Why be normal?- a weird turn on mathematics IA

Coming up with a topic for my mathematics SL IA was difficult. Teachers tell you to do something you're interested in or curious about, but how much does that really help? I like singing, writing, animals, and Harry Potter, but what does that have to do with mathematics? Then, it hit me. Please enjoy this example of a mathematics SL IA, and see if it can inspire you in the weird ways you can get the job done! Money in the wizarding world of Harry Potter- does it make sense?

Introduction


In our world, most money systems are based on the base 10, meaning for example 1 Euro= 100 cents, 1 pound = 100 pence, 1 SEK = 100 öre, and so on. It is simple for most people to remember, and coherent throughout the big currencies around the globe, leaving little room for misunderstanding. In the world of Harry Potter, however, it is quite different. In Harry Potter and the Philosopher's stone (J.K. Rowling) Rubeus Hagrid explains the money system to Harry;


"The gold ones are Galleons,' he [Hagrid] explained. 'Seventeen silver Sickles to a

Galleon and twenty-nine Knuts to a Sickle, it's easy enough" (p. 58).


But is it? It is obviously not organized the same way our euros and pounds are, but is their money system really organized at all? What is the simplest, most efficient way for me to express having 5 Galleons, 7 Sickles and 2 Knuts? This is something that I have wondered for years, ever since I first read the book years and years ago. Now I finally have the mathematical skills necessary to find out; is there a base that the Galleons (G), Sickles (S) and Knuts (K) are based on?  In the same way that we write 5,5 pounds (5 pounds, 50 pence) in the pound-system, can we use an equivalent to the decimal system to express the amount of eg Knuts in Galleons?


Calculations and exploration


First, I have to mathematically disprove the 10-base system.

As Hagrid said:

For the 10- base system to work, the number of eg Sickles to a Galleon must be divisible by ten, and the quota must be a natural number. On the level between Galleons and Sickles, the number of Sickles to a Galleon must be divisible by ten. 17 cannot be divisible by ten, or anything, as it is a prime number. On the level between Sickles and Knuts, the number of Knuts to a Sickle must be divisible by ten. As 29 also is a prime number, it is not divisible by ten either. It might be a given at this point that the 10- base system is not at work here, but we can also check whether the Galleon- Knut comparison has a 10- base. It does not.

Now that I have effectively proven the fact that the system is not that of a 10- base, we know that I cannot use the decimal system to express how much  5G+ 7S + 2K is in Galleons. I therefore need to figure out what the most efficient way to say it is without the use of decimals. Another thing that makes the magical money different from ours is the fact that they have three 'levels' of currency unlike the normal two. This allows for several levels of investigation:


Is there a base-correlation throughout the currency?

Are there different bases between all of the levels of currency(ex. Galleons-Knuts is different from Galleons- Sickles)?

Can we use one base to efficiently express the value of 5G, 7S and 2K?


Investigation 1: Is there a base-correlation throughout the currency?

As the number of Sickles to a Galleon (17) and the number of Knuts to a Sickle (29) are both prime numbers, there is no base for the entire money system in Natural numbers. Thereby, the first question is answered with a simple 'No'.


Investigation 2: Are there different bases between all of the levels of currency(ex. Galleons-Knuts is different from Galleons- Sickles)?