# Mathematics

### From Mizahar Lore

**Mathematics** is the science of quantity and relationships between quantities. Mathematicians find structure and patterns in numbers as well as physical reality.

At lower levels, math helps perform basic operations with numbers; at higher levels, the user becomes a full-fledged researcher capable of discovering new laws and theorems. Mathematics has limited use on its own, but becomes a precious support skill in a variety of fields, ranging from construction and engineering to record keeping and even several branches of magic, such as Alchemy.

## Contents |

## Prerequisites and Related Skills

Mathematics has no prerequisites; however, training from an academy or academy alum greatly enhances your ability to learn mathematics. It is not impossible to learn mathematics directly from books and other texts though. Despite the possibility of learning mathematics on your own, nearly every single skilled mathematician is academy trained.

As mentioned above, many fields are greatly augmented by math. Some even need mathematics in order for a practitioner to operate at all. Areas that most heavily require mathematics are Architecture, Construction, Physics, Sea and Land Navigation and Gadgeteering. Some other areas that utilize math to a lesser extent are Drawing, Medicine, Philtering, Cooking, and essentially any other area that involves measuring dimensions or quantities.

In addition to the countless non-magical applications for mathematics, magic can also greatly benefit from the application of math.

## History

Mathematics was donated to mankind by Eyris, goddess of knowledge, shortly after Qalaya had gifted them with writing. Rather than giving them all the facts, however, she provided them with a few basic axioms - statements that are considered to be true - and ordered people to derive all the laws of the world.

The Axioms were given to scholars of both Alahea and Suvan, however it was Suvan that truly utilized the axioms. Despite this, Alahea receives a vast majority if not all of the credit for mathematical strides; however it was actually Suvan that initiated the progress of math. The Suvan were excellent engineers, and were the ones to devise algebra, geometry, trigonometry, and calculus. However, the Suvan had no use or need to develop any theoretical mathematics. In fact, the concept of theoretical mathematics had not even been considered until the Alaheans stole the four major subjects of math from the Suvan. The discipline of math had been reinvigorated within Alahean borders, and it was not long before magi began intertwining mathematics with philosophy. It was for this reason that Alahea is credited with advanced mathematics.

There were ten Axioms of given by Eyris, they are as follows:

- 1 - The whole is greater than the part.
- 2 - If equals are added to equals, then the wholes are equal.
- 3 - If equals are subtracted from equals, then the remainders are equal.
- 4 - A straight line can be drawn between any two points.
- 5 - A line segment can be extended indefinitely.
- 6 - All right angles are congruent.
- 7 - Things that equal the same thing also equal one another.
- 8 - Things that coincide with one another equal one another.
- 9 - Given any straight line segment, a circle can be drawn having said segment as a radius and one endpoint as center.
- 10 - If a pair of lines are drawn which intersect a third in such a way that the sum of their interior angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side upon sufficient extension.

Collectively these laws are known as Eyris’ Axioms, and most are common knowledge among anybody having attended an academy or had some formal training in mathematics.

### Logic

While much of early mathematics was derived from Eyris' Axioms, there was an entire branch that had been found lacking in the area. That of logic. Eyris had the knowledge and the passion, but it was Gnora that inevitably instituted logic within the area of mathematics. This has been a point of contention between the two deities. Eyris begrudges Gnora's addition of the laws of logic, even though she recognizes it as necessary. Gnora, remains firm on her position that it was a necessary addition. Though Eyris secretly agrees, one would be hard pressed to get her to agree with her emotionless counterpart.

The method with which Gnora presented logic was different. Instead of Axioms, she gifted the world with a single book. Logikos, the title of the book, is an incredibly complex manuscript detailing the many forms of logic. The book is designed as such that it needs to be read many times over to comprehend it fully. After reading through the book, you will have grasped certain concepts. When you read through it again, you will understand new nuances Gnora worked into the text. The exact contents of Logikos are unknown at the moment, as it was lost during the Valterrian. However, many of the books that stemmed from Logikos can be find, but they are rare. The greatest known compliation of logic texts can be found within the Academy of Zeltiva. Many scholars who are aware of Suvan's mathematical proficiency have searched extensively for Logikos and related texts in ruins; however none have revealed themselves. The lack of texts on Logic in Suvan ruins has led scholars to believe that the Alaheans were the primary developers of logic, as opposed to the Suvan.

### Development of Mathematics

There have been countless formulas stemming from the Eyris’ Axioms, and as many laws of logic from Logikos; although most were from before the Valterrian. While many of these have been erased from the occurrences of the cataclysmic time, some have been salvaged from ruins. Even among those salvaged, some are only pieces of larger works, and even more are simply useless to the average post-Valterrian mathematician. Perhaps one of the most famous of these pre-Valterrian mathematicians was Ramisis (Rah-Me-Sés). His theorem is one of less than fifty pre-Valterrian formulas that is used today. The theorem that Ramisis proposed is as follows:

“In any right triangle, the area of the square whose side is the hypotenuse is equal to the sum of the areas of the squares whose sides are the two legs.”

Much like the axioms given by Eyris, these formulas derived from mathematicians led to new and more complex theorems and idea. For example, Ramisis' Theorem allowed later mathematicians to discover real numbers as it showed that the ratio between the side of a square and its diagonal cannot be expressed as a fraction of integer numbers. This is largely the entirety of mathematical progression, newer mathematicians building off of previous works.

Ramisis was just one of many theoretical mathematicians of the pre-Valterrian era. Back when magic and science were at its peak in Alahea, theoretical mathematicians were not an uncommon occurrence. These were often the magi and scholars, or less often engineers, of Alahean society. However, since the Valterrian, theoretical mathematics has become a lost science. Of all the mathematicians since the Valterrian, an overwhelming majority have only been applied mathematicians – someone who uses math to bolster other fields.

The height of mathematics was reached in late Alahea, when the subject was a prominent part of the curriculum of paramagical Studies at the Imperial Academy of Magic. Most of its finer points have been lost with the Valterrian, but scholars salvaged what they could. With the forced shift towards practical application brought by the cataclysm, Mathematics and Logic is a niche field at best; finding a high-level practitioner is a mark of impressive education.

## Present

Mathematics is currently a recovering area of study. A majority of people with a mathematical background are the tradesman and engineers of the world. Most occupations involve basic measuring of various things. For example, a cook that cannot accurately measure ingredients will be a poor cook. A seamstress that fails to measure her customer’s dimensions will not stay in business long. So despite its relative niche status at higher levels, basic math is still employed in everyday life in post-Valterrian Mizahar. To this day, archaeologists and explorers are discovering caches of knowledge regarding math. Of course, these finds are relatively rare considering they are of much less interest and import to the explorers than say, a weapon.

## Skill Progression

Note that since every Mizaharian has a basic education, things such as measurement and basic operations are within everybody's grasp. However, the more advanced concepts under the novice ranking come with training.

Mathematics is used in many different disciplines, but not always in the same forms. When one begins training in mathematics, they choose between one of three areas of math to focus: Algebra, Geometry, or Trigonometry. At each new rank of math, another area is chosen to augment their understanding. Upon achieving Expert in mathematics, the mathematician can choose both the last of the three and calculus.

The same system as above applies for the mathematician's logic progression. At each rank gained, the mathematician chooses another branch of logic. The only restriction is that one must have grasped Symbolic Logic before attempting Mathematical Logic. As well, though not required, it is recommended that one take Formal and Informal Logic in succession. The succession is mildly different though, in that there is only one new branch of logic understood per rank, even at master.

At each level of math, a mathematician has achieved a skill level to be able to work with a new type of number set confidently. These number sets are detailed as "Number Proficiencies".

Novice (1-25) | |

Number Proficiency: Natural numbers (Including 0)
A novice level mathematician usually has memorized all of the ten postulates. However, their understanding of said postulates is not guaranteed. The advancement of a novice mathematician is usually in regards to their abilities with basic mathematical operations such as addition, subtraction, multiplication, and division (e.g. the mathematician will be able to multiply double or triple digits, divide by an imperfect divisor, etc...). At this level, the mathematician has grasped the basis of one numeral proficiency, one logic branch, and one domain of mathematics. However, the comprehension of anything beyond basic operations and the numeral proficiency requires ranks in Mathematics. This is the most common level of mathematician, and the ability to devise proofs or theory is beyond them. | |

Competent (26-50) | |

Number Proficiency: Integers and Rational Numbers
At this stage, all basic operations (add, subtract, divide, multiply) have been mastered by the mathematician. The mathematician of Competent ranking has mastered the areas of logic and math that he took up at novice level, and begun to explore the demesnes of two new areas. Theoretics are still beyond this level of mathematicians reach. However, at this stage the mathematician can begin to understand the patterns inherent in math. | |

Expert (51-75) | |

Number Proficiency: Real Numbers
Mathematicians of expert level are at the theoretical level of mathematics. They have a thorough understanding of every mathematical area: geometry, algebra, trigonometry, and calculus in all its applied functions. In addition to the areas of applied mathematics,another branch of logic is available to the mathematician. They have begun to see the inherent patterns and connections in the world, that everything can be represented with mathematics. Although they can understand these things, they do not yet have the skill to apply this information. As well, at this level the mathematician knows basis of most of the theoretics that have been devised, and can create simple formulas of his own. As well, concepts such as real numbers are grasped at this level. | |

Master (76-100) | |

Number Proficiency: Complex Numbers
This is the level of mathematics where the world takes on a new light. Mathematicians of this level have mastered virtually everything in the area of applied mathematics, and almost everything within theoretical mathematics is known to them. The final branch of logic is mastered at this level as well. Concepts such as imaginary and complex numbers are easily within the grasp of a master level mathematician. The formulas a master mathematician creates are revered by academia, and often of staggering complexity. At this level, a mathematician can not only see and explain the mathematics inherent in everything, but he can apply them and devise formulas to represent them. Master level mathematicians are nigh unheard of in present day Mizahar, but the few that are this accomplished are literal celebrities of the academic world. |