When should the new millennium begin?

Härmel Nestra
Eesti keeles

If the question were “When does the new millennium begin?” together with an adjustment of the system of chronology assumed by the inquirer, it would have a unique answer and greater discussion would therefore be senseless. In the Christian era for instance, it is defined that the second millennium and the 20th century finished with 31 December 2000 and the third millennium and the 21th century started with 1 January 2001. So neither the centuries change simultaneously with the hundreds digit nor the millenniums change simultaneously with the thousands digit of the year number. This is affirmed by many authoritative sources, among of them by the Royal Observatory of Greenwich.

This writing analyses whether this definition itself is good enough or would some other be substantially more reasonable. As we remember, there was an enormous celebrating by huge masses when year 1999 was replaced by 2000 and this turn of the year was continually considered the start of the new millennium. But when year 2000 was replaced by 2001 and the new millennium started officially, almost nothing specific happened. This fact on its own clearly justifies the question in the title.

Thereby, the question can be only about the moment of changing centuries and millenniums, not about their length. There must be precisely 100 consecutive years in any century and precisely 1000 consecutive years in any millennium, because the general conception of notions ‘century’ and ‘millennium’ specifies it so. It is also natural to require that the first century and millennium begin at the beginning of the era and every year belong entirely to precisely one century and millennium. These requirements have caused also the described incompatibility between the centuries and hundreds digits in the Christian era since the Christian era has been defined to begin at the beginning of year 1—year 1 AD is preceded by 1 BC and no year 0 exists. So the first century of the Christian era is made up by years 1–100, the second by years 101–200, the third by years 201–300 etc.; millenniums are made up by years 1–1000, 1001–2000 etc. The year numbers of the Christian era are ordinals indicating which in order year it is from the beginning of the era (to one or another direction).

But as told before, majority of people has preferred to consider the century and millennium turning a year earlier. It became obvious in the clearest way the last time, but also in past, people have perceived the last turn of the year before the turn of the century more significant than the turn of the century itself. This preferring some other definition to the official, being as widespread in countries with high educational level as in the others, indicates that there may be serious reasons for this trend and the authors of the official definition may themselves have missed their way. In the following, it will occur that there are several arguments indeed which support placing the turns of centuries and millenniums—and simultaneously the beginning of the era of course—a year earlier than they are set officially.

Firstly, it is widespread nowadays to group years into decades with designation such as “nineteen sixties”, “nineteen seventies”, “nineteen nineties”. By these, the years 1960–1969, 1970–1979 and 1990–1999, respectively, are meant. In case of this grouping, digits from left up to the tens digit are the same precisely within the bounds of one decade. Any different grouping (e.g. 1971–1980) is exceptional. But if a decade is defined as a collection of years with the same number of tens, then why the centuries and millenniums should be defined in some different in principle manner? What can be more natural than taking 10 consecutive decades together to form one larger group and getting therefore 1900–1999 to be the century? If one takes 1901–2000 for century, then the bounds of centuries do not respect the bounds of the decades, which leads to inconveniences.

Secondly, it is widespread nowadays that on screens where numbers running are measuring the motion of time, every number indicates how many total units being measured by this number there are, not which in order unit is running. For this reason, the account begins with 0 units of course. Among the rest, such system is used in observing the clock time that is essentially an adjustment of calendrical timekeeping. At the beginning of a new day, the hour, minute and second readings are all 0, the latter becomes 1 only when the first second has passed. When it has been 1 for the entire second second, it becomes 2 precisely when the third second begins etc. The hour and minute readings behave the same way: the minute reading becomes 1 precisely when 1 minute has passed from the beginning of the day, 2 when 2 minutes have passed etc. Therefore it holds that the 20th hour from the beginning of the day ends precisely when the digits 20.00.00 appear to the screen. It would be fully natural by analogy if also the year number indicated the total years passed from the beginning of the era and the 20th century ended precisely at the moment when the digits 2000 appear to the year number. The year before year 1 with which the era begins would naturally be denoted by 0.

These two arguments are grounded on the natural carrying some habits of everyday life over to the case of year numbers in which they turn out to be violated. The next third argument comes from science. In astronomy, denoting years with integers 0, –1, –2 etc. has been used for several hundreds of years already. The correspondence with the Christian era is set so that all the years AD are denoted by the same number as in the Christian era, but 0 is written instead of 1 BC, –1 instead of 2 BC, –2 instead of 3 BC etc., generally n+1 BC is called –n in the astronomical system. This system of chronology can often be found in the literature pertaining to astronomy. It has won more and more countenance and is now also the ISO standard. Moreover, astronomers use fractional year numbers in order to express any time instant uniformly with the amount of time measured in years having passed from some fixed start moment. But for the start moment, the beginning of year 0 has been choosed, i.e. earlier by one year than the beginning of the Christian era. Why should people prefer the Christian era if even the scientists use others?

The three arguments presented so far all strive for retention of some traditions and extending them to analogous situations. Such an extension is strongly desirable in itself since it creates a more uniform and comprehensible system. Thus it can't be said that abandoning the Christian era means ignoring the historical traditions and is therefore condemnable. It violates one tradition but supports three whereby these three are much more modern, much more established themselves in the minds of people and hence graver than the one which it violates. The tradition of the Christian system of chronology comes from year 532 and was brought into being by a so primitive society which did not know about zero.

But these three traditions—grouping, measuring and astronomical—are not only customs been formed spontaneously during the windings and turnings of history. It turns out that they all have a rational connection with one fundamental tradition—our positional number system.

I used the word ‘our’ in the sense that also a different one would be possible in principle. I don't mean here that a positional number system can have some other radix instead of ten. All positional number systems which one can find in number theory books—decimal, as well as binary and all the others—are actually of the same kind and are grounded on the addition of digit multiples of powers of the radix. But different in principle positional number systems being grounded on ordinals would be possible. Since I don't know better words, I call these positional number systems ordinal and the traditional ones additive.

Ordinal positional number system has practically vanished by today, but in olden times, it was used in natural languages. Traces of it have persisted in some languages yet nowadays. Ordinal decimal number system is built up according to the principle that in the representation of a number, digits must show which in order unit of grade is running. Sorting the higher place values to the left, eleven is 21 in this system (one of the second ten), twelve is 22 (two of the second ten), thirteen is 23 (three of the second ten); twenty-one is 31 (one of the third ten), twenty-two is 32, thirty-one is 41. For example about larger numbers, a hundred and one is 211 (one of the first ten of the second hundred), a hundred and two is 212, a hundred and eleven is 221, a hundred and twelve is 222 etc. Only one question arises: how the last, tenth unit of every grade should be denoted? Choose 0 for this. Then a hundred and ninety-one is 201, a hundred and ninety-nine is 209, while two hundred is 200 as usual (ten of the tenth ten of the second hundred). Ninety-nine is 09 in the ordinal system and hundred is 00 or 100 because in the ordinal system, one may add leading 1-s to any number without changing its value, like one may add leading zeros to any number represented in traditional number system.

When the rules of this system have become clear, write year numbers in it. The year number 1999 would be written as 2009 in this system, the number 2000 is 2000 as usual, and the number 2001 aquires the form 3111. Observe that the four digits of year number would change simultaneously with the moment when year two thousand ends and year two thousand and one begins.

The ordinal system can be used also for writing fractional numbers: for instance, 17.5 is 28.5 ordinally (half of the eighth one of the second ten). In general, the ordinal representation of any non-negative real number can be deduced from its traditional representation very simply by adding 1 to every digit except the last non-zero digit and the trailing zeros (thereby 9+1=0 of course). If we added 1 also to the last non-zero digit and the trailing zeros, we should append the infinitely repeating 1.

Hence the general principle is the following. In the representation of a number in the traditional additive number system, each digit denotes how many total units of grade there are, but in the ordinal system, it denotes which in order unit of grade is running. Only the last non-zero digit and the trailing zeros can be exceptions. These are the same in both additive and ordinal system and their meaning depends on the concrete use of the number represented.

It follows from this principle that, independently on how many digits the additive decimal representation of a number contains (including these after the decimal point), the following condition is satisfied: before a unit becomes total, the digits with less place value are nines and the unit becomes total when these turn to zeros simultaneously. Completely analogously, it holds for ordinal decimal system that, independently on how many digits the number contains (including these after the decimal point), the digits with less place value are 0-s before a unit becomes total and the latter happens when these turn to 1-s simultaneously.

Also centuries are used as time units, whereby it fully harmonizes by its length with the decimal number system. Thus it can be claimed that not the simple matter that the hundreds digit changes simultaneously with coming of year 2000 has insomuch influenced the masses' convictions about turns of centuries, as far as the specificity of the number system in use has. It makes sense to believe that, if the ordinal number system had won in the history, then masses would have celebrated the arrival of the new millennium at the end of year two thousand even if it were defined to happen a year earlier officially.

This implies also why the decades are defined namely in the way described above. Moreover, grouping the years with the same number of tens to one decade is the most convenient way to handle decades. But in the case of traditional positional number system, this gives just the result considered above.

The traditional manner of handling clock time is in principle itself an additive positional number system. This is not the decimal system but mixed system—there are 60 possible readings of both seconds and minutes while 24 possible readings of hours—but the maxim and kind of positional number system are the same. Thus namely such a manner of handling clock time harmonizes with the traditional way of representing numbers. The alternative manner via ordinals which was used in olden times would create a situation where both additive and ordinal positional number system would be used within one and the same reading. Note that such a situation actually occurs in writing dates with numbers: each number separately is represented in additive system, but the whole date (e.g. 07-26) is an ordinal representation because each component denotes an ordinal number.

At last, also the account of years of astronomy can be motivated by the use of the traditional positional number system. It is simple and convenient if the part before decimal point of the number expressing time instants persists equal to the year number for whole year instead of remaining behind by 1. But this is achieved by declaring the beginning of year 0 to be the starting point. In case of ordinal number system, the same goal would be reached if one choosed the beginning of year 1 for starting point.

Also a direct rational argument supports denoting years by 0, –1, –2 etc. In treating history, it is often necessary to compute how many years there were before some two events. If both events happened after the beginning of the era or both before, then the time remaining between them is found naturally by subtracting the smaller year number from the greater. But if the earlier event took place before the beginning of the era and the other after while the chronology does not use year 0, then the natural arithmetic operation is not sufficient: also 1 must be subtracted. Hence a thing simple in its nature has become involved (subtraction of 1 tends often to be forgotten because this is not needed in analogous circumstances). Among the rest, this situation arises always when one is computing the number of years passed from an event of antiquity by today. Thus it would be strongly reasonable if the chronology contained also year 0.

But then, year numbers would not be interpretable as ordinals any more and so the reason for locating the beginning of the era at the beginning of year 1 would disappear. And not only this: since zero can't be expressed as an integer in ordinal system (it is possible to write as a repeating decimal 1.(1)), the necessity of using zero supports the choice in favour of additive positional number system which, as we saw already, gives three different reasons to put the beginning of the era to the beginning of year 0. This leads to the summary that, whenever the era contains year 0, it is rational to position the beginning of the era namely at the beginning of year 0.

Studying the ordinal system more closely, it becomes clear that we are actually lucky of this system not being in everyday use. The mentioned absence of integral representation of zero is one strong disadvantage already. Moreover, even elementary arithmetic operations are considerably complicated to perform in ordinal system. Addition and subtraction are not very much harder than in the additive system, but the simplest way to multiply and divide numbers expressed in the ordinal system is to transform the numbers to our system, perform the operation and transform then the result back to ordinal system. Hence the ancient matematicians made the right choice in inventing the positional number system. The ordinal system is justified only in writing ordinal numbers.

In measuring a continuous quantity such as time, the additive and ordinal positional number system are equally usable in principle. Of course, the measuring can't be started from one instead of zero, but zero can be represented in ordinal system as 1.111111... which is not a bit worse than 0.000000... in the additive system. Therefore I don't agree with the standpoint sometimes expressed that the fact on its own that time is a continuous quantity supports denoting the time units starting with zero. This is a wrong impression caused by the habit to use additive system in measuring.

Now what can be said in favour of the Christian era? At first, the Christian era is a very long tradition. Also the other calendar units, such as days, months and weeks, are traditionally denoted by ordinal numbers, which supports by analogy the numbering of years of Christian system of chronology. Many languages interpret year numbers as ordinals. But these traditions don't have any rational motivations. Nothing explains in what respect denoting the calendar units by ordinal numbers is better in principle than denoting them by numbers measuring the total units passed from some starting moment. All the rational arguments support the astronomical system of chronology. The Christian era has been insulated nowadays, being divergent from other general trends.

This refutes also the standpoint widely used by conservatives when they understand that the question concerns the reasonability of the construction of the Christian system of chronology itself: when counting, it is natural to start labelling the elements with one, not zero. I agree entirely, but what has chronology to do with counting?

It can be claimed also that the symmetry of year numbers with respect to the beginning of the era speaks in favour of a system without zero, but under closer consideration, this doesn't seem to speak much in favour. There is no reason why the year numbers would have to stand symmetrically because even time itself runs asymmetrically: towards the beginning of the era before it and away from the beginning of the era after it.

Now what can be concluded from all this? As we saw, using ordinal numbers in chronology conflicts with the principles of the positional number system in use which are rationally motivated. This means that the conflict is inevitable, an enlightenment of the people does not help here. On the other hand, the conflict is not substantial to chronologies or even to measuring of continuous processes. The conflict is deeper and lies actually between the traditional additive number system and ordinal thinking. Take an arbitrary discrete set of 2000 elements for instance. Let these 2000 elements be numbered with ordinals 1, 2, ..., 2000. Grouping these elements to “centuries” according to the increasing order of the numbers, one gets the first century containing the elements 1, 2, ..., 100, the next century containing the elements 101, 102, ..., 200 etc., leading directly to the above conflict. When using the ordinal decimal system, the conflict does not occur:

  1  21  31  41  51  61  71  81  91  01 211 221 231 241 251 261 271 281 291 201
  2  22  32  42  52  62  72  82  92  02 212 222 232 242 252 262 272 282 292 202
  3  23  33  43  53  63  73  83  93  03 213 223 233 243 253 263 273 283 293 203
  4  24  34  44  54  64  74  84  94  04 214 224 234 244 254 264 274 284 294 204
  5  25  35  45  55  65  75  85  95  05 215 225 235 245 255 265 275 285 295 205  ........
  6  26  36  46  56  66  76  86  96  06 216 226 236 246 256 266 276 286 296 206
  7  27  37  47  57  67  77  87  97  07 217 227 237 247 257 267 277 287 297 207
  8  28  38  48  58  68  78  88  98  08 218 228 238 248 258 268 278 288 298 208
  9  29  39  49  59  69  79  89  99  09 219 229 239 249 259 269 279 289 299 209
  0  20  30  40  50  60  70  80  90  00 210 220 230 240 250 260 270 280 290 200

Whereas counting does not matter in chronology, endeavour to avoid this conflict in chronology is justified nevertheless.

It is justified even to say that the official system of chronology including the definition of centuries is in so many respects out of date and evidently unreasonable that it is almost axiomatic that young people who do not know the official definition of centuries adopt a different point of view. People's brains are soaked through with grouping years to decades and everyday use of standard representation of clock time. Therefore it is not to be wondered at the fact that most people did not guess by themselves that somebody could consider the turns of the centuries not coinciding with the turns of the hundreds digit. Additionally a part of people who familiarize themselves with the nuances of official system of chronology and understand when the centuries officially turn simply ignores the official definition as something very queer. It goes on in such a way now and will presumably proceed in future, too.

With regard to the possible changing the official definition better, I think the circumstances must be analysed widerly. In the calendar used in Christian world, the absence of year 0 is far not the only disadvantage. The Christian calendar has been strongly criticized from many aspects, many proposals of modification have been suggested during its history. For example, many projects have been suggested for tying concrete weekdays to the dates and also for dividing the year to months of the same length. Also some methods have been invented for making the calendar more precise by modifying the rule for leap days. The present leap day rule is thereby also unreasonably complicated (every 4 years a day is added, except every 100 years when the day is omitted, except every 400 years when it is added however), a much simpler rule (every 4 years add a day, except every 128 years) would give a far more precise calendar. Hence it is probably not thoughtful to take up changing the official system of chronology for adding year 0 only.

If however the changing of the calendar happens to become a common topic, then the new calendar enforced will prospectively be a fully modern thing which was not conceivable centuries ago. Adding the year 0 would seem to be a small detail against the background of the whole reform. There are a lot of other elements worth to change.

But till this, let everybody feel free to celebrate the turns of the centuries according to the system best according to one's view. Besides it, listen to the opinions of the famous and the pompous. For example, Bill Clinton said in his speech on 19 January 1999, “But I want all the folks at home listening to this to know that we need every state and local government, every business, large and small, to work with us to make sure that this Y2K computer bug will be remembered as the last headache of the 20th century, not the first crisis of the 21st.” . Boris Yeltsin mentioned within his resignation speech on 31 December 1999 that he does it “on the last day of the outgoing century”. But Arthur C. Clarke who bravely fighted in favour of celebrating the arrival of new millennium at the turn of year 2000 to 2001 compared our system of chronology with weighing machine whose scale begins with 1 instead of 0. “Would you be happy when your grocer claimed he'd sold you 10 kg of tea?" he questioned, showing metaphorically that the century does not end when year 2000 starts. A good example indeed. If some grocer started to weigh goods with a machine whose scale starts from 1 instead of 0, then this droll grocer would soon be without clients.

July 1999: the first draft
December 1999: cosmetic changes (e.g. “present” → “20th”)
July 2001: renovation and translation into English

A page of millennium texts.

nestra@math.ut.ee