Worldbuilding 103: Mathing
Greetings and satisdation! I am not the one providing it, but maybe that big guy you know will. A thing I often see people struggle with in writing and worldbuilding is mathematics, or more, the ability to calculate things! So, in today’s blogpost, the magnificent, eloquent, exceptional, amazing Anne Winchell will be providing questions or exercises for me to calculate, and I will show you how I do to figure things out.
Rules of ease
Before I do, however, a few rules on how to make it easier for ourselves:
Rule 1: First digit is the only one that matters
By that, I mean if you find values such as 241, 7742, 0.004123, and so on, you can round to the first digit only. So you get 200, 8000, and 0.004. This is because we are interested more in the magnitude of a number, and all digits beyond the first one rarely affect the magnitude enough. Wait, you don’t know what a magnitude is? Luckily we have…
Rule 2: Magnitude over value
Magnitude says “how big” a number is, not caring about the value. That is, basically, what power of 10 is closest? In the example above, 200 has a magnitude of about 2 because 10 to the 2nd power is 100, 8000 has magnitude of about 4 because 10 to the 4th power is 10 thousand and 8000 is almost 10 thousand. And 0.004 is… -3 in magnitude because 10 to -3 is 0.001 (I use the trick of writing 1000, 3 zeros, reverse it 0001, then add a point after when it comes to negative powers).
When it comes to a number, for example 3214, you have 4 digits, so it is either magnitude 3 (number of digits minus one) or 4 (number of digits), which is an easy way to remember it! For 0.00… numbers you count how many 0’s before the digit, which for 0.004 is 3, so that minus 1 is 2, so it is either -3 or -2. A rule of thumb is that if the first digit is 3 or less, it belongs to the lower magnitude of the two options, so 100 to 300 are magnitude 2, and 400 to 900 belong to magnitude 3, 0.001 to 0.003 are magnitude -3, and 0.004 to 0.009 are magnitude -2; a bit counter intuitive for people, but trust me on this, the 3 being the cut off point instead of 5 is for a very important reason.
This all matters because most often we’re interested in the size of a number over the exact value. After all, what matters most? If something costs 25 thousand dollars, 26 thousand dollars, or 24 thousand dollars, or does it matter more that all values are closer to 10 thousand dollars rather than 100 thousand dollars?
Rule 3: Half a magnitude above or below needs no explanation
Let’s say we have gotten a value at the end of our calculation, and it is, let's say, 150 days for whatever we calculated. We have done lots of rounding and all, but do we need to use exactly that value? Of course not! We can fudge the numbers somewhat. Life is complicated, with many factors unaccounted for, and thus, I have said that if you do one half magnitude above or below the value calculated, you can do whatever you want that fits you. How do you calculate it? Technically, you multiply and divide by square root of 10, but you can get it close enough by multiplying or dividing by 3 instead. The square root is 3.16… Yeah, close enough. (I wonder if I made 3 special in a previous rule… hmmm). In our example here, that 150 days could be between 50 and 450 days without raising questions.
So as you can see, it gives a pretty decent range that you can land where you want. But if you want to go beyond this min and max, you generally need to justify it in your calculations before somehow, but not always.
Rule 4: Multiplication is addition, division is subtraction
One thing to remember is that when you count with a lot of these calculations, addition and subtraction of pure numbers tend to take a back seat. That is because they often do not change the magnitude, and when they do, the numbers are already close to the upper magnitude anyway. For example, 500+600=1100, but 500 and 600 are already magnitude 3 by the previous rule, and so is 1100.
Multiplication and division are the important ones, to the degree that, very often, you can throw away additions or subtractions and ignore them completely. I will give examples, hopefully, in the exercises! But if you count in pure magnitudes of numbers, when you multiply the numbers, you add the magnitudes, and if you divide the numbers, you subtract the magnitudes.
For example, multiply 234 with 7432: using Rule 2 we get that the magnitude of 234 is 2, and the magnitude of 7432 is 4. The number's product is 1739088, 7 digits, which with an initial digit of 1 means the magnitude is 6, and 2 plus 4 is 6! Now take the number 3996840 and divide it by 456, using Rule 1, 3996840 is about 4000000, which by rule 2 has magnitude of 7. 456 has magnitude 3. So by what I have said, we should get a number with magnitude 7-3=4. Well 3996840/456 = 8765, which does indeed have magnitude 4!
So, this allows us to do quick maths with less accuracy by converting it to pure magnitudes. If you have, like in the first example, figured out that the result should be of magnitude 6, how can you use Rule 3 to find the range of numbers you can use without needing to justify it? Well, we know magnitude 6 means 1 followed by six 0’s, so 1 million. To get a half magnitude above, you do 1 million times 3, which means 3 million, and to get the lower half magnitude, you divide one million by three, that’s a hassle, so a quick and dirty way is to take the magnitude below the one in the given number, which would be 5 here, or 1 hundred thousand, and make it times 3, so 3 hundred thousand. Overall, then, we get between 3 hundred thousand and 3 million, which, given the example was 1.7 million almost, is not a bad estimate.
As a last thing that I will not go too deeply into, but it is important to remember, is that when it comes to magnitude, when you do power, as in 5 to the 4th power, 5^4, and so on, you do multiplication of the magnitude. For roots, square root, cube root, etc, you divide the magnitude by two, three, etc, respectively. Keep in mind, though, the magnitude error gets greater from the roundings. So the full name should be ”Multiplication is addition, division is subtraction, power is multiplication, roots are division,” but that got too long.
Let’s use 20 as an example, which, by previous rules, has a magnitude of 1. We want to cube it, that is, 20^3. We can calculate that by hand and get 8000, but if we use the magnitude rules I have laid out here, we take the magnitude of 20, that is 1, and multiply it by the power, that is 3. So we get 3 times 1 = 3. The astute of you will notice, however, that 8000 is by my rules of magnitude 4, which is one whole magnitude wrong! Why is that? Magnitude counting is not accurate as we round, so while I have used integers for magnitude, you can have decimal magnitudes, hence the whole half a magnitude in previous rule. For 20, we have that 20 = 2 times 10. The 10 is the magnitude of 1, but the 2 alone also has a magnitude, about 0.3. And remember the title, multiplication is addition, so the more exact magnitude of 20 is 0.3 + 1 = 1.3, 2 times 10 = 20. And if we take 1.3 times 3, we get 3.9 which rounds to 4.
But calculating these other decimal magnitudes is a hassle, so a more efficient way to think is this: calculate the magnitude as before and do multiplication, 1 times 3 (magnitude of 1 times the power of 3), then look at the first digit, 2, and take it to the 3rd power, which is 8, 2 times 2 times 2, and 8 is by previous rule magnitude 1, meaning we add it to the result. So, the final equation becomes 1 times 3 plus 1 = 4.
Exercises
Next, we’re going to get into some concrete ways to use maths in your worlds. I’ll explain how I calculate these problems so that you can hopefully see the process and be able to do it for yourself. These are all supplied by the glorious Lady Verbosa, aka Anne Winchellius–I forgot her in Rome, okay?
Exercise 1:
How long would it take an average-sized army at American Civil War levels of technology to move 100 miles, and what would be involved in the process if they had to go straight into battle at the end?
First off is that I work in kilometres, or km, so I will be using that. 1 mile is about 1.6 km, which as per my rules, I can round to 2, so 100 miles is about 200 km of distance. I know from the top of my head that the U.S Civil War is around 1880, which means trains were a thing. So, I first check out how fast trains go back then. I find this link: How Fast Were Trains In The 1800s?
Which gives 112 mph or, as per previous rounding, about 200 km/h. But that presumes a city, as trains require infrastructure. But as per my post on communication, we can essentially count this as “instantaneous” to the point that we only count distances from the nearest train station town or city. This is alone important because it means train cities are important for the war effort.
Of course, no enemy is going to willingly go to a city to be slaughtered in that era, so we can assume it is away from a city by some distance, like the 200 km we started with. I find this info on forced march: 30 km in 5 hours carrying 20 kg. 20 kg is not much, but when you are marching, it is a lot. So horses are likely to be used to carry, and horses can easily match any speed humans can for sustained periods of time. So the speed is 30/5=6 km/h without the humans having to carry the load. Let’s round it to 10 km/h because they’re in a hurry.
That means with enough horses to carry all the loads and humans marching fast, we need to do 200 km at a speed of 10 km/h, or 20 hours. Which means you are going to die from exhaustion, metaphorically. If you expect the enemy to be there, you don’t want to be completely exhausted and tired. So let’s say they do 8 hours each day, which rounded gives us 3 days. And remember rule 3, divide by three for the low end and multiply by 3 for the high end of easily justified numbers, so between 1 and 9 days are not unbelievable. The 1 day is really straining it here, but hey, people have pushed for longer periods of time marching when necessary, so you can do it!
Exercise 1b:
Assuming the same size army, how fast could they travel 200 km with energy to end in a fight if this were a Medieval level of technology and they were travelling through extremely mountainous terrain in unfriendly territory with the occasional nearby village defending itself but also offering potential for plunder of various sorts?
Darn difference! Well, for Mediaeval travel, we can maybe use this helpful fellow, which gives a person with luggage about 15 km/day in ideal conditions. Of course, we are not doing that, so we can say instead that it is 15 km/day without luggage. And with luggage, we halve it. Let’s say 8 km/day then when it is mountainous.
If we cannot find the link I had found, we could go to the earlier estimate of 10 km/h. With 8 hours of walking that is 80 km/day. That was without luggage, and now we have it on, so we halve it to 40 km/day. But that is still on mostly roads and flat terrain. From personal experience, you can say that in hilly uneven terrain, or in mountains and such, your speed is reduced by 60-80%, so we say ¼ of your original speed, which gives us 10 km/day, which, as a quick estimate, is not too far off the other. Are any of these very accurate? Probably not, but they are close enough.
So those 200 km will then take about 20 days; Rule #3 again, divide by and multiply by 3, we get between 6 and 60 days. If you go for the lower end of that, it essentially means you leave anyone behind that falls behind, which is going to be A LOT of people, and they are going to die. If you go for the upper limit, well, then you’re more merciful and try to get those sacks of meat with you. Let’s hope the barbarian natives are not too savagery.
Exercise 1c:
How many supplies would these people need?
Ah, Medieval warfare, nothing beats it? Except lack of cannons, BOO! Let’s say, for this problem, that we have 1000 men and 200 horses. The site linked there says 1 kg of grain for the humans and 5 kg for the horses, per day. That means 2000 kg of grain per day. That is a lot! You can see why modern armies use ration bars and such. But you know, a lot of pillaging was done in that era, and for ages, so let’s say they only need to bring 50% of that, which is 1000 kg for everyone, the rest is pillaging. We can then use Rule #3 again and get that they should be carrying between 300 and 3000 per day and to have at the onset of the campaign.
But it is better to carry more just in case! A horse can carry about 200 kg each, and we had 200 horses, so if every human is walking, and we stick to terrain that a horse can walk on enough with human help, that means we can carry 200 times 200, 40 thousand kg. Assume 50% is grain and the rest is equipment, that gives us 20 thousand kg of grain, or about 10 days of food for the convoy. With our usual divide and multiply by 3, we get between 3 and 30 days.
Exercise 2:
How much time has to pass after a new government is put in place (through a violent revolution) before the society/nation is stable again?
This one is considerably harder to interpret, but we will try, right? Well, historically, we have two revolutions to pick from, the French or the Russian revolution, and we can choose to mimic either. I’ll pick the Russian one for this, which lasted from 1917 to 1923. I chose also to understand “stable” as in “things are rebuilt and working again”... Which for the Soviet Union, happened roughly by the early 30s. Modern estimates give railroad building times at about 7 years, factories and buildings take a couple of years each. We can safely assume they cannot reasonably do everything at once, but a lot can. So rounding up on the Soviet time, we get about 10 years. And Rule 3 again gives us 3 to 30 years. Sure, if you do it in 3; it is going to require a lot of work and investment, but it can be done!
Exercise 3:
What would a good range of populations be for a planet similar to Earth in terms of liveability but twice the size?
For this, it is important to ask ourselves what era of human society in terms of food production we are dealing with. I am going to take the late 19th century because I like that era a lot. Using wikipedia for an estimate, we can see around 1900, we had about 1.6 billion people. Given we are doing twice the area and assuming the same climate and abilities, we get 3 billion people then, rounded. Usual rule, you know by now! Divide and multiply by 3! 1 billion to 9 billion people.
9 Billion might feel like a lot for such a society, but keep in mind, they could have extremely fertile land in huge quantities, and with the recent developments in agriculture and machines, they could have skyrocketed the population and still be growing.
Exercise 3b:
Sticking with this planet and assuming advanced technology at or greater than our current level of technology, at about what population amount would life be unsustainable for that population?
Why do these random subexercises keep popping up!? Oh well. Using this resource, we can, with more advanced tech, say they use the hydroponics alternative, but we round down a bit and say 8000 people per square km per day. From here, we see that about half of all land is used for agriculture. For Earth, this amounts to 48 million square km. Rounded to 50. Multiply those, we get 400 thousand million, which is 400 billion people in normal speech. Now, that is Earth, and this is double the size, so 800 billion people, or almost a trillion people on this planet. Which we all can agree feels a wee bit high. After all, is it reasonable to think that you humans will use the same amount of land with more efficient farming? No. So lets reduce it by one magnitude because of shifts in politics and priorities and say 100 billion, and, with the usual divide by and multiply by 3, we get between 30 billion and 300 billion. Still a lot, but it feels way more reasonable.
Can anyone now see why even with this I call horseshit on Star Wars and their one trillion people ecumenopolis?
Exercise 4:
If I wanted a population of 3 million people on a continent with limited food supplies and inhospitable, icy terrain, how large a landmass should it be, and what else do I need to know to calculate that?
First of all, we need to decide how they sustain themselves. Are they like the Sami or old Indians in the U.S.A? Where there is no agriculture and they mostly herd animals and use them? I will choose to go with that. For that, we can go to the Nordic countries, and we’ll take the northern region of Finland because only Sami would be tough enough to endure it. It has a population density of 2 per square km. Which is still probably more than just the Sami and all, but let’s say it is all Sami.
The calculation is fortunately fairly rudimentary, 3 (from the desired population of 3 million) divided by 2 equals 1.5, so we need 1.5 MILLION square km of area. Or about 1/6 of the USA.
Which is about the dark cyan colour. The usual multiply and divide by 3 and we get 0.5 to 4.5 million square km which is between 1/18 th and ½ United states. If you want modern societies, you can take Norwegian, Swedish, and or Finnish population densities of the appropriate times for harsh but agricultural societies. If you want really tough ones, you can look for Siberian ones instead. The calculation is the same, population divided by density.
Exercise 5:
How long would it take a ship to go between planets if there is a portal in the Kuiper belt of every solar system that allows instantaneous travel, but outside of that, the speed is the speed of light, and the start planet is at Earth distance and the target planet in the other system is at Jupiter's distance from its star?
Now, this is one in my favourite flavour, space opera! Well, the crucial point here is the distance to the Kuiper belt, which we can say is about 40 AU. One AU is the distance between the sun and Earth, which is about 150 billion metres. Given we are zooming around in the solar system at the speed of light, which is about 300 thousand km/s, or 300 million metres per second, and Jupiter is a distance of 5 AU from the sun or 45 from the Kuiper belt, let’s do the magnitude style I explained.
The speed of light has a magnitude of 8, the distance to the Kuiper belt in AUs has magnitude of 2, and AU in metres is a magnitude of 11, and the Kuiper belt to Jupiter is of magnitude 2. So to calculate, we would want to take the distance to the gate, to the Kuiper belt, times the size of an AU, divided by the speed of light, to get the travel time out there. By Rule #4, we then add and subtract magnitudes, or 2+11-8=5 magnitudes in time travel. If you pay attention, it is the same on the other side. But this is seconds, and we likely want hours. There are 3600 seconds in an hour, which gets a magnitude of 4. We would do division so it is subtraction, 5-4=1. This is 10 to the 1st power, or just 10.
So it should be between 3 and 30 hours, but it is identical on the other side, so in total, between 6 and 60 hours. If we crunch the actual numbers we get 11 hours, which is remarkably close to the original magnitude answer, isn’t it? Of course, light is a hard limit that you cannot break, except here we can by our portal gate and much else, so you know, enough wiggle room!
Exercise 6:
Say a conquering army decides that the only way to prevent a return to the fighting is to kill everyone old enough to oppose them in a wee bit of genocide. How long would it take them to kill everyone between, say, 20-35 years old if the territory is approximately the size of the United States with slightly lower population density, and what would be the best ways to accomplish this?
Mini disclaimer here: this is of course a horrific thing, and me throwing around numbers is not trivialising real world atrocities, it is to show how even with these things you can use estimates and rounding and the likes to make atrocities more believable in your stories in order to make the impact greater for you as an author.
To calculate this requires us to know the life table of it; that is, a table that shows how many have survived since birth up to a specific age, which is not the easiest to get for a specific level of technology and era and conditions. But one can definitely search for it in the West that goes back quite far. But if we assume that, for simplicity, a modern comparison is good enough, it is a remarkably easy calculation. Up to about age 70, the line of how many have survived since birth is essentially flat with little changing. For simplicity that will soon become apparent, we will say it is flat up to 75 after which people die like flies.
We can estimate that about 25% of the population is above 75, we can do this because humans on average live about 85 years and the death rate is close to zero before it picks up fast due to age and it will make our calculations easier and won’t be too wrong. With this, we can say that 75 and below, each year contains 1% of the population at 50:50 between the sexes (we are assuming humans). That means between 20 and 35 years of age equals 15% of the population.
The USA has about 300 million people if we round it down, which should cover the slightly lower population density. I did find a graph ages ago, when I was helping Anne for her book, about the Holocaust and the efficiency of the Nazi genocide, which is extremely disturbing, to say the least. But I found there that at the peak, they killed 650 people per hour. I told you it was horrific. I will call this the PIG rate, Peak Industrial Genocide Rate. With this, we now have enough information to answer this question.
15% of 300 is the same as 15 times 3, as 3 is 1% of 300, so 45. So it is about 50 million people. We will want per day instead as we are more likely to count days than hours, but for ease we round the hours to 20h/day and the PIG rate up to 1000 per hour. Why this? One is going up and one is going down, which should just about cancel each other out. That gives us 20 thousand killings per day.
50 million divided by 20 thousand, we can do 50 divided by 2, which gives 25, so we have 25 million now divided by 10 thousand, which is the same as removing 4 zeros (or 4 magnitudes) of 25 million, which gives 2500 days. We can divide by 400, instead of 365 as that is a nasty number, 400 is much nicer. That gives us 2500/400=25/4=6.25. Let’s round it to 6 years to kill them all. The usual multiply and divide by 3 gives us 2 to 18 years. Which, given the Holocaust happened between 1941 and 1945, a total of four years, with 17 million victims dead, falls within the realm of horrifying atrocities.
Summa Summarum
I hope you have learned a lot here! The important thing to remember here is that we do what is called Fermi approximations: he famously used pieces of paper during the nuclear detonation of Trinity to estimate its yield quite exactly. The thing to do there is do rounding and estimates up and down so over all they will cancel each other out.
If you are exceptionally uncertain on the values or cannot find even a ballpark estimate online, then you can work on magnitudes instead and pick ones that are reasonable. If you keep them within reason, the answer you get is likely to be somewhere close to what you will need. And the half magnitude up or down will give you the wiggle room needed.
You may hate maths, but if your estimates are totally off, you’ll destroy the suspension of disbelief and lose your readers. So I hope this helps you make better estimations and not fail like many famous writers have with entirely unbelievable values.
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