# How do I divide a year into months?

In my story, I have an alien planet with a long and eccentric orbit that takes 1018 days. Now I'm trying to make a calendar out of it, dividing these 1018 days into shorter time units - months (inhabiting civilisation, of course, will name these units differently). But whether I divide year with a number of months to arrive at the number of days per month or with days per month to arrive at the number of months, or divide seasons into equal parts, I get decimals like 84.83 days per month with 12 month per year or 21.21 days per month with 48 months per year, or 15.42 months per year taking shortest of seasons, 66 days, as one month, 30.85 months per year when dividing this season in half.

How can I get a whole number of months with a minimal decimal for the number of days per month (which might be remedied with a leap year)?

Are there maybe some tools that may help with this?

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– L.Dutch
Commented Feb 16, 2023 at 3:00
• There's a bit more to a calendar than just math. Does this planet have a moon? If so, how many, and is this calendar a lunar-based calendar or a solar one? Seasons is additionally a whole other nut to crack, especially with an eccentric orbit. Are you asking that seasons be a functioning part of this calendar, instead of the common Gregorian calendar where the seasons are more a separate thing that is applied to the calendar and not a part of it directly, as in, does this calendar have a set number of days for each season similar to a month?
– INPU
Commented Feb 24, 2023 at 1:15
• @INPU, other comments, and answers, have indicated that historically lunar calendar has had importance for hunters. As these people are herbivores and developed agriculture, I think their calendar would rather not be lunar. Commented Feb 24, 2023 at 8:31

## Use Euclidian divisions...

First, quick maths recall: All integers can be divided with the Euclidian division. An Euclidian division is quite simple, you divide a dividend D (your big number) by a divisor d (your small number you divide with) as much as you can, and when you cannot anymore you have both the quotient q (your result) and a remainder r (the rest). Mathematically, this is commonly shown like this:

D = d x q + r

In calendar terms, D is the total number of days every year, d is your number of months, q is your number of days per months and r is the annoying days you'll need to add back once in a while.

All spreadsheet software (Excel, OpenOffice, Google Docs...) include functions to calculate the dividend, quotient and remainder. With this you can automate your calculations and test many values at once:

Above is an example using Excel, which took me about 10 minutes to set up. In Excel, the functions you'll use are QUOTIENT and MOD (modulo operation for the remainder). While there may be some differences between software, it should be quite similar across them.

Now to have the minimum remaining days every year, choose the lowest remainder all while keeping a meaningful quotient and divisor according to the size and the seasons-to-month relationship you wish. The remainder is what you'll need to add every once in a while. The simplest would be to add it every year on one or two months (like January having 31 days instead of 30), but you can also cumulate them every year and "spend" them later: If you want to add a 40 days month once in a while and your remainder is 2, then you'd wait 20 (40/2) years before adding a whole month back.

Remember however: the more you cumulate those days before paying the bills, the less the months will be bound together with the seasons. For instance, if you wait a full year before adding back a year, you'll drift off slowly over a 1018 year period having "January" be in winter at first, then being in summer then back to winter. It's perfectly doable but you need to take this into consideration.

• Addition to answer: If the idea of cumulating days and then adding them back every few years seems weird, remember that Earth has 365.2422 days per year, and so due to that fraction, every january 1 we deviate a little bit from the sun, and then every ~4 years we have a leap year to pay it back. Commented Feb 17, 2023 at 21:37
• This is completely artificial (as is OP's question), and not how months are determined by early members of a species. Commented Feb 18, 2023 at 4:26
• @RonJohn By definition everything people create is artificial, both calendars and fictional worlds. You can write an all-detailed "natural" evolution of calendars through time of early societies (and have multiple calendars, btw), but the fact is... Most people don't care, as much as most people don't care that years are composed of a round 1018 days. In other words, it is -especially regarding the querent's admitted knowledge in mathematics- a complete waste of time. It's better to focus on core components you're strong at, than working on details you're weak at. Commented Feb 20, 2023 at 9:29
• People looked at the sky, noticed that the Big White Thing in the night sky grew bigger then smaller on a consistent basis, and called that “month”. They noticed that there were four main steps (gone, one side there, full, the other side there) and called those “weeks”. Everything would be different if the Moon orbited at a different distance. Commented Feb 20, 2023 at 14:11

Division with Remainder

The year has $$1018$$ days. If you want twelve months then you divide the year length by twelve. $$1018$$ divided by $$12$$ is $$84$$ remainder $$10$$.

To turn this into a calendar, start with twelve months with $$84$$ days. Then distribute the remainder ten among the months. For example ten months with 85 days and two months with 84 days. Or one month with 94 days and eleven months with 84 days.

How do you calculate the remainder using a calculator? Like this:

$$\frac{1018}{12} = 84.833\ldots$$

In other words, how many twelves are in 1018? The answer is $$84$$ twelves and an extra $$0.833\ldots$$ of a twelve. To see how much that is multiply twelve by $$0.833\ldots$$. We have

$$12 \times 0.833\ldots = 10$$

So the total is $$84$$ twelves and an extra ten.

• Note this technic works with 12 months in a year, but also with 2, 3, 4, 5... 4212... 7983... Commented Feb 15, 2023 at 15:11
• Those extra days can be used for lots of interesting things as well. Festivals, mystery days, errors in the calendar producing interesting results over the centuries, huge drama over people disagreeing about the exact length of the year, etc. Commented Feb 15, 2023 at 16:25
• +1, but also those 10 extra days don't technically need to be part of a particular month. It seems weird to us today, but many ancient calendars considered holidays and such to be outside the normal concept of months/weeks Commented Feb 15, 2023 at 16:56
• @BThompson There is evidence that the current 7 day week goes back to the dawn of recorded history, without interruption, in the middle east. Ie, Holidays didn't skip a weekday. Months? Years? Those change. But the 7 day week somehow survived uninterrupted.
– Yakk
Commented Feb 16, 2023 at 3:57
• @Yakk The Romans had an 8 day week. Ancient Chinese calendars had 10-day weeks. Other cultures used 9, 6, 5, or 4 days. I think the 7-day week being special is more survival bias than anything else. It's our current week, and when we look for its origins we can trace back to "the dawn of recorded history" in the middle east. But humans have come up with loads of other options, probably including others that went back to the dawn of recorded history (in places other than the middle east). Is 7 days really "special", or just the one that happened to end up dominant?
– Ben
Commented Feb 17, 2023 at 3:32

# Go Egyptian

I see a healthy amount of math in the other answers and I love that. However the Egyptians, which were a civilisation known for their advanced mathing for their time, prioritised simplicity.

Their calendar was the simplest one I have ever seen From Wikipedia:

The ancient Egyptian calendar – a civil calendar – was a solar calendar with a 365-day year. The year consisted of three seasons of 120 days each, plus an intercalary month of five epagomenal days treated as outside of the year proper. Each season was divided into four months of 30 days. These twelve months were initially numbered within each season but came to also be known by the names of their principal festivals. Each month was divided into three 10-day periods known as decans or decades.

To do something similar, with 1,018 days you could have ten months of a hundred days each, or 50 months of 20 days each, and then an intercalary month of 18 epagomenal days.

• In other words : Use the remainder of the division as a special month :). I couldn't help but add maths in it ^^. Commented Feb 15, 2023 at 16:34
• The epagomenal days is exactly what the Mayan Uayeb was. Commented Feb 15, 2023 at 19:38
• @JeffZeitlin Or for that matter the Roman Mercedonius, or any number of other intercalary months. Commented Feb 15, 2023 at 21:37
• I would use the epagomenal days as some sort of end-of-year festival or something. Make them really stand out. Maybe even create a myth about them. I always hated the fact that we don't have a nicely divided gregorian calendar. 12 simple months of 30 days each, and having a big new-year vacation celebration lasting the epagomenal days. Hell, we could have had 3, 10-day, weeks with 3 days off a week... Commented Feb 16, 2023 at 14:34
• fwiw, I'd do 20 months of 50. Commented Feb 16, 2023 at 19:38

# Remainder month(s)

Though I understand the desire for equal months, note that having everything nice and equal is rare. Because why would a planet rotate an exact amount of times around it's axis before it has travelled around a star?

If not equal, you can do near equal. Of your twelve months you have ten with 85 days, two with 84. A day difference is unlikely to make a difference for day to day use. Especially on 84 days.

If this doesn't agree with you, why not change the amount of days? A planet can spin faster and slower with no difference in the circle around the star. That way you can make it fit any lenght of month you want. Like why not 1020 days in a year? 12 months, 85 days exactly.

But again, planets do not work that well. Even on Earth we're using weird time shenanigans so we eventually do not end with winter in August. So such discrepancies make it more realistic.

• I agree that such discrepancies would make it more realistic and fun. Unfortunately, I'm not quite good at mathematics; I graduated secondary school with unsatisfactory grades in one of two math subjects. So, for now, I choose the easier way; this also isn't the only alien planet in my story. Commented Feb 16, 2023 at 7:58
• Addition to answer: The Earth has 365.2422 days per rotation around the sun. That fraction is why we have leap years. Commented Feb 17, 2023 at 21:38

On Earth, length of a month is not an arbitrary decision: it is loosely based on moon cycles, but then also subjected to egotism and politics and whatnot. Week, too: it is approximately how long each moon phase lasts.

Why the year on your planet is divided into smaller units?

Just to divide the year into smaller subunits? Then you justify them somehow.

Based on a moon or something similar? How does it orbit your planet?

• As indicated in some chat comments, one reason to divide a year into smaller subunits is seasons related to agricultural activity. Both Earth and my fictional planet this question is about has distinct seasons due to axial tilt and elliptic orbit. Commented Feb 16, 2023 at 10:55
• @KrišjānisLiepiņš: I think you're missing what this answer is alluding to. The answer is not asking your reason for subunits, but hinting that on Earth subunits came about based on the reasons. Thus, if the reason for subunits is because there are 3 seasons, then there will be 3 subunits, possibly further subdivided. If the people have identified that the moon has 5 phases, and there's 11 moon cycles during the year, then you'll have 11 months, each subdivided in 5 weeks, etc... Thus, you need to think about these elements (seasons, moons, etc...) and then use them to subdivide. Commented Feb 17, 2023 at 7:59
• Once you've thought about those reasons, and used them to subdivide, this is also a good time to add backstory. Maybe there used to be 10 months of 30/31 days with 50 days at the end of the year, but then "2700 years ago, King Numa Pompilius shortened each month to 29/30, and added two new months at the end of the year." Commented Feb 17, 2023 at 21:48

There were two traditions that gave rise to most earth calendars: the solar calendar and the lunar calendar. The year captures a solar cycle. The sun returns to the same place after a year. The month captures a lunar cycle. The moon returns to the same place after a month (approximately).

Solar calendars make sense to farmers. It helps them get seed in the ground at the right time. Lunar calendars apppeal to hunter gatherers. It helps them figure out when the night provides a little light.

The Babylonians developed a solar calendar. They noticed that 365 is awfully close to 360, and 360 has a lot of factors. The rest is details.

The Hebrews developed a lunisolar calendar, and they had an extra "leap month" occcasionally to take care of the remainder.

Many early civilizations developed lunar calendars. They noticed that 29.xxx is close to 30, and 30 is a convenient number.

In the case of your 1018, the farmers might consider that 1018 is awfully close to 1024. 1024 is two to the tenth power. If you like splitting things in half, this is great. I'm not sure what the hunter gatherers are going to home in on.

One option would be to have 32 months of 32 days each, with six months being shortened to 31 days.

• "two traditions " --> maybe 3 with the day being the most common unit. Commented Feb 17, 2023 at 15:18

Get creative.

Not all months are equal. You specifically call out the eccentric orbit, so the months near aphelion are going to be warmer, and other months will be colder. (This is in addition to any effect of your axial tilt, which could exaggerate or moderate the effect for one or the other hemisphere.)

Also, the time between equinoxes will be unequal, because the orbit moves faster near the star, and much slower further away.

Depending on what the crop growing season looks like, (is the warmer time when they can grow crops, or do they grow crops most of the year and have to stop when it's too hot?) So you might have a few short months of frenzied activity, and much longer months where there is little agriculture activity.

Calendars were first invented to help farmers plant at the right time of the season.

In our case, we have a short February because July and August are named after Roman emperors who wanted an extra day in 'their' month. Invent lore around the length of your months if you desire.

• Some people debate that the oldest calendars might have actually been monthly to track women's periods, though this does not appear to be a mainstream view. It's also hard to tell, since women's periods appear to correlate with the lunar calendar. Commented Feb 17, 2023 at 21:53

One approach is the Roman calendar style. The Roman calendar had 10 months. The first 9 were about equal length (approximately 30 days) and the last one (December, as the name implies) was the entire winter (consisting approximately of the 3 modern months of December, January, and February). Of course, this system had problems and had to be constantly adjusted with leap days.

• It was even simpler than being adjusted with leap days. March 1st, the first day of the civil year, was whenever the Great Priest ("Pontifex Maximus", Supreme Pontiff, a title which nowadays belongs to the Pope of Rome) said it was. This, of course, gave the P.M. great political power, because that was when the old consuls were replaced by the newly elected ones. Commented Feb 16, 2023 at 8:50

Brute force.

Small assumption: There should be at least 10 days in a month (at most 100 months). The computer could easily handle much larger numbers but they probably wouldn't be useful.

Put numbers 3 through whatever you want in column A, and the following calculates how far this is from being a round number: ABS((MOD(1018, A2)/A2)-ROUND(MOD(1018, A2)/A2,0)) The computer does sorting.

Ignoring 1 and 2 as not-very-interesting numbers, here's your top 11:

Months  Days per month
85  11.97647059
68  14.97058824
60  16.96666667
51  19.96078431
78  13.05128205
93  10.94623656
73  13.94520548
34  29.94117647
92  11.06521739
30  33.93333333
35  29.08571429


with the most reasonable (IMO) being 34 months, 30 months, or 35 months.

• 34 months of 29 days leaves 32 days leftover, which probably becomes 32 months of 30 days and 2 months of 29 days. 35 months of 29 days leaves 3 days leftover, which probably becomes either (A) 32 months of 29 days and 3 months of 30 days, OR (B) 35 months of 29 days, and 3 "non-month holidays" at the end of the year. Commented Feb 15, 2023 at 22:41

Prime Factorization

The "math thing" you need to know about is called "prime factorization". All numbers can be written out uniquely based on what primes can evenly divide them.

6=2x3
64=2x2x2x2x2x2
210=2x3x5x7

This means that 6 boxes can be arranged in 2-rows-of-3 or 3-rows-of-2; 210 boxes on the other hand can be arranged in (2)-rows-of-(3x5x7), or (3)-rows-of-(2x5x7), (5)x(2x3x7), (7)x(2x3x5), (2x3)x(5x7), (2x5)x(3x7), etc. which gives far more options to choose from.

Unfortunately, this means that if a number has a big-prime rather than multiple smaller-primes in its factorization, then the number of perfectly-divisible options goes down.

1018=2x509
365=5x73

This is bad for your world because it means that the only two choices of evenly-divisible months are (2)-months-of-(509)-days or (509)-months-of-(2)-days.

That's not the worst thing in the world, though. After all, our 365-day year is just as bad in this regard - it would need to have (5)-months-of-(73)-days or (73)-months-of-(5)-days to divide evenly... Instead, we opted to have (12)-months-of-(30.4)-days and just decided to fix the 0.4-day problem by pushing them around and making un-evenly-sized months.

Your world is the same. You need to realize that perfectly-even months are impossible (other than the ridiculous 2x509 or 509x2 options) and just pick a nice round number that that works for your world's seasons/moon-phases/etc. that is at least "pretty close" to 1018. (And then you just need to remember the exceptions: "Every-other-month-has-an-extra-day" or whatever.) The "Division with Remainder" and "Brute Force" answers essentially being how you might go about choosing a nice "round" number.

First, ask yourself why we do have 12 months....

It has to do with moon cycles and star constellations. What to do if your planet has no moon or perhaps has multiple moons? Or the moons aren't tidaly locked or have an ecliptic orbit? Then your math could potentionally break.

Another point is that people began to notice that with each moon cycle the sun rose a bit higher over the horizon or sank a bit more. And once each year the sun was at exactly the same position than some time ago - not necesary months, that's only how the egyptians began counting days.

Others here propose elegant mathematical solutions that make sense from a modern day's pov but wouldn't neccesary from a person's pov 3500 years ago. So always think practical first I guess. We have to ask ourself Do the people on this world know that their planet orbits a sun, and do they know the math for rotary angles?

If yes, then a possible solution would be to split the year into different angle positions, like 1/2 pi, pi, 1-1/2 pi, 2 pi for a 4 month year. It does not really matter how you split the year as long as the planet arrives at the same location after a full orbit.

If not, the people would still be able to measure the effect of the orbit by looking at the stars. Between the 2 twilight zones there is a phase of complete darkness, where you would look away from the sun into space. During that time you can see fix-stars , and some wandering stars. The fix stars change over time but ultimately come back to the same place where they started. Now choose a random divisor for the time until that happens.

You see, there really is no need to force egyptian calendar heuristics onto an alien world and use fancy math. Often in history stuff has been determined by random observations and chances.

You can move the extra days outside of the rest of the months and I have seen several fantasy novels use this type of calendar system. These days can be put at the end of the year or between months at various parts of the year. I will provide an example based on a 365 day year.

365 days in a year
12 months in a year (30 days each)
5 weeks in a month (6 days in a week) - Not asked for but how they did it
5 days at the end of the year that don't count for a month or week
2 days to remember the past year (counts as a single day)
1 day of rest
2 days to celebrate the new year (counts as a single day)

This system ensures that every month has the same amount of days and each day/week falls in the same time throughout the year. in your case you can go with 12 months of 84 days and split the 10 extra days between months throughout the year.

## 9 Months with 113 days per month

9 * 113 = 1,017

Well, I know you specified that your year has exactly 1,018 days each year but this only sums up to 1,017 days. There are a few ways to solve this.

1. Change the number of days in a year from 1,018 to 1,017. Yeah, I know this is not what you are looking for.
2. Make the excess 1 day to be an extra/special day of the year. For example, first day of the year, last day of the year, or any exceptional event day of the year. That day is not in any month.
3. Make each months may have different number of days, so that you can fit this extra day anywhere.

An earth year is 365.24 days or something like that. 12 months of 30 days would be 360 days. 7 x 30 + 5 x 31 days would be 365 (what we do with February is quite strange actually).

So a year is 0.24 days longer than that. Four years are 0.96 days longer, so we make every fourth year a leap your, changing one month from 30 to 31 days. Now four years are 0.04 days shorter. A correction of one day every 25 x 4 years or 100 years fixes that.

You can do the same with any year length. Pick a nice round number for the number of months. Divide the year into longer and shorter months, so the error is less than half a day. And then create leap years.

Now if you have a big moon like we do, or at least an obviously visible one, say more than one tenth the size of our moon, then it is tempting to make the length of a month kind of similar to the length of a moon cycle.

Some more things to consider: You have seasons. You'd probably want a season that is cold, one where it warms up, one that is hot, and one that cools down. Every season about the same length. And you want every season to be whole months. So you would have four seasons of three months each, or four months each, or five months each, but you wouldn't want 18 months split amongst four seasons. Maybe if the year is long you'd have six or eight seasons.

• it's even more complicated. a day in spring is not the same length as a day in fall because the earth lingers on it's path. at the end of the year, the remainder occurs because the earth spins clockwise and did not finish a full 360° rotation. so the problem actually stems from humans forcing a day to be exactly 24 hours. If we would overthink this assumption, we would not need leap years.
– user59660
Commented Feb 17, 2023 at 15:36
• also there is a timezone (i think it's -14 ? ) on earth which has 13 months, where the last month only has 4 days or so. calendars are weird
– user59660
Commented Feb 17, 2023 at 15:37

The remainder is a normal phenomena, as it occurs in our calendar. The remainder of .25 days exists in our own orbit around the sun, hence a leap year (extra day) to realign the counts every four years. Otherwise the calendar would drift 1 day off every four years. After 100 years that would be a 25 day shift, resulting in the calendar months correspond to different times of the earths orbit (i.e. the seasons).

Also, have you considered that the “day” is not set by the planet’s orbit around a star, but the time the planet completes a rotation on its own axis? In that way, the 1018 days only means that it completes its orbit around the star close to the time it takes for the planet to rotate 1018 times. Consider whether you want a planet that spins faster or slower and you can always scale it to your orbital speed.

• That's because our years are not a whole number multiple of our day lengths, but in the question OP makes it sound like there are exactly 1018 days in each year, therefore there is no need for leap years Commented Feb 15, 2023 at 16:56
• As @theonlygusti says, we have a leap year because Earth's days and year don't synchronise perfectly. On my alien planet in question, I might consider this detail later, but it is not the subject of this question. Commented Feb 16, 2023 at 8:11
• @theonlygusti thanks Commented Feb 17, 2023 at 13:26