/98 at 15:58:19
From: Ann Warger
Subject: "Year 0"

While working with my eighth grade Algebra class, we came across the 
fact that between 10 B.C. and 10 A.D. there were really 19 years rather 
than 20. The book stated that there is no year 0. The only information 
I could find on this was from a History college professor who said that 
this is accurate, because the year between 1 B.C. and 1 A.D. is , 
historically, the 14th year of the reign of Augustus.  Do you have any 
more information that you could pass on to me?  

I am also now unsure how to answer the questions in my other textbooks 
that really use B.C./A.D. as an example of a number line with 0 as the 
origin.  Our social studies teacher is also wondering how she should 
treat this.

Thank you for your help.
Ann

Date: 10/19/98 at 18:24:38
From: Doctor Rick
Subject: Re: "Year 0"

Hello, Ann. 

Thanks for asking this question, it's a good one. To answer it, we need 
to look at the history of the calendar, and the history of math.

BC dates were not used in the BC era. That's pretty obvious if you 
think about it. :-) As a matter of fact, AD and BC were not invented 
until around AD 525, by Dionysius Exiguus. 

Still, AD 525 is far enough in the past that people in Europe didn't 
have a clear idea about negative numbers. In fact, it wasn't until 1657 
that a mathematician (John Hudde) used a single variable to represent 
either a positive or a negative number. 

For all those years until 1657, positive and negative numbers were 
handled as separate special cases. For instance, quadratic equations 
were solved by separate methods for 6 cases: 

   ax^2 = bx, ax^2 = c, bx = c, ax^2 + bx = c,
   ax^2 + c = bx, and bx + c = ax^2 

-- zero was a special case, too. (This was the approach of al-Khwarizmi 
in the Arabic book that gave algebra its name, about AD 830.)

Let's get back to Dionysius. He identified the year 1 AD ("1 Anno 
Domini," the first year of the Lord) as the year that Christ was born 
(he was probably off by 4 to 6 years). The previous year was 1 BC, 
the first year "Before Christ." (I have not figured out why one 
abbreviation is in Latin while the other is in English.) 

Dionysius would not have thought of 1 BC as the year -1; he would not 
have thought of putting BC and AD together on a number line (time 
line). BC and AD were two separate cases. To find the time between two 
AD dates, you would take their difference; to find the time between an 
AD date and a BC date, you would add them and subtract 1. This would 
not be seen as a problem; it's just the way they solved quadratics - a 
set of special cases.

It's a challenge to try to see the world as people saw it before key
concepts were developed. It also gives perspective in dealing with 
students who are having trouble grasping concepts like negative numbers 
- we are asking them to lay hold of ideas that were missed by many 
generations of intelligent people!

To answer your specific questions, it is incorrect to introduce 
negative numbers using a time line. If we had it to do over again, we 
would certainly make it so that a time line would work - but we are 
stuck with a year-numbering system that predates the integers, and we 
can't pretend otherwise. On the other hand, we sure have a great way to 
bring up the history of a key mathematical concept!

ONLINE BIBLIOGRAPHY

Here is an interesting Web page about the origin of the calendar (see 
the section "Century and Millennium").

  http://ourworld.compuserve.com/homepages/Larry_Freeman/calendar.htm   

Here is another (see the section "The start of Anno Domini Dating").

  http://ourworld.compuserve.com/homepages/khagen/Easter.html#AD   

Here is a Web page about the origin of negative numbers.

  http://www.seanet.com/~ksbrown/kmath298.htm   

This page is my reference for John Hudde.


  http://jeff560.tripod.com/variables.html   

And this is my reference for al-Khwarizmi.

  http://www-groups.dcs.st-andrews.ac.uk/~history/Mathematicians/Al-Khwarizmi.html

- Doctor Rick, The Math Forum
  http://mathforum.org/dr.math/