A TV game show

I recently saw a little video clip where a TV game show asked the question “On what calendar date did the 20th century start?”. All three contestants felt so sure about their answers that they bet all their game points on it. All three were wrong. Moreover, all three had the same incorrect answer.

Why is this difficult? Numbering (or at least numbering in the Western World) got off to a bad start. We're taught incorrectly at an early age. Attempts at doing it correctly are misunderstood. Or met with belligerent resistance. Let's examine one small part of the situation, namely the surprising elements of the question from the TV game show.

A year (again, in the Western World) starts with January 1. So the 20th century started on January 1, but of what year? If you grew up in an English-speaking country (or when you were learning English), you probably remember as a child learning that, say, the 16th century and the 1500s referred to the same thing. This clash between the “16” and “15” comes as a surprise. Oh, if that child in each of us had only objected more vehimently, saying, “Look, this is nonsense! Surely, numbers and how we use them weren't meant to be this complicated!”. Because of this complication, we learn that you have to subtract off 1 from “16” to get the “15” in “1500s”.

If you remembered this extra cognitive burden about historical time lines, then you would realize that the answer to the game-show question is not the “obvious” January 1, 2000. And then, perhaps you would have, just like the TV show contestants did, placed all your game points on the answer January 1, 1900.

That kind of sounds tricky enough that a game show might consider the question. But the answer is even trickier than this. A long time ago, someone (in the Christian part of the Western World) decided to number years in relation to the birth of Jesus Christ. Never mind that they likely didn't get the year right (according to Biblical scholars, King Herod, who played a role in the birth story of Jesus, died a few years earlier), what's interesting here is that the first such year was named “AD 1”, where AD is an abbreviation for tha Latin “Anno Domini”, that is “in the year of the Lord”. So, with this numbering scheme, the first hundred years are named 1, 2, 3, etc. up to, and including, 100. Let me say 1 through 100 to refer to this inclusive range of years.

In other words, the first century comprises the years 1 through 100. The second century comprises the years 101 through 200. The third century comprises the years 201 through 300. If you continue this, you'll find that the twentieth century comprises the years 1901 through 2000. So, that gives us the correct answer to the TV show's question: January 1, 1901.

Math is hard beautiful

As 2-year-olds and 6-year-olds, we humans all questioned things. But if your propensity to question things has now waned, then your reaction right now may be “oh, that's tricky, I'll remember that piece of trivia for the next time I'm hanging out at the coffee shop or shooting the breeze with some friends at dinner”. Or, if (perhaps because of this cognitive burden placed on you from a young age) you've felt that math wasn't your thing, then your reaction right now may be “math is hard, my head is spinning, and I don't actually care about math”. But if you, like me, think that math is beautiful and describes things in real life, then the TV game-show question should have you up in arms. It's not math that's the problem. The problem must be that numbers aren't applied in a good way.

Decades

If you're not yet convinced there's anything wrong, let me give a follow-up question to the one from the TV game show: On what calendar date did the one hundred and ninetyfirst decade begin?

If answering this question were completely straightforward, then you'd think: Alright, 191^st decade. A decade is 10 years. So, the answer ought to be January 1, 1910.

Or, maybe you now remember that years start at 1, so perhaps you should add 1 to answer and make it January 1, 1911.

We have become accustomed to the “16th century” vs “1500s” issue with regard to centuries. But we don't often speak of decades in the same kind of way. If you do work it out, you'll notice that the first decade comprises the years 1 through 10. The second decade comprises the years 11 through 20. Etc. After a while, you'll find that the one hundred and ninetyfirst decade comprises the years 1901 through 1910. So, the answer to my follow-up question is January 1, 1901.

In other words, the 20th century starts on the very same day as the 191^st decade. WTF! Surely, this can't be normal!

How to solve this problem

Let's take a moment to consider how to calculate answers to questions like the ones above.

If it's 11 o'clock in the morning and you want to figure out how many minutes you've been awake, you can't just multiply 11 (o'clock) by the number of minutes per hour. No, if you're going to multiply by 60, you must first work out how many hours you've been awake. So, you need to subtract off the hour you woke up. Suppose you woke up at 7 o'clock. By 11 - 7, we get 4, which we multiply 60 to get the answer 240.

Likewise, if we want to multiply by the number of years in a decade, we first need to know how many decades we're talking about. When we hear something like “191^st decade” in common language, we thus first need to subtract off the starting point, namely the “1^st decade”. By 191 - 1, we get 190, which we multiply by 10 to get “1900 years”. This is a duration measured in years, but it isn't yet the answer to the question of “what year?”.

If it's been 3 days since you last had coffee and you want to know what weekday it is, then you can't just answer “3”, which isn't even the name of a day. No, you need to “add” this “3” to the name of the day at the start of that 3-day period. For example, if you last had coffee on Monday, then apparently today is “Monday + 3” (that is, “Thursday”). My point is not to define addition on weekdays. My point is that we need to add the duration to the starting point.

If we add a duration of 1900 years to the starting point “AD 1”, we get “AD 1901”, which is the answer we're looking for.

In summary, the formula that gives us the answer is

(decadeOfInterest - startingDecade) * yearsPerDecade + startingYear

In our case:

(191 - 1) * 10 + 1     =     1901

Applied to centuries, we have

(centuryOfInterest - startingCentury) * yearsPerCentury + startingYear

or specifically

(20 - 1) * 100 + 1     =     1901

Simplifying the math

The complicated parts of the formulas we just saw are the subtracting and adding of starting points. These come about in two different ways. The startingDecade and startingCentury are 1, because in common language we start with “1^st”. And the startingYear is 1, because someone a long time ago decided that the first “year of the Lord” would be denoted by “AD 1”.

If we have to do calculations like this a lot, then it would be nice to have starting points that are easier to work with. It is especially easy to add or subtract 0. Let's consider using 0 instead of 1 as those two starting points. I'll start with the less tendentious of the two.

Anno Domini, Retractavit

It is unfortunate that years (in the Western World, and certainly influenced by Christians) start with “AD 1”, for several reasons.

One reason is that the year immediately preceding “AD 1” is “1 BC” (for “Before Christ”). Or, to use the secular terms for these, the year immediately preceding “CE 1” (for “Common Era”) is “1 BCE” (for “Before Common Era”). That is, there is no year 0. This adds another wrinkle to our calculations. For example, if a distant relative of yours was born in the year 20 BCE, then how many years ago is that? A straightforward subtraction would be to take 2024 minus -20, which gives a duration of 2044 years. But subtraction expects there to be a 0, and since there isn't, we have counted one year too many. So, the actual answer is 2043. (Okay, there are other complications, too, like the switch from the Julian calendar to the Gregorian calendar. But if we're just counting years, then you can argue that that calendar switch was just a corrective action.)

We know from his song that Prince considered partying in 1999. If you were feeling festive at that time and wanted to focus your partying energy on the big turn of the millenium, then would you party (with Prince) on New Year's Eve of 1999 (when there was still a whole year left of the second millenium) or would you party on December 31, 2000? Good math shouldn't stop you from having a good time.

Let me, rather lightheartedly, suggest a simple fix.

It doesn't sound very appealing to rename each of the last 2000 years. But if you (or your parents) feel bad for having had the big party in the wrong year, here's a mental exercise that might ease your mind: let's rename the year “1 BCE” “CE 0”, or perhaps “RCE 0” (for “Revised Common Era”). So, instead of the timeline

..., 3 BCE,  2 BCE,  1 BCE,  CE 1,  CE 2,  CE 3, ...

we'll have

..., 3 BRCE, 2 BRCE, RCE 0, RCE 1, RCE 2, RCE 3, ...

In this revised numbering, there's still a missing year (this time between 2 BRCE and RCE 0). But it has some advantages: One advantage is that there is just one difference between the BCE/CE years and the corresponding BRCE/RCE years, namely that “1 BCE” is now “RCE 0”. Thus, when you read a history book that talks about a particular year (using the BCE/CE numbering), it's only this one year that you'll have to transpose in your head. A second advantage is that your partying is now synchronized with Prince and the rest of the world, since New Year's Eve RCE 1999 (which coincides with the day when Prince partied) was indeed the day that ended the second (revised) millenium. A third advantage is that, with this revised naming scheme, the answer to the TV show question “On what calendar date did the (revised) 20th century start?”, the answer is January 1, RCE 1900, so the three contestants would all have doubled their game points.

Ordering names

Rebranding years is not a serious proposal, since it doesn't have too much influence on everyday math. But ordering words like “twentieth” and “one hundred and ninetyfirst” do. This I find most regrettable.

Suppose you have been given three pieces of chocolate that you have placed in a row on the table in front of you. In describing each one, you'd like to give them names. Numbers will do. From what we have learnt above, we should not make the mistake of skipping 0, so we'll refer to the chocolate as piece 0, piece 1, and piece 2.

As much as this makes perfect sense, I find the proposal of such a numbering to be met by objections. Apparently, this simple idea is so contrary to world beliefs that it is met by closed minds, almost like the kind of dead-end conversations you'd steer into if you'd try to discuss climate change with a conservative or explain to an intellectual that the Earth is flat. In my analysis of the situation, the major mind-blocker seems to be that ordering words like “first” and “second” (abbreviated “1^st” and “2^nd”) are, in the minds of so many people, associated with the numbers 1 and 2. This is a really hard association to break, but let's try.

An online search of the definitions of “first” from Oxford Languages gives

first
   number
1. coming before all others in time or order; earliest; 1st.
2. foremost in position, rank, or importance.

Not only does this show the abbreviation “1st”, but it also numbers the two definitions using “1” and “2”. The definitions themselves, however, are just saying that “first” means there's nothing preceding it. There is something preceding 1, namely 0, so the definition is also a bit contradictory.

Note that the word for the ordering degree “first” itself has no common part with the word for the number “one”. Indeed, this is the same in many languages:

So, the abbreviation of “first” as “1st” is regrettable. Furthermore, the English words “second” and “two” also have nothing in common, and the same is true in many languages (German is an exception, where “zweite” and “zwei” look similar). The abbreviation of “second” as “2nd” is thus equally regrettable.

A language can have some special words like “first” and “second”, since these are used often. But as we get to higher ordering degrees, it is unreasonable to expect a brand new word for each, because that would be too difficult to learn and remember. So, sooner or later, we'll want to use ordering-degree words that are based on number words. I'll refer to those words in common usage today as legacy ordinals. That is, the legacy ordinals are the words that follow “first” and “second” in English. The choice of legacy ordinals in English (and similarly in many other languages) was unfortunate, because they were misaligned with numbers. Case in point:

Let's consider how we might construct words if we were given the chance to rewrite history. This is a delicate thought experiment, because it is easy to get confused with the legacy ordinals. It would best if we could just forget the legacy ordinals, but that's too difficult, so let's not just repurpose the legacy ordinals themselves. Instead, let's do something different.

To denote ordering degrees, let's introduce the notation 0°, 1°, 2°, etc. (Portugese uses a notation like 1°, but for legacy ordinals.) We also need a pronounciation for these that avoids confusion with the legacy ordinals. Let's not use the “-th” ending of legacy ordinals. Instead, how about pronouncing ° as “-ik”? Then, we have

zero-ik, one-ik, two-ik, three-ik, four-ik, five-ik, six-ik, seven-ik, eight-ik, nine-ik, ten-ik, …

Now, when we talk about our chocolate, we'll say the 0° piece (“zero-ik piece”), 1° piece, and 2° piece. We'll also talk about the 19° century and the 190° decade.

Another way to stay clear of legacy ordinals is to think of the numbers as being the names of the things we're describing. I already did this above when I mentioned “piece 0”, “piece 1”, and “piece 2”. Using such names, we would say “century 19” and “decade 190”. Such names make sentence structure slightly more awkward, but they do make the right connection with numbers.

Note that the words “first” and “second”, which are not legacy ordinals, can still be used. As usual, “first” still means there's nothing preceding it—that is, 0°—and “second” still refers to the one that follows the first—that is, 1°.

Concluding remarks

I have argued that numbering makes better sense if we embrace 0, rather than skipping it and always having that ugly offset-of-1 that we constantly have to add or subtract in our heads. But in making this argument, I too have been under a bias.

The strangeness of the legacy ordinals was revealed because I—and the TV show—asked about the first day of the 20th century. Had I instead asked about the last day of the 20th century, the answer is December 31, CE 2000 (or RCE 1999). This is also the last day of the 200th decade! When considering the last day, it is the ordering degrees that are strange: the 19° century ends on the very same day as the 199° decade (namely December 31, CE 2000).

So, the legacy ordinals actually seem perfectly reasonable, provided we prefer to associate names with endings. If you find that you prefer to celebrate your, say, 23rd birthday on the last day that you are 23 years old, then the legacy ordinals are for you.

For the rest of us, we'd be better off starting with 0 and using ordering degrees based on that numbering.

Acknowledgements

A different way to retrofit AD years to make them include AD 0 is to split the year currently known as AD 1 into two, shorter years, AD' 0 and AD' 1. This numbering, suggested to me by Nick Klose, seems reasonable, since it renames even fewer days than the Revised Common Era numbering I mentioned above.

Denis Carnier suggested the “-ik” pronounciation of the ordering degrees n°.