As you wait outside the Arithmancy classroom, listening to the general chatter of your fellow sixth-years, a tall man in a dark red sweater comes hurrying down the corridor. “Those staircases…” he mutters to himself before he unlocks the door, throws it open and steps aside. “Alright, let’s get to work. Find a seat and make yourselves comfortable. We’ve got a lot to cover this year,” he addresses the assembled group with a wide smile. He waits for everyone to file into the room before he steps in and locks the door once more. 

Introduction

Welcome to your first Arithmancy lesson. Throughout the next two years, we are going to cover a myriad of tools that will help you use numbers to modify the future and uncover hidden truths that just need a little digging. Differently from most forms of Divination, the beauty of Arithmancy lies in how direct interpreting numbers can be: you won’t need to deal with vague images and faint predictions, but values that you can count, add and from which you can extract meaning by applying precise formulas.

Divination usually relies a lot on your Inner Eye, and those who have that special ability can often ace the class without much effort. In my classroom, hard work and interpretation of what is right in front of your eyes will be needed - you can’t just rely on your innate skills to thrive over here. As always, my office will be open in case you have any questions or need some extra practice with your calculations.

Having said that, let us cover some of my class procedures and future plans before we delve into the actual content for today’s lesson.

Class Procedures

In order to give you the opportunity to apply your newfound knowledge in a meaningful way, all lessons will have a blend of theoretical and practical assignments. Practical assignments may take any format - including quizzes, essays or any other type of alternative submission that will require you to engage meaningfully with the concepts explored.

When giving the answer to an Arithmantical question, you may either express your answer directly or show your work step-by-step. If you choose to do the former, you will get marks exclusively based on your final result: either full marks or no marks at all. Conversely, if you opt to break down your work step-by-step, you may be entitled to partial credit based on how correct your thought process was. For that reason, I strongly urge you to show your work at all times - not only is this good practice, but it may also increase your grade in the long run.

For essays, detailed rubrics will be given directly inside the assignment. I should also make a note that plagiarized or improper content will not be accepted - if my PAs or I come across a submission that was directly copied from the lesson, an external source, or otherwise goes against the rules we have here at Hogwarts, a 0 will be given with no chance to resubmit.

Furthermore, if English is not your primary language or if you have a learning disability, you may, respectively, add the codes [NES] or [LD] to your assignments. By doing so, you will be granted exemption from any spelling or grammar rubric element, although you’ll still be held to the same standards when it comes to content!

Having said that, here’s our content for the entire course:

Year One - Basics of Arithmancy

Lesson 1 - Basic Magical Meanings of Numbers
Lesson 2 - Classes of Numbers and Their Properties
Lesson 3 - Basic Numerical Operations
Lesson 4 - Numerical Systems
Lesson 5 - Personality Numbers
Lesson 6 - Advanced Numerical Operations
Lesson 7 - Ratio and Proportion / Important Ratios and Their Meanings
Lesson 8 - Notable Arithmancers and Discoveries
Lesson 9 - Symbolic Representations

Year Two - Applications of Arithmancy

Lesson 1 - Combat Arithmancy
Lesson 2 - Date Selection
Lesson 3 - Arithmancy and Engineering
Lesson 4 - Arithmancy and Brewing
Lesson 5 - Spell Development
Lesson 6 - Persuasive Speechmaking
Lesson 7 - Art of Naming
Lesson 8 - Hidden Numbers All Around Us
Lesson 9 - Historical Arithmancy

Why Study Arithmancy?

Most people seem to think that Arithmancy is not a particularly useful field of magic - that it’s just a branch of Divination, or something that’s nothing more than a novelty to be played with. These people are wrong; Arithmancy is what differentiates the good magic users from the very best, particularly when applied in the correct way.

It’s true that Arithmancy is neither a field used by itself, nor something that you need to know in order to be minimally adept at your elected future career (this being the reason why it’s an elective course) - and yet, when you look at those who thrived in their chosen fields, it’s clear to see how Arithmancy had a lot to do with their success. I can tell you that the numbers added to the most famous broom models - such as the Comet 260 or the Nimbus 2000 - or the chosen wand movement proportions in effective spells are not random, differently from what one may think at first.

With time, you too may make use of the power of these numbers and harness the strength that links us to the basic rules of this universe. Numbers are naught but a representation of all that connects us to the most basic level of understanding, and this is why they are so effective in adapting everything we know.

That, my attentive students, is why Arithmancy will be useful regardless of what path you decide to take.

Number Sets

Now that we have laid down the backdrop for our course, let us begin to explore more practical aspects of our content. Today’s class will focus mostly on a basic foray into what numbers there are, as well as their magical meanings.

Before we begin, though, we need to discuss the concept of number sets. Simply put, a number set is any group of numbers, which may be selected manually by a person or according to a specific rule. Number sets are represented by curly brackets containing all of its constituent numbers (also called elements) separated by commas - for example, the notation {3, 7, 10, 12} represents a number set containing four elements. These elements are, respectively, the numbers 3, 7, 10 and 12.

Conventionally, number sets are written in order, with smaller numbers placed to the left of larger numbers. Repetition of numbers within the set, likewise, is usually discouraged; however, nothing prevents you from breaking these rules if needed - for example, if you want to create a set containing a score for all students, it’s absolutely fine for you to repeat the same number in case two assignments were given the same grade.

At this point, I’d like to focus on the most common number sets. As I said before, sets can be either defined manually or based on specific rules. The sets I will introduce below belong to the latter category: they are defined exclusively in accordance with a basic rule.

1. Natural Numbers - The natural number set contains all of the numbers you use to count whole things - any number that may be used to illustrate a full quantity, such as 7, 19 or 612, belongs to this number set. Another way to describe this set is as the set of positive integers.  It should be noted, however, that 0 does not belong to this group (it is neither negative nor positive) - rather, it belongs to the set called…
2. Whole Numbers - Take the natural numbers group and add zero to it. This is the whole number set, so-called like this because it contains all quantities that exist naturally without having any lingering broken bits. You can also think of this as a set of all non-negative integers (i.e all positive integers and zero). 
3. Integer Numbers - Sometimes, we may need to use negative numbers, which represent something that is missing from the world rather than a physical presence. For instance, if I want to indicate that I owe you 16 Galleons, I may use the number -16 to show that I have a pending debt that needs to be solved. The integer set contains every whole number as well as their negative opposite, but it still does not contain any fractional parts.
4. Rational Numbers - This set contains the result of any division made using two integers, which means that broken bits may already be found in this set. For example, the number 0.125 belongs to the rational number group, as it can be represented by a division between two integers (1/8). However, even though every broken number that has an end is a rational number, not every rational number has an end to it - as an example, the number 0.333… is still considered a rational number, even though it never ends. What matters is whether the number can be represented by a division between two integers (0.333… may be represented as 1/3, which makes it rational).
5. Irrational Numbers - If a number can exist in reality but not be represented by a division between two integers, then it is an irrational number. For example, some of you may be familiar with the number pi (π), which represents the ratio between the circumference and the diameter of a circle. The number π, which is approximately equivalent to 3.14, has no end and cannot be represented perfectly by a division. Therefore, it is irrational. 
6. Real Numbers - Get the rational number set and the irrational number set, and put them all inside the same group. That’s all there is to real numbers.
7. Imaginary Numbers - Understanding imaginary numbers requires us to have knowledge of facts that will only be explored in a future lesson, so let me explain this group in very basic terms for the time being. Simply put, there are a few numbers that cannot exist in reality because no real number can generate that specific result. Therefore, in order to account for these numbers that can’t exist in the real world, a separate group - aptly named imaginary numbers - was created. The most important imaginary number is i, which represents the square root of -1 (square roots and other related operations will be discussed in a future lesson).
8. Complex Numbers - Get the real number set and the imaginary number set, and put them all inside the same group. That’s all there is to complex numbers. This set also includes numbers that are not fully real, nor fully imaginary: for example, 2 + 3i is a complex number, but it’s neither real nor imaginary.

For those who prefer visual references, please take a look at the nifty diagram at the front of the class.

One thing that you should note is that all of these sets are infinite: one cannot answer what is the last natural number or give the average to all real numbers. However, some infinities are larger than others - it’s true that the natural number set is infinite, but there are more real numbers than natural numbers out there! It’s impossible to pair every real number with a natural number, making the first infinity larger than the second one.

Numbers And Their Magical Meanings

Now that we know what a number set is, let’s take a closer look at some special numbers. Naturally, given the fact that all standard number sets are infinite, there’s absolutely no way for us to cover every number in this course - and much less in our first class, no matter how hard I try! For that reason, we will only go over the basic digits in the decimal system in this class.

Zero (0) - As you may expect, zero represents nothingness, or the vacuum before creation. For that reason, zero is a number with a lot of potential, as it represents the beginning of a new adventure and all the possible changes that we may experience.

One (1) - One represents unity and creation, the first starting point from which everything else must surface. If zero represents the point before the beginning, one represents the first step in this journey and the decision to make something new.

Two (2) - Two represents duality and opposition, as many concepts come in opposite pairs: think about light and dark, day and night, hot and cold as a few examples. This idea of complete opposites is a strong aspect of the number, as well as the idea of joining two distinctive partners.

Three (3) - Three represents stability and the junction of physical, emotional and psychological aspects of humanity. Those who ascribe to religious beliefs may also perceive three to be a spiritual number, as it may be associated with the three natural realms (land, sea and sky) as well as many other concepts in different religions.

Four (4) - Four represents the material world, the world of tangible possibilities, as it is connected to the four classical Aristotelian elements (fire, air, water and earth), as well as the four cardinal directions and the four seasons. Additionally, four may also be connected to a feeling of homeliness and comfort.

Five (5) - If four means stability, five means creativity and spontaneity. In the same way some consider ether to be a free-flowing element (lending credence to a looser five-element conception), the number five may also be seen as a creative twist of the rules that guide this world. However, this freedom may also lead to conflict and struggle at times.

Six (6) - Six represents accountability, care and the process of healing, usually through a patient and understanding approach. In the same way that three represents the entirety of a human being, it’s no surprise that double that number represents making a person whole once more - explaining the properties borne by this digit.

Seven (7) - Ah, the quintessential magical number. Perhaps many of you already know that Bridget Wenlock was able to uncover the properties of this digit, which are intrinsically connected to the formation of magic itself. Furthermore, seven also represents intuition, luck and light (as represented by the seven colors of the rainbow).

Eight (8) - This number represents drive and passion, and the pursuit for self-improvement and achievement. It’s no coincidence that an eight, when rotated ninety degrees, can represent the concept of infinity - this number is deeply connected to the constant flow of the world and the battles that we fight therein.

Nine (9) - Nine represents the process of completion as well as the end of things, as it is the digit that comes immediately after a new cycle with zero - it’s no surprise, therefore, that classes here at Hogwarts is Here all end after the ninth lesson. Additionally, nine may also represent cooperation (owing to its formation as three times three) and both abundance as well as generosity (owing to the fact it is the digit with the highest value).

Closing

You may have noted that I mentioned the use of the decimal system in this class. The decimal system refers to the number system in which every movement to the left represents a multiplication by 10 - for example, the digit 1 in the number 169 is in the third slot from the right to the left, which means that it actually represents the value 1 * 10 * 10 = 100. There are other numerical systems, such as binary and hexadecimal - however, these are outside the scope of this course as they relate to more advanced forms of Arithmancy. Nevertheless, you are free to ask me questions on the topic if you wish.

Dismissed.

Original lesson written by Professor Vaylen Draekon
Improvements made by Professor Epona Salvatrix and Professor Calum Buchanan
Image credits here, here, here, here and here