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)?

We could, between just two 'stages' of our money, use a system similar to the hexadecimal system (base 16, using 1-9 and A-F). The Hexadecimal system replaces the binary code in computer programming since one hexadecimal replaces four binary digits. It is therefore much shorter and more comfortable to write and understand, as presented below:


This shows how the Hexadecimal system works compared to the decimal system and the binary system.

For example:

In Sickles, the base would be 17, and we can then use 1-9 and A-G for Sickles (sixteen 'numbers'). This would allow us to express part of the value of our 5G, 7S and 2K:














Presented above is the decimal system compared to our Sickle based system (sik).

This system means that 5G and 7S would be expressed:


57 would then be the value of our money in Sickles, excluding the Knuts.

This system can also be used in the reverse. If I want to express in the Sickle base for example, I could do this:

We could manufacture a similar system to express the value of the 7 Sickles and 5 Knuts:

Presented above is the decimal system compared to the Knut system (knu)

With this system, we could express the Sickles and Knuts like this:













75 would be the value of the money in Knuts, excluding the Galleons.

In answer to our second question, there are different bases between the levels of currency, as

Investigation 3: Can we use one base to efficiently express the value of 5G, 7S and 2K?

We now want to find a base that can combine the Knut system and Sickle system. To do that, we need to find their LCM (lowest common multiple) and see if we can use that base to represent all. As both numbers (17 and 29) are prime numbers, to find their LCM we must simply multiply them by each other:

To make a base system using the '_493', I would need 493 separate symbols to express each digit. I would then have to use 1-9 (9 digits), the latin alphabet lower case (a-z, 26 digits) and uppercase where they differ (A-z, excluding c, f, j, k, o, p, s, v, w, x, z; 15 digits), the greek alphabet (24 digits), the unique letters from the Gothic, Cyrillic, Coptic, Armenian, Georgian, Glagolitic, Arabic, Hindu(…) alphabet.

It would take many different alphabets to fill the 493 slots. The memorizing of the specific value for each unique letter would take a very long time and would be extremely difficult. The possibilities for misunderstandings are of such magnitude that it would make life very complicated, if you were to express the contents of your money pouch in this way.

The answer to the third investigative question, there is no base system that can be efficiently used to express the value of 5G, 7S and 5K


Thoughts


My aim with this exploration/investigation was to find the simplest way to say 5 Galleons, 7 Sickles and 2 Knuts. I've found a fairly simple way to express 5 galleons, 7 Sickles with the Sickle system, and one to express 7 Sickles, 5 Knuts with the Knut system. However, since their LCM is so high, to express my entire 'fortune', I would need to use a base system of 493, but that base system would only make it more difficult to express the value.

The easiest and most efficient way to express the value of my money would therefore be to simply say "I have 5 Galleons, 7 Sickles and 5 Knuts".


Bibliography

Mathematics for the international student-  Mathematics HL (Option): Discrete Mathematics (Haese Mathematics, First Edition  2014)

Chapter 1: number theory

Harry Potter & the Philosopher's stone (J.K. Rowling, Bloomsbury, 2013)

quote from chapter five, "Diagon Alley", p. 58

https://en.m.wikipedia.org/wiki/Hexadecimal?fbclid=IwAR3rQHK0IZavUZZtwckHQ1FgwrIK9CrZC0Y1JqXTTQRxEIJvW4LyMlaTeQE (11 february, 2019)

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