Reintroducing Arithmetic
Addition, Its Derivative Operations, & Properties of Those Operations
I have entitled this section "PreMath" because most of the topics we will cover in this course are not "math" as a mathematician might consider it; rather, these topics are the rudiments that are required to do true mathematics, namely proof and less conceptually concrete problems. Many of the topics in this course will be familiar to most who have gone through a public school mathematics curriculum: prealgebra, algebra (I and II), Euclidean geometry, Trigonometry, and other topics in "precalculus." Additional to these traditional topics, we will discuss topics in Number Theory, Combinatorics, Graph Theory, and Game Theory.
For most of the past 150 years, mathematics has been taught with calculus as the seeming apex of understanding, causing all course materials to work towards producing a student extremely proficient in doing calculus. This model has many historical reasons for existing, but it need not be this way and perhaps should not be this way anymore for similar historical reasons. Thus, though this course will certainly prepare one for doing calculus, it will also give them many other mathematical tools, perhaps not useful to calculus, but other modern and more relevant fields of mathematics that the student will study in later years at Aegis Institute.
In many ways, arithmetic is not math as it is understood today. Rather, arithmetic is a tool used in mathematical thinking. The term got originated from the Greek word “arithmos” which simply means numbers. We will consider this our definition of arithmetic:
Arithmetic is the elementary branch of mathematics that specifically deals with the study of numbers and properties of traditional operations like addition, subtraction, multiplication, and division.
While we are familiar with four primary operations in arithmetic – addition, subtraction, multiplication, and divisions – it is important to realize and understand that all these operations are merely variants of addition. Let’s examine this claim by first intimately examining each of the arithmetic operations.
Besides the traditional operations of addition, subtraction, multiplication, and division arithmetic also include advanced computing of percentage, logarithm, exponentiation and square roots, etc. Arithmetic is a branch of mathematics concerned with numerals and their traditional operations.
Addition and Arithmetic Operations:
Properties of Arithmetic:
The main properties of arithmetic you must know are:

The Commutative Property

The Associative Property

The Distributive Property

The Additive/Multiplicative Identity Property

The Additive/Multiplicative Inverse Property
The identity and inverse properties you have already encountered in our discussion of additive and multiplicative inverses and identities. So, what we really need to focus on are the first three.