When people go about making their own magical “systems”, there tend to be two popular routes they take: making their own magical language without formalizing their ontology or they create random sigils. The problem, though, is that those two popular methods alone technically don’t create a magical system. A magical system is an abstraction comprised of a body of axioms, rules, and operations. If you were to look at classical and historical “magical systems”, you will see a formally articulated set of axioms and rules; an example of this is Western occultism. The problem with informally creating your own magical system is that the strings in an informal language are unordered and thus do not consistently lead to consequences such as what the magic is supposed to do. A string is:

A string of lengthnon an alphabetlofmcharacters is an arrangement ofnnot necessarily distinct symbols froml. There aremsuch distinct strings.^{n}

Strings, in this context, include words, symbols, concepts, sigils, etc. An informal system technically doesn’t lay out any rules that allow for the formation of magical objects, so you can’t really build any consistent structures out of it. There is no formalized notion of magical objects. When you create magical symbols called sigils by crossing out letters and rearranging lines and such, you are only changing the symbol for a concept. It has the same meaning, conceptually, as if you did not turn it into a sigil. To make your own magical system, you need to come up with a formal ontology. Having a set of valid and sound axioms creates a consistent theory for your magical system. The easiest route to go about creating a solid theory is to use equivalence classes and equivalence relations as a backbone.

# Division and Modulo Operations

Say that you were given an equation: 3k=9. How would you solve it to find k? This one is easy, you would solve by dividing both sides by 3. You would have 3k/3=9/3. When you simplify, you get k=3. This means that 3*3=9, so 9 is a multiple of 3. What if instead of giving you the equation 3k=9, I gave you the equation 3k=10. You would do something similar: 3k/3=10/3. What does that give you when you simplify? You’ll get k=10/3. If you put that in a calculator, you’ll get something like 3.33… as a repeating decimal – it’s not exact because you have to truncate somewhere. You can write 10 as 3*3+1, so you would have 10=3*3+1. Wait… What?!

Now, in the age of calculators, people typically don’t think about how they learned how to divide. More than likely if you learned how to do Math in the West, you initially learned how to divide through long division. If you followed that process, for 10/3, you would say that the closest number less than 10 that 3 can divide evenly would be 9. Since 3 can divide 9 evenly, 9 is a multiple of 3. So, you would write 3 at the top and then subtract 9 from 10, right? That would leave you with 1. What do you do with that 1? If you were to write it as a mixed number, you would have 3 and 1/3. Remember how we got 3.33…? You can simply write this as 3 remainder 1, so 10/3=3 remainder 1. That, in turn, implies 10/3 is 3 one off. 10/3=3 remainder 1 is equal to 10=3*3+1. Does 9/3 have a remainder? It does; the remainder is 0, or 9/3=3 remainder 0 which in turns means 9=3*3+0. That’s called a modulo operation. You don’t need to do this by hand every time, because that gets complicated for complex Algebraic problems. On most scientific calculators, there is a button labeled mod. If you were to do 10mod3 in your calculator, it will give you 1, which is the remainder. It’s just important that you understand this concept because this is essential to the topic of equivalence classes and their relations.