The arrival of 2020 has brought to mind the various controversies at the start of the year 2000, also called Y2K. As a software engineer responsible for date-sensitive communications within large computer systems, I well recall being on call that Saturday, in case something went wrong. I also recall all the questions we got in 1999 about the coming year and the new millennium – or was it? Today we hear the question, is 2020 the start of a new decade? Let’s look at some of those questions, for auld lang syne …

## The Y2K problem

First, a 1998 question about the coming software apocalypse:

Year 2000 (Y2K) Problem Can you explain the Year 2000 problem in layman's terms? I keep hearing about code that has to be rewritten and how it's just not possible in the time left. I don't understand why it just can't be rewritten.

Doctor Jeremiah summarized the problem:

This is a very good question. The short answer is "It can be rewritten" but of course then you realize it will take more than two years. Here is why: When you write down a year like 12/23/34 you really mean 12/23/1934. Notice the two extra digits. For the last 40 years all the dates stored in computers have been missing those two digits. Now that the year 2000 is coming, we have two problems: 1)The data in the computers will be wrong.Say you were born on 12/23/34. After the year 2000 the computers will probably think that means 2034 when it really means 1934. So all the data have to be fixed. 2) All the software that people use for entering the data and doing calculations on the data tries to use dates with two-digit years. Once all the data are fixed we can't continue to pretend dates have two-digit years. Soall the software has to be fixed. There are a lot of data and people have been making software that uses two-digit years for 40 years. These two things put together will take a very long time to fix. The problem is that we only have two years left. If we had fifteen then we would be okay. If we had 5 times as many computer professionals we would perhaps be able to do it in two years. But there just isn't enough time or enough people.

All this because the first computers and programs were developed by people with 50 or even 90 years of assuming “’60” meant “1960”, so they (no, “we”) weren’t planning ahead to when it no longer would!

Pretend that it is 12/31/99 and the computers think that means 12/31/1999. Now, when the date changes to 01/01/00 the computers will think that this means 01/01/1900. Say your bank is giving you 3 percent interest annually and you have $1000 in the bank. The interest is 8.1 cents per day. But going from 12/31/1999 to 01/01/1900 is -100 years (-36500 days instead on just +1 day). If they added the interest for this many days they would have to subtract $2956 from your account and then mail you a letter saying that you owed them $1956. So you can see that it is a big problem. It can't be fixed fast enough and the consequences are really difficult.

But nothing big happened, in part because of all the work we put in to make sure everything was either fixed or didn’t matter. And we were all relieved. For details, see Wikipedia.

## New millennium? The Y0 problem

But most questions were about whether to call 2000 the start of a new century or millennium. Here is the first such question in our archive, at the start of 1999 (there had been a few others that didn’t make it in):

Millennium and the Year 0 I am doing a science project on when the new millennium really begins. I already know thatit truly begins in the year 2001becausethere was no zero year between B.C. and A.D.I was wondering if you knew why or could give me some sites about why the early mathematicians didn't let a whole year pass before naming it one. I would also appreciate any other information you could give me about my project. I have already looked it up under many different subjects in Encarta 98 and tried tons of different ways to get information on the Web, but I'm having trouble finding information.

Katie understood that because there was no year 0, the first 1000 years A.D. were 1 through 1000, so the second millennium started in 1001, not 1000; and the third millennium started in 2001. (We’ll see some other opinions on that!) Her question was, **why was there no year 0?**

Doctor Rick answered that question:

Thanks for asking this question, it's a good one. To answer it, you will need to look at thehistory of the calendar, and thehistory of math. BC dates were not used in the BC era. BC means "before Christ"; that is, before the birth of Christ. Before he was born, people (even the relatively few people who were expecting his birth) did not know just when it would be, so they couldn't date their calendars that way! As a matter of fact, AD and BC were not invented until around AD 525, by Dionysius Exiguus. In AD 525, 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 separately. Nobody drew a number line with positive numbers on one side, negative numbers on the other, and 0 in between.

So positive, negative, and zero were not thought of as three parts of one number line, but as three separate cases: before, at, and after. There was no thought of using a single quantity that could mean any of those.

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 (timeline). 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 other kinds of problems (like quadratic equations, if you've seen them) - as a set of special cases.

So there was no year 0 because there was no 0 (yet).

For a similar answer later in the year, see

First Day of the 21st Century

## But what if there *was* a Y0?

Here’s another question, near the end of 1999:

The Second Millennium I have heard debates aboutwhether the second millennium actually starts on the year 2000or 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?

I answered this time:

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.

This just recaps what Doctor Rick said. But I’d thought a little further:

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 thatordinal 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 99 the "zeroth century." Yet no one questions that the second millennium starts after, not in, the 20th century.

By focusing on centuries or millennia alone, we fail to see that “fixing” one layer does not “fix” the next. If we make the years include a Year 0, we are still probably not going to call anything “Century 0”, or the “Zeroth Millennium”. If we count starting at “1” or “first”, things called “0” are not going to fit.

To look at this another way, the integers really represent onlypointson a number line, notintervals. "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 callTime 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.

In other words, the names we give to years (or centuries or millennia) don’t behave like numbers, so it doesn’t really make sense to treat them as such. This is even more so when we look *inside* a year:

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).

Negative mixed numbers like \(-3\frac{1}{4}\), as we’ve seen recently, are treated like \(-\left(3+\frac{1}{4}\right) = -3-\frac{1}{4}\); both the whole part and the fraction are negated. But “negative mixed dates” like April 1, 3 B.C. don’t work that way – the month is always measured positively, going forward in time.

I had more to say, but you get the point. I closed,

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.

It’s a matter of words, not of math.

Today, I see similar questions about whether 2020 is the start of a decade. To me, the answer to this one is simple: If you are asked when the **203rd decade** starts, that’s 2021. But if it’s about the **2020’s**, they definitely start in 2020! (What would you call a decade that starts in 2021 and includes 2030?) *Every* year is the start of *a* new decade; it’s what you *call* that decade that counts.

## One more view

In the first month of Y2K, we had a question from a public radio host:

The Third Millennium Hello Dr. Math, My name is Bruce Hutchison. I am the anthropologist co-host of a new, nationally syndicated public radio program called "The Big Picture" with Bruce and Melissa. Anthropologist Melissa Farncomb and I look at why people do the things they do, across time and cultures (usually in a lighthearted way)! We're recording this week's show on The Millennium and Prophesies. We have a question for you for our show, to which we'd be delighted to give you credit: When is the "true" millennium, and why?

Doctor Ken answered:

Hi Bruce, Glad to help out, I'm a public radio fan myself. My name is Ken Williams, and though I can't speak for all the Dr. Math volunteers (there are over 200 people who can answer Dr. Math questions these days), I'm the founder of the service, so sometimes I take on these kinds of spokesperson duties. Regarding the question about when the "true" millennium begins, I actually believethat's a better question for an anthropologist than a mathematician. The word "millennium" simply means any period of 1000 years, though it's natural for us humans to want to start some millennium at a known point in history and keep dividing the eons into consecutive millennia thereafter. Therefore, if we're going to talk about a "true" millennium, we should probably fix some important event in the past and count forward 1000 and 2000 years. Supposedly, we've done this with the birth of Christ. Seems simple enough - just count forward 2000 years from the nativity, and pencil in a millennium celebration on the calendar. Well, the problems with that are numerous. First, since our years are actually enumerated as "the 1999th year of our lord," it seems we actually started counting at the year 1, i.e. "the first year of our lord."That's essentially the argument of people who say the millennium begins next year.

(That is, in 2001.)

Did we actually start counting like this, from the year 1? Of course not - the present-day Gregorian calendar, which is reasonably accurate and reliable, wasn't adopted until the year 1582 AD (for English-speaking countries). Before that, people used the Julian calendar, which had been in use since about 4 A.D. and was recalibrated in about 527 A.D. to count years from the birth of Christ. Previously it counted from the founding of the Roman Empire. To perform this recalibration Roman scholars did the best they could, but modern scholars seem to think they were off by a few years. Furthermore, 10 or 11 days were actually deleted from the calendar in 1582 when we shifted to the Gregorian calendar - should we adjust for those too, and have the big party on Jan. 10, 2002? I think not. And finally, if Christ was actually born on Dec. 25, then _that_ year can be considered "the first year of our lord," i.e. the first year of the new millennium, and the following year (year 1) is the second. Which would imply that 2000 ushered in a new millennium.

This assumes the original Christmas was just *before* the Year 1. But since the years are something of an approximation anyway, who cares?

In short, since the historical/calendric situation is so messy, I believe thatwe should measure the millennium by noticing when the big party is, and Prince doesn't party like it's 2000. I think we're in the new millennium now.

He went on to answer another question they’d asked. And we can move on, too.

I’ll be following this up with a few topics on dates, starting with another 2020 issue: leap years.

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GlennI think 1 BC should become 0 BC/AD Jan 1 through Dec 24 would be 0 BC and Dec 25 through Dec 31 would be 0 AD and the new year would become 1 AD because it would be the 1st year of our lord and 0 BC would be like 2 months BC 7 months BC up to 11 months BC then 1 BC would be 1 year BC. 0 BC would belong to the 1st century BC and 0 AD would belong to the 1st century A.D.

0 BC/AD will be a special year because it is the only year that can be divided into 2 first 51 weeks being 0 BC and the last week 0 AD and the only time a new century and millennium would start on Christmas. 0 AD would only be a week but it still belongs to the new century as if it were a year of its own even though it’s not that’s why it would be 0 BC/AD. The year Christ is born is special unlike any year.

First decade AD would be the years 0 AD through 9 AD though 0 AD is only a week. So the 1st decade would only be 9 years and a week and the 1st decade BC would only be 9 years and 51 weeks just 1 week shy of 10. Then -1 can be the same year as 1 BC and 1 can be the same as 1 AD.

Dave PetersonHi, Glenn.

Ultimately, what you are saying is that we can call 1 BC the year zero (part BC and part AD), and then call 2 BC “1 BC”, and so on. This is actually done, in the astronomical calendar, except that it just uses numbers, so your 1 BC is just called -1. This makes year numbers work mathematically.

Your approach of including 0 as the first year of a decade or century in either direction is a nice adaptation, allowing the natural names of decades or centuries to make sense (the second century is the one-hundreds, and so on, from 100 to 199). It doesn’t fit terribly well with mathematical thinking (where including 0 in two different periods is unnatural), but it is a reasonable compromise with human language.

You are assuming that Jesus was born on Dec. 25, 1 BC, which is neither factually nor theoretically true, as I understand it. But since we don’t know the actual exact date, we can consider your ideas hypothetically.

Of course, you’re not going to get all historians to change their nomenclature, or change all the old books, so this isn’t going to go anywhere; but it’s an interesting thought experiment!