Professor Fox nodded. "Ah yes Ms. Tan. Characteristic universalities are magical properties that all numbers have regardless of the culture that cast the spell. Some curses are the same across cultures, many are not. So when young Arithmancers are attempting to bypass the curses left on ancient Nubian pyramids, if they can identify the numerical characterstics of the curse, they can attempt to find a way to bypass it using other known spells and potions and whatnot. So, in essence, the magical charactersitics of the number 7 does not change no matter what culture one is in. However, for the life of me I have yet to see a Vietnamese bat-bogey hex.

"Now then, on to the third question. 'Are numbers such as one or zero ever used in the same way runes are - that is written directly for their magical characteristics?'" He looked up at the class, "The answer is yes. There is a sub-field of Arithmancy known as iteromancy, in which witches and wizards make marks on wands and doors and add ingredients to potions and wave their wands a certain number of times. The number of different motions, ingredients, and marks if done properly can have an effect on the magic. This is why Professor Braxton carefully asks that you stir your cauldron three times and not four. Indeed, ancient wand makers used to use only ancient runes and iteromancy for wands until magical cores were discovered. Just to calm everyone's minds first however, wizards cannot simply carve the number zero into something and make the wards and magic of that place decay quickly. So everyone is quite safe here at Hogwarts. At any rate, we will be covering introductions to iteromancy next term. Again I stress that anyone who wishes to be a successful curse breaker need know Arithmancy, Ancient Runes, Defense Against the Dark Arts, Charms, Herbology, and Potions. Are there any brief questions about iteromancy?"

Erica strained her head and glanced around the class. It sure was quiet, with the exception of Nadia and a few other older students. She briefly wondered how many students were actually understanding what Fox was telling them. From the confused and lost looks of most of her classmates, it wasn't difficult to guess. Erica noted down most of what Fox had lectured and raised her hand.

"Professor, these marks that you speak of, how exactly are they made and what kind of effects can be brought upon by doing this, besides decaying quickly, as you already mentioned?"

Professor Fox held up a hand as if to slow Erica down. "Now, I understand you students are eager to learn everything in one lesson, and I appreciate it. However, as I said we will be studying iteromancy next term. If you do not understand the basics of numbers and their magical meanings, you will simply have a horrid time in future endeveors. So in short, you will learn how they can be made and the effects vary rather widely depending on what one writes." He flipped the parchment he was holding in his hands, "Now, on to the last question, then we can finally get this lesson underway. 'How does an Arithmancer use identity numbers to locate missing magical objects such as the Book of Mysteria?'

"Well, this is another sub-field of Arithmancy known as topogramancy (pronounced tahpah-grah-man-see), and a rather over-looked field I would mention. Many people don't realize that bypassing the curses on an ancient tomb is actually often easier than finding the tomb. Many ancient wizards were quite adept at hiding their tombs. Why my friend once found a tomb in Israel that had unknowingly been built over by muggle pediatricians. I myself specialized in this area for a number of years. I digress, it is suffice to say a rather advanced form of Arithmancy in which as always what you know about what you are trying to find is very important. Topogramancers often have a rough idea where something might be, say a tomb, then proceed to examine the magical properties of various candidate places and rule out those whose magical properties a) fit in with the surrounding area b) are not consistent with the properties one would expect for hiding an object. The methodology varies significantly, and I'm afraid this is one of those areas students where you have to learn about it after you leave Hogwarts. Often, like the puzzles one has to attempt many different techniques before one can discern the appropriate place. Are there any questions on this particular subject that are not 'How or what magic does an Arithmancer use to do this?' If not, I will finally begin the real lesson, composite numbers."

Erica didn't have anything else to add since her question wasn't going to be answered until next term. She simply noted down the fact and thought to herself how she was going to be able to be proficient in Charms and Herbology, her worst subjects by far. She inwardly groaned at the thought.

Professor Fox turned and walked around to the black board. He then began searching around as if for an eraser before realizing that there was of course no eraser. Shaking his head to himself, he pulled out his wand and tapped the blackboard, the contents of it then seemingly disappeared.

"Now then, on to today's lesson. We will be examining composite numbers. So, can anyone tell me which numbers between 0 and 9 are composite, and what the definition of a composite number is?"

'If you do not understand the basics of numbers and their magical meanings, you will simply have a horrid time in future endeveors.' Hence why I want to know! Ugh with this guy...

However, she kept quiet and attentive at the last question, the one she was looking forwards to hear. What you know about what you are trying to-- narrowing her eyes, Nadia wrote something on the margin and groaned out loud at receiving yet the same answer; students have to learn about it after you leave Hogwarts. What was the use of school when-- Ah, the futileness of it!

Nadia raised her hand, "A composite number is a positive integer over 1 (n>1) which is not prime(i.e., which has factors other than 1 and itself). The first few composite numbers are 4, 6, 8, 9. One (1) is a special case that is considered to be neither composite or prime.

Professor Fox nodded. "Good answer Ms. Rey. Yes, composite numbers are numbers which have more than two distinct divisors. The numbers we will learn about today are 4,6,8, and 9." He wrote the numbers then on the board. "Now, obviously, once today's lesson is complete everyone will have an introductory, basic understanding of the properties of all the individual numbers between 0 and 9. After that, should we be able to fit it in before the term is done, we will have an introduction to magical number representation. So again, I highlight that in this class we start from the basics and work our way up."
He flipped a page of notes then looked up at the class, "Now, to begin with the number 4. Can anyone guess as to what magical properties might be associated with this number?"

"The thing number 4 is more famous about is that Four (四, formal writing: 肆, pinyin si4) is considered an unlucky number in Chinese, Korean, and Japanese cultures because it sounds like the word "death" (死 pinyin si3). Due to that, in said places many numbered product lines skip the "four". Some hospitals do not have a 4th floor."

Nadia chuckled. "Four is first and foremost related to Earth. The simplest solid object — a tetrahedron — has four sides; hence four denotes solid matter in general and the Earth in particular, as the Earth is a solid object bounded by four cardinal points (North, South, East and West). The association with Earth means that four symbolizes being practical and "down-to-earth," while the fact that four is the first composite number is linked to the idea of an other-directed, "composite personality" that takes cues from different and often conflicting sources, leading to fiercely independent, "out-of-the-box" thinking. So, like five, it seems to be related to materia.

"Time is another concept strongly associated with four, on the grounds that the year has four seasons and the month has (roughly) four weeks. The life story of Jesus is told in four gospels, each of which is in turn linked to one of the four classical "elements" of fire, air, earth, and water. Other important roles of four include, the four seasons, the four parts of the day, suits of playing cards and even the four horsemen of the Apocalypse."

Professor Fox sighed and nodded, perhaps the students would interact more onc ethey got to the actual lecture. "Good points Ms. Rey, good points. However, always try and avoid culture-specific references when possible in Arithmancy. The realm of influence of cultural aspects of numbers is simply to those who believe in them. You are right however, 4 is associated with the four cardinal directions.

"For that reason, four is very associated with location. There are four pieces of information one need always know about an object to determine its absolute location: It's latitude, longitude, altitude, and time. Four pieces of information. Thus, four is a number strongly associated with space and the first number to include time, so time as well to a minor extent.
"Four is also a number of organization. If two is the number representative of human understanding, four is simply two times two, and two plus two. Furthermore, both non-prime divisors of 4 (2 and 2) lead through division to 1 and subtraction to 0. So four is strongly associated with prime and identity numbers, making it a very stable number in numerous arithmantic operations. Organizational schemes within the magical world with four parts therefore tend to be more stable and well balanced. There are four houses at Hogwarts, four directions, four arithmetic operations, four appendages, the list goes on.
"Lastly, as two is the number associated with understanding, and four is strongly associated with two, four is associated with information. Typically, while three pieces of information are enough to help one understand a pattern, four pieces are what is required to extend the pattern further, for greater comprehension." He stepped over to the board and wrote clearly on it:

4 - organization, space/location, information

"Now, are there any questions concerning the properties of the number 4 before we move on?"

Erica sat back in her chair and ran over what the professor had gone over in her mind many times before raising her hand.

"So, since the number four relates to both space and location, I'm guessing that it's properties are strongly related to the field of Topogramancy. Is this correct, Professor?

Professor Fox nodded as always. "Good point Ms. Arbon, good point indeed. Yes, for some of the more simple spells in the field all one needs are units of four pieces of information to find an obfuscated object. Indeed in common topogramancy parlance the pieces of information about an object's location are often called 'four-blocks.'" He seemed to reflect on this last comment he made rather intently for a moment. Professor Fox quickly returned his attention to the class, as if from a day dream. "At any rate, the number four is indeed important to the field. Now then, on to the number 6..."

He again drew an underlined 6 on the board and then turned to face the class. "Now, the number 6 is a composite number associated with numerous things. Before we get on to those however, would anyone be able to contribute anything interesting about the mathematical aspects of the number or theorize about its magical properties?"

Nadia raised her hand, since it wasn't a question, feeling safe. "Could it be that, composite numbers, for being composed of two numbers in divition, represent some kind of team-work? Harmony number. I have heard it does relate with that and beauty. The most common appreciation is that 6 is a most powerful alternative for 3. Like the 2 + 2 = 4, 3 + 3 = 6. More powerful conception of the body-mind-spirit trilogy, and more of a representation for "wholeness". Can this be correct?

"Lastly, Six can also denote perfectionism, as it is the first "perfect" number, in that the sum of its divisors, other than itself, is equal to itself: 6 X 1 = 6 and 3 X 2 = 6, and 1 + 2 + 3 = 6 (the next such number is 28)."

Zekke nodded and raised his hand, "And 6 is the sum of the two 3s and three is symbolic of the trinity - the triad. So if 3 is a powerful number, then 6 could be powerful for the same reason because it is comprised of two 3s. If such is the case, then would 9 be a magical number for similar reasons. Also, 6 squared is 36 and -" Then he stopped.

"Errrr... nevermind, I lost my train of thought." He gave a rueful grin and shook his head. He actually hadn't forgotten the thought, but he just realized that it probably wasn't an appropriate observation.

"Yes, good answers both of you. Now, as Ms. Rey pointed out, 6 is a mathematically 'perfect number' which means that the sum of its divisors is 6. This of course also means that the number six minus its divisors is 0. Furthermore, 6 divided by the sum of its divisors is 1. Six is therefore highly connected to both identity numbers. Lastly, as six is a mathematically perfect number it is associated with completion and fulfillment. Hence six has the characteristics of completion and stability. Thus many magical systems associated with 6 tend to seek some kind of stability.
"As was also pointed out, six is associated with the numbers 2 and 3. In this case, where three represents problem solving and intelligence, 2 represents human understanding. As a result 6 often is representative of intuitive understanding and intuitive problem solving and natural intelligence.
"As a last note, as 4 is the first number to properly incorporate time dependency, and three does not. 6 has a characteristic of timelessness as well." Professor Fox then wrote on the blackboard.

6 - natural and intuitive intelligence, stability, timelessness

"Are there any questions students?"

Erica did another glance around the classroom and didn't see anyone wanting to enter in discussions, so she raised her hand again.

"Um, Professor, could it be possible to get a little more in depth as to how the number six relates to time? I'm having a bit of trouble understanding that."

Professor Fox nodded. "Certainly Ms. Arbon. Well, the number 4 is the first number to incorporate time specifically. As I mentioned earlier there are four pieces of information one needs to know about any object, latitude, longitude, altitude and time. Thus, four and multiples of four possess a dependence on time. Whereas three is independent of time. Therefore, multiples of three have a tendency to be invariant over small time scales, whereas multiples of four tend to be variant in time. This combined with the fact that six is a perfect and stable number means that six has the property of being largely independent of time. This is different than the 'permenance' of seven. The permenance of seven makes it difficult for spells and magic associated with seven to reverse or break, the timelessness, perfection, and completion of six just means the magic associated with the number decays much more slowly than those of other numbers."
He thought for a moment then said, "An odor hex strongly associated with seven will be difficult to reverse. An odor hex strongly associated with six will remain over longer periods of time. Does that make sense?"

"Yes, I think so. It's just a lot to take in all at once, I guess."

Erica scribbled down quite a bit of notes at this point and reached over to her bag to get another clean scroll of parchment. Unrolling it, she continued until she had finished her notes up to that point and noted somewhere that further study might have to be involved.

Nadia took some notes, wondering if nine would maybe be the last to last large periods of time... if that made any sense. She faned herself with her quill as she a bit about pondered this.

Finally she raised her hand to ask a quick question. "So, in your opinion, which is more likely to decay if helped with the magic of zero, the six magic or the seven magic?"

It was an interesting concept, because while the six had more stablity to work against seven, zero was acting to break a spell and seven was more powerful to those attacks. Who would, so express it someway, win and last longer?

+[edit]+
"Responding to Ms. Rey's question, it is not quite that simple. I will do my best to answer it nonetheless. The zeroeth law indicates all magical systems tend to lose potency if left unattended, excluding certain exceptions. What you are confused about is that we are describing magic spells, not making them. A spell represented solely by a 7 will be difficult to break or even reverse, a spell associated solely with 6 will have less dependency on time. A spell intended to break or bypass another spell will work just as easily on a spell associated solely with 6 as with 4 or 2. Indeed, a spell associated with 7 is largely a 'one time deal' as it were, and if properly maintained can last indefinately, if not it too will lose potency in time.
"If say, a spell represented only by 6 transformed you into a piece of toast, it would slowly lose potency and by the time the toast had decayed and rotted, it would be too late. But, it would be easy enough to reverse it. If however the same spell were solely represented by a 7, it would slowly lose potency though faster than the first case, and again by the time the bread rotted, it would be too late anyway. Though in this case it would be rather difficult to reverse the spell. Does that answer your question?"
+[edit]+

Professor Fox flipped his notes and continued his lecture. "Now then, moving along to number 8. The number 8 is the only basic composite number which can be represented as a cubed number. That is, 8 can be expressed as 2x2x2. As such, just as 4 was characteristic of more complicated thought, 8 is characteristic of highly complex thought process. "Super-logical" systems such as those which ar enot altogether mystic or mysterious but complex nonetheless, can be represented by eight for this reason. Examples might include systems which govern many different charms all at once, say for a tomb.
"Furthermore, just as four is the number of pieces of informaiton one needs to locate that object, eight is the number needed to plot a course to or from an object or place. 'Eight-blocks' in advanced Arithmancy therefore refer to sets of information which dictate a course between two points. Again, I make note that the given course from one object or place to another may change in time. As such eight is characteristic of navigation and movement. For similar reasons the number is largely dependent on time and has a certain characteristic of being fluid, mutable, and somewhat flighty. Indeed, I'm not certain if it would be entirely proper to mention this here..." He looked around a moment then continued, hurrying along to the next subject. "Many Arithmancers have claimed that Veela magic is strongly associated with the number 8."
Professor Fox then flipped the parchment again and continued reading. "Finishing the lecture section, we will move directly to the number 9. I apologize for hitting everyone with both of these at once, but everyone seems a little eager to go. Now then..."
"The number 9 is the square of the number 3, being represented as 3x3. It therefore is representative of very advanced thought processes, as well as concepts difficult for humans to comprehend. Many divination systems are strongly associated with this number. Many advanced spells have a 9 buried somewhere in their core set, but that's another lecture entirely.
"Also, the number is representative of the highest value in the basic numbers. It therefore represents strength, power, and potency. However, the number itself has little relation to identity numbers and has a tendency to be less stable than either 3 or 6. Magic associated with 9 therefore tends to need other magic to help stablize it.
"Furthermore, the number 9 is strongly associated with the idea of growth and as such with many plants, potions, and magic that slowly builds up and at some point begins to decline." Professor Fox then wrote on the blackboard.

8 - complex thought/reason, fluidity, movement, navigation
9 - power/strength, growth, very complex thought/mysticism

"Are there any questions?"

"Professor, to return to the properties of 7, would the Avada Kedavra curse be associated with the powers of 7 since it is not reversable?"

Professor Fox raised an eyebrow at this odd question. "Well Ms. Weasley, again, when representing spells by numbers often the spells are not simply 1, 3, 7, or 0 but numbers like 9059 or what have you. The issue is how the different numbers interact with one another as we mentioned earlier in class. We will not be discussing the representation of unforgivable curses in my class as the only thing you students need know about them you will learn in defense against the dark arts as Headmistress Rae finds appropriate. Suffice it to say, a spell of that nature has a somewhat odd arithmantic representation. Now, are there any other questions students?"

Nadia frowned as she wrote down, then grimaced as she raised her hand again, "Well, not completely. I understand the concept of decay means in time."

So it was only understandable her doubt, "If a spell could speed up the decaying, by force of the zeroeth law, and seven is strenght while six is non-decay or slow decay, which is the one to decay first? I'm still doubtful." Maybe it dealt with how strong the spell represented by zero was.

In the same topic, she was curious if nine would be the most easily influenciated by the zeroeth rule. She would address the rest of her doubts later ( ).

Professor Fox sighed quietly to himself. He nodded grimly as if almost in defeat, thinking to himself that he will from now on spend a single class period on a single number. "Yes Ms. Rey, I understand your confusion. There is no 'zeroeth law spell,' the zeroeth law is simply a statement of the way magic works. A wizard could no more do what you are suggesting than they could resurrect Ben Hurr. Unfortunately, there are limits to even what wizards can do, and this is one of them. Thus, the statement 'if a spell could speed up the decaying, by force of the zeroeth law' is a moot argument. The 'zeroeth law' is a natural magical property not a force. A spell which causes other magic spells to decay does not 'use' said law, it simply causes magic to decay. There are numerous spells of that nature, each of them operates and behaves slightly different than the rest, but they too are bound by the same laws which govern all spells, thus there are ways to prevent that kind of sinister motive.
"To illustrate the point, there are a number of different ways and spells to locate an object. Each of them may be described partially by the properties of four as well as eight and other numbers, but they are charms and spells nonetheless. As such, they too can be overcome by other spells. There is no such thing Ms. Rey as a 'perfect spell' which cannot be overcome, all spells are made by humans, and therefore all spells are not perfect." Professor Fox sighed, looking rather tired. He somehow thought he had managed to just make the students come up with some new thing to misinterpret. Again, sighing in heavy resignation he paused for a moment to enjoy the silence of the classroom before inevitably and regretably asking "Are there any further questions?"

Totally confused by what the professor has tried to explain, Terry raises her hand to ask a dumb question. "If I follow what you are trying to explain, professor, the numbers we have been studying magical characteristics represent the properties of magic. If that is the case, do the numbers have any magical powers of their own?" Terry's head is starting to pound. The more Professor Fox tries to explain thing; the more confused she seems to get.

Peony winced, holding her head between her hands. God, I hate numbers. Her head really did hurt. All the words, spilling out... Ye Gods. And Professor Fox wasn't exactly warm and fuzzy when it came to students asking questions. Sometimes he got a look on his face that went something like: Oh God, I wish I could just slip these overactive little brats a potion to make them sleep till December. There were times she really wished he would do just that too.

She didn't understand how numbers were woven into spells, except maybe in potions, seven scales of a newt - or whatever.