The Second Millennium

Date: 11/29/1999 at 20:11:32
From: Steve Hesketh
Subject: Year 2000

I have heard debates about whether the second millennium actually 
starts on the year 2000 or if in fact it starts on the year 2001. My 
belief is that it starts on the year 2001 because the number system 
starts at the number 1 and not 0. I think about counting from 1 to 10. 
I think of 10 as the last number of that series and 11 being the 
beginning of the next series. The beginning is where the counting 
starts which I thought was number 1. 

Another way for me to look at it is in the area of grades in school. 
In school, if a student does not do an assignment assigned to do he/
she will have a 0. If students do the assignment, they may receive 
between 1 and 100 [not 99] based on how much they completed and how 
near to perfect it is. 

If the start of the millennium is not based on math or logic, then 
what is it based on? My wife says it's not based on our modern 
mathematical system but rather on the basis of when 1 BC ended, that 
the years started counting up from 0 and then on up to 1 AD and so on. 
She says that the zero was counted by the ancient people in their form 
of counting.

I would like to read or hear your view on this before I go any 
further. Is there a right or wrong way to look at this debate, or can 
it be a never-ending debate?

That is all for now.

Date: 11/30/1999 at 09:26:43
From: Doctor Peterson
Subject: Re: Year 2000

Hi, Steve.

Yes, this can be an endless debate, because it's really about words as 
much as math, and words always leave room for disagreement, even 
though mathematically there's only one answer. But there are some 
interesting ideas attached to it, and I'll take this opportunity to 
write out some things that I haven't heard anywhere else.

We have an answer to some of your questions in our archives at:

   Millennium and the Year 0
   http://mathforum.org/dr.math/problems/katie1.3.99.html   

This deals with the fact that there was no year zero, because people 
didn't understand the concepts of zero and negative numbers yet when 
the current system of numbering years was started. The year "AD 1" 
meant "the first year of [not 'after'] the Lord," and the year "1 BC" 
meant "the first year before Christ." Neither they nor we talk about 
the "zeroth year," so you can see why 0 didn't seem necessary. It's 
only when you try to do things like calculate the number of years 
between 5 BC and 5 AD that you find it would be more convenient if 
5 BC could be treated as -5. But one result of this is that, with no 
year zero, the first century had to be years 1 to 100, not 0 to 99.

On the other hand, having a year zero wouldn't quite solve all the 
confusion. Suppose there had been a Year Zero, in which Christ was 
born, say on Jan. 1 to keep it simple. Then Jan. 1 in the Year +1 
would be one year after, and Jan. 1, Year -1, would be one year 
before the birth. But what would you call the first centuries AD and 
BC? If you include 0 in the "first positive century," so it consists 
of the years 0 - 99, then it can't also be in the "first negative 
century," which would still have to be 1 - 100 BC (or -1 to -100). 

Or you could make a case, especially if the birth were, say, on 
July 1, that the Year Zero was neither "before" nor "after" Christ, 
but should be kept separate, as 0 is neither positive nor negative. 
Something would still be wrong. Maybe it's really better not to have a 
Year Zero, so we don't have to debate over whether to think of it as 
AD or BC.

The real issue is just that ordinal numbers like "first" don't fit 
well with negative numbers and zero. As you pointed out, we count 
starting with 1, not 0 (though 0 is lurking in the background, as the 
number we had before we started counting); yet that gives you only 99 
counting numbers before you need a new digit, and naturally want to 
say you've started a new century. Notice, by the way, that there is no 
"Century Zero" either; and I can't imagine calling all the years from 
0 to 999 the "zeroth century." Yet no one questions that the second 
millennium starts after, not in, the 20th century.

To look at this another way, the integers really represent only points 
on a number line, not intervals. "Zero" doesn't last a whole year, but 
is just a moment in time. We could diagram the years like this:

                            |
                      <---BC|AD--->
          5   4   3   2   1 | 1   2   3   4   5        year names
    <---+---+---+---+---+---+---+---+---+---+---+--->
       -5  -4  -3  -2  -1   0   1   2   3   4   5      time in years

There was a point in time, midnight Jan 1, 1 AD, that we can call Time 
0. The year from Time 0 to Time 1 is the first "positive year" and 
the year from Time -1 to Time 0 is the first "negative year." 

These years are "read" in opposite directions: it's the _end_ of 1 AD 
that is actually one year after 0, and it's the _beginning_ of 1 BC 
that is actually one year before 0. Year names are not coordinates of 
moments of time (we don't talk about date 1999 years, 11 months, and 
30 days) but labels (the 30th day of the eleventh month, called 
November, of year 1999) for intervals of time. Since the first century 
covers the time from Time 0 to Time 100, it includes all of the years 
1-100 in this scheme.

This scheme doesn't allow for negatives; we don't reverse the order of 
the months in BC years, calling November "negative February." Yet we 
do reverse the significance of the year's name; so dates don't work as 
neatly as numbers - there's no way they could and still make sense. 
The 0 doesn't divide "negative time" from "positive time" in any real 
sense (seasons don't reverse when you go negative).

Suppose we did label years to include a Year Zero:

                            |
                     <---BC | AD--->
        5   4   3   2   1   0   1   2   3   4   5    year names
    <-+---+---+---+---+---+---+---+---+---+---+--->
        |   |   |   |   |   |   |   |   |   |   |
       -5  -4  -3  -2  -1   0   1   2   3   4   5    time in years

Then our "time 0" would be, say, July 1 in the Year Zero, in order to 
keep things symmetrical, and the Year Zero would be the one-year 
interval including this defining moment, lasting from time -1/2 to 
time 1/2. But that seems awfully unnatural, doesn't it? And as I said, 
it doesn't allow the centuries to have more reasonable names anyway. 
All we would gain would be the ability to subtract year numbers. The 
first century, from Time 0 to Time 100, would cover the years 1-99, 
plus half of the years 0 and 100. Seen this way, years like 0, 1000, 
and 2000 would straddle two millennia.

But I don't think any of this really matters. I say 2000 is the start 
of the next millennium for all practical purposes; we just have to be 
careful to call it "the 2000's" rather than "the 21st century" or "the 
second millennium." Similarly, 1900 was the start of the 1900's, 
though 1901 technically started the 20th century.

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