Fantasy races might have diferent math systems to what we are used to, and this might also lead them to have other interesting units for length and weight.
So what kind ofmath system might the diferent races have ?
The foot, is a god (WOTC) given mesurement so all races would have the foot as the base unit for distance.
Dwarves might use a base 5 system.
that might have arisen from some cultural or spiritual reason why they only used one hand for couning (5 fingers) instead of the 2 hands (10 fingers) humans use.
This would also have led to dwarves having special names for distances of 5, 25, 125, 625 and 3125 feet.
Dwarves are especialy adept at architecture and masonry, so much so that thse dwarven units of distance are also used by most architects of other races (and why so many buildings can be divided in neat 5 feet squares)
,
the counting system of Elves also evolved from using their hands.
when counting on their hands thelves usualy hold their hands with the palms down and don't use the tumb.
This is becouse they use a more complex way of counting on their fingersthen just counting the number of fingers.
but having values from their base 4 counting system assigned to diferent digets,
Elves often mark their fingers with dots while working out math problems or noting down a number they need to remember.
The black dots on the ilustration above would be written as 2101, and represent a value of 145 in the human base-10 system
this means they can count upto 255 on one hand and upto 65525 on 2 hands
Ooh, math theory! I like! Wait, this is my birthbay, why am I writing articles on a forum? Oh well!
Null and Void - The Question of Zero
So here's a question: do these fantasy races have zero in their numbers? The reason is simple - zero allows logical calculation of negative numbers; and with the fact that it literally means "nothing", we can use it to mark an empty position in a number, which gives positional/place-value notation (which is important because it adds flexibility and reduces the total sysmbol set we require to read the number).
So, do they have a positional notation, a system of negative numbers, a Zero?
"Your Total Comes Out To Bork-Bork-Bork" - Learning Math
How easy is it to learn and use the math system? Does one require special training to even know how to multiply and divide, or is it so simple that the average commoner can learn basic algebra? In the real world, Europeans used the Roman system of counting before they switched over to our modern Hindu-Arabic number system - Roman numerals are easy to use for adding and subtracting, but require a lot of effort to multiply and divide; conversely, the Hindu-Arabic system is so simple that advanced mathematics began to be learned in Europe, and today the average person learns algebra during their schooling. This also ties into how simple the system is - how many symbols (numbers and operators) do you need to know to read an equation? How easy is it to solve an equation? This doesn't have to require a Zero or a simple system: the Egyptians had a heiroglyph-notation system with no Zero (from what we can tell), but recovered documents from their era shows us that they could solve quadratic equations by 1300BC, and used a system of powers of two to help solve multiplication and division problems - meaning that their system was capable of complicated equations.
"And The Gods Said, 'Let Two Plus Two Equal Four' And Math Was Created" - The Effects of Ancient Cultures and Their Math
How much of an effect does the math of past cultures have on the current era of your game world? Depends on how much was remembered, and that depends on how much of it was used in daily life - which typically means measurements of weight, length, and volume (typically either a function of weight or length, depending on the culture). Looking at historical examples, we can trace the Foot of twelve inches back to the Romans, whose foot was split into both 12 ad 16 sections (depending on which function you wanted), and from there to the Egyptians - although we're not sure how the Egyptian Cubit got turned into the Roman Foot...
As a side note, I'm fairly sure that the use of the foot in D&D is a combination of two factors - 1: players will best recognize measurements if written in real life units, and 2: WoTC is American and Americans use the Imperial (US) set of measurements. So feel free to replace it in game with something else (some days I'm tempted to replace it with the metric system). Just keep a conversion chart around.
Another historical example of ancient math being used today is the Babylonian system - a rare double-base system which used both Base 10 and Base 60. We see base 60 in the measurement of time, and is the entire basis of our angular measurements (60x6 = 360 degrees). AS you design a world, feel especially free to pick an ancient culture that dominated the area (there's always one, we assume there's one, history says there should be one because Romans and Europe), take its math, and let it continue through the ages in modified forms.
Factor Of What? - The Base System
Why do we describe number systems as Base-X? Because that is how many numbers make up one higher unit. We use Base-10, so we count by 10's, and numbers up to 10, so amounts beyond 10 are stated as 10+X. A Base 5 system would have numbers past 5 be 5+X, and you get the general idea.
A quicker way to describe it is to use the exponent system (which would never have existed if we hadn't freed our math thinking from the roman and greek obsession with math being only used to describe the physical world). Any number to power of 0 is 1, and any number to the power of 1 is itself. With this, we can easily describe an entire number system with a series of exponent notations (the egyptians had their heiroglyphs go up by a power of 10, so there is historical evidence for the fact that I'm not pulling out just modern math theory that was never used in the real world). It's especially useful in positional notation as well. Let's look at Base-Ten: 10^0=1, 10^1=10, 10^2=100, 10^3=1000, 10^4=10000, 10^5=100000, and so on. What about, let's say, Base 5? 5^0=1, 5^1=5, 5^2=25, 5^3=125, 5^4=625, 5^5=3125... As we can see, a higher Base also gives us more numbers for less symbols in a single sequence. Let's look at Base 16, which requires us to add symbols to Base 10; A=10 B=11 C=12 D=13 E=14 F=15 10=16 (that's our place-value notation at work right there!). So, let's just kinda ignore that though, we don't need it to describe just how high a base-16 system goes in a given amount of symbols. 16^0=1, 16^1=16, 16^2=256, 16^3=4096, 16^4=65536, 16^5=1048576. Yes, we get that high with 5 symbols or five places in a number that denote a power of 16.
Speaking of which, I can now easily describe Positional, or Place-Value, notation now! In a positional notation system, the total number of symbols needed drops. We simply have a set of symbols that go from 1 to X-1, where X is the Base System (whether it be 5, 10, 12, 16, and so on), with a symbol that denotes an emtpy column (this could be zero, and it can evolve into Zero if the need is recognized). With positional notation, numbers are read as a row of these symbols, with the position in the row (called columns) showing which power of X it is worth. A one column number is worth up to X-1, a two column number is worth X+0 to (X^2)-1, a three column number is worth X^2+0 to X^3-1, and you get the idea.
Knuckle Down On The Beancounters - Creation of Systems of Numbering and Examples
I had to make a joke based on the campaign world that I'm building - the elves have culturally dominated the world, and their numbering system is based off of the number of knuckles your fingers have. Which easily segues into me providing a sample to describe the logic you might use.
Elves, having stereotypical long fingers and dextrous hands, can easily count using their knuckles. Disregarding the thumb, they went with the fingers, counting 12 knuckles on one hand by folding each finger one joint at a time, and then doing the same with the other hand to mark how many groups of twelves they have counted. Early writing notations had 5 symbols; A Knuckle symbol for 1, a Finger sysmbol for 3, a Hand Symbol for 12, a second type of Finger symbol for 32, and second type of Hand symbol for 144. They then started adding symbols for greater powers of 12. After a while, somebody creaed positional notation, and added additional symbols. It is still trying to catch on.
Dwarves, by contrast, went with a base 5/25 system, accounting for shorter fingers, and the need to quickly count creatures and items in mines and on battlefields. They count up to five with one hand, and count groups of five with the other. This lead to an early system of 3 symbols: 1, 5, and 25. Eventually, with the adoption of place-value notation (it was efficient, thus they used it), 25 became a place-holder for an empty column, and eventually their zero. (yes, I stole this from the D'ni of the Myst series). Their place value system still has the original 3 symbols, but with additions to show greater value. Their symbols are a vertical line for 1, and additions of up to 3 short lines coming from its right side for values up to 4; a horizontal line for 5, with up to 3 short lines coming off its bottom for values up to 20, and a Mountain for Zero/Empty Column. Showing a value up to 25 is done by stacking the symbol for 1-4 above the symbol for 5-20.
Humans, on the other hand, simple counted to 10 with the digits of both hands, and then placed a marker before them (or in their mind) to represent groups of 10. Then they make a mark for groups of 10 groups of 10, and so on.
When creating a system, decide what method the race or culture used in their past to do basic counting. Was it fingers? Knuckles? Fingers and knuckles? Fingers, knuckles, and the hand itself? How was the method used? Did they use the entire set, then place a marker before them? Or did they use just part of the set, and used the other part to mark how many times they had counted that first part? Then make some symbols for logical divisions of their numbers, decide if they went to place value notation, and then test it. Sometimes it's bulky and horrible, other times it's efficient and quick.
"Is That A Five or A Giant Ant?" - Introducing Alternate Number Systems to Your Players
Sometimes an alternate numbering system is just background fluff, noticed and used on occasion. Sometimes its something cool, a challange to overcome in the course of an adventure. Imagine the players having to go on the trail of an ancient artifact, which requires going to an ancient temple and using that culture's numbers to suceed. But how do you introduce something like that. Obviously you want to draw some attention to it, either subtly ("You feel like the columns have too little decoration on them"), to the blatant ("The columns have a series of symbols on them, each one showing a regular grouping of the symbols on the column before it") to the mixture ("the columns are blank except for a small series of symbols carved into them"). Make sure to have a handout (could be hand drawn) for the players to reference, but make them work to figure out what it means.
RustedKitsune wrote:
Sorry I thread-jacked to write an article on the math theory and stuff behind counting systems...
RustedKitsune wrote: