## How we count years, decades and centuries

I’m writing this on 1st January 2020 (2020–01–01). It’s a new year, but is it a new decade? Some people object to that claim, insisting that the previous decade does not end until 2020–12–31 and, therefore, the new decade does not begin until 2021. Are they right?

This blog post was inspired by one such discussion (with Marco Heimeshoff and J B Rainsberger), because — in addition to being fun! — it helped reveal some of the assumptions that surround calendars. I recently wrote that

Date–time handling is harder than people expect — even when they’re expecting it. It is a truth universally acknowledged that a single programmer in possession of the belief they understand dates and times, must be in want of a code review.

If you think you know everything you need to know about date–time handling, you are almost certainly wrong. If you think you don’t, you are almost certainly right.

Unsurprisingly, this topic peaks every 10 years (normally around the divisible-by-10 mark), with an Everest-like peak in the run up to the year 2000 — new decade, new century, new millennium!

The question of when the decade ends is intimately bound to the question of when the decade begins. Likewise for the century and the millennium. If the last decade began in 2010 then, yes, the current decade kicks off ten years after that in 2020. If, however, the decade began in 2011, we have another year to go.

If you go back and reread the last paragraph you may notice extensive use of the word the. That’s the first assumption: decades have a single and unambiguous way of being referenced. Decades (and centuries), however, have different start — and, therefore, end — points depending on how they are referred to. When talking about a child’s first decade, for example, the referent is their first ten years of life, not the calendar decade they were born in.

When talking about centuries, the 1900s span 1900–01–01 to 1999–12–31, because those dates fall in years that match the pattern 19??. Whether the twentieth century is equivalent to the 1900s or is considered to have run from 1901 to 2000 is a separate question. Sometimes a century is a specific hundred-year interval with respect to an epoch or frame of reference; sometimes it is simply a hundred years.

Returning to decades, in English we most often refer to decades by terms such as the 1980s or the eighties. This referencing convention is a zero-based system, i.e., you start counting from 0, which means that each cycle of 10 begins with a 0 and ends with a 9 in the last digit; that digit is also equivalent to the years completed since the cycle began. Using that system of reference, the last day of the eighties was 1989–12–31. Any claim that 1990 was in the 1980s but 1980 was not can be considered a solecism. The position that the ninth decade of the twentieth century is (or is not) equivalent to the 1980s is, however, different and depends on our definition of the twentieth century.

So, yes, “Happy new decade!” is perfectly fine — welcome to the 2020s! If someone specifically claims that the twenties do not begin until 2021, however, neither language nor numbers lend them much support.

We are still left with the question of whether or not we might also consider ourselves to be at the start of the third decade of the twenty-first century. This is a question of beginnings… beginning with what calendar we are using and whether we begin counting from 0 or 1. A common assumption is that there is only one calendar. And the corollary assumption is that it starts counting from 1.

There are many calendars and many starting points to choose from. In practice, though, given that I’m writing this in English and have already said “I’m writing this on 1st January 2020”, the Gregorian calendar is a safe bet. This is the calendar of current affairs, of business and, for many people, it is simply the calendar.

So when did the Gregorian calendar begin? It began in 1582 (1582–10–15, to be precise). 1581, 1580, 1579, etc., are not years in the Gregorian calendar. Strictly speaking, therefore, the Gregorian calendar has neither a year 0 nor a year 1. We are currently living in the fifth century of the Gregorian calendar, a century that began in the 1980s.

This might not quite have been the new year resolution you were expecting.

That said, although there is no 1581 in the Gregorian calendar, there is a 1581 in the Julian calendar — the calendar system the Gregorian calendar was introduced to fix with a leap-year calculation that better aligned calendar years with solar years — and in proleptic Gregorian calendars — the projection of the Gregorian calendar back in time before its origin.

If we accept the Gregorian calendar as a bug-fix release to the Julian calendar, i.e., we take the AD/CE system as a whole and in all its flavours, then by definition and consensus the first year of the Common Era is numbered as 1 and there is no year 0.

So that means, in the usual AD/CE system, the first century ran from 1 to 100. And the twentieth century ran from 1901 to 2000. And the third decade of the twenty-first century runs from 2021 to 2030. Anyone who wants to claim the decade starts next year is — like anyone claiming the decade started today — not wrong in doing so.

It is worth keeping in mind that when we refer to the Julian calendar, we are not referring to the exact calendar system introduced by Julius Caesar in 46 BCE. The Julian reform realigned the start of the year, reconfigured the months and introduced an intercalary day at the end of February every four years. The year labelling, however, remained a ragbag, i.e., the year of an emperor’s reign, the year since Rome was founded or the year in which consuls took office.

The year numbering we are familiar with didn’t come along until the sixth century. It didn’t start being widely used in Europe for another three centuries. Zero itself didn’t arrive in Europe until the twelfth century. Although zero makes calculations easier and is implied in how we count a person’s age — a child turn’s one at the end of their first year — it was not an explicit concept when Dionysius Exiguus devised anno Domini nostri Jesu Christi numbering. A year 0 would have been a conceptual impossibility. The year number is also an ordinal rather than a cardinal number: it counts not “completed years” (i.e., age or quantity) but “year of” (e.g., first year).

Are we done? Not quite. Exploring dates, times and calendars means never having to say you’re done, but let’s close with just two more observations.

The first observation is that these days not only do we have zero, but it is integrated and retconned into everything we do. That someone invented a non-zero-based counting scheme 1500 years ago to account for an epoch event over 500 years before that matters far less than the conventions, preferences and biases that we have around numbers.

We like things that match. We see greater significance in 1999 and 2000 than in 1998 or 2001. After two millennia, did it matter that the third millennium was ‘supposed’ to begin in 2001 rather than 2000? Not at all. The arrival of the year 2000 was greeted with more celebration than 2001. Culture, society and cognitive bias eat arithmetic for breakfast. From a more numerate perspective, the difference of one year is not particularly significant given the margin of error in the start of the epoch: historians reckon Jesus of Nazareth was born 2024±1 years ago, not 2019 (and not 2020).

Including zero in the number line makes everything from programming to accounting to astronomy easier and less error prone. Astronomical year numbering, for example, has a year 0: it is Gregorian from 1582, but Julian before that, and includes a year 0.

The second closing observation is that, reading this, you are currently reliant on a calendar that has a year 0. This is true of anyone living fully in the digital age. You thought you were using the Gregorian calendar, but in practice you are using ISO 8601. This is the standard of the web and officially, by country, most of the world’s population. ISO 8601 is an international standard for date and time representation. And it has a year 0.