Algebra Essentials

Algebra is the generalization of arithmetic that introduces variables and algebraic operations. This ability to manipulate algebraic structures is relavant to many branches of mathematics and empirical sciences.

Types of Numbers

Name	Symbol	Set	Description
Natural Numbers	$\mathbb{N}$	$\{1, 2, 3, ...\}$	Counting numbers
Whole Numbers	$\mathbb{W}$ or $\mathbb{N_0}$	$\{0, 1, 2, 3, ...\}$	Includes zero
Integers	$\mathbb{Z}$	$\{..., -3, -2, -1, 0, 1, 2, 3, ...\}$	Includes negative numbers
Rational Numbers	$\mathbb{Q}$	$\{\frac{m}{n} \mid m$ and $n \in \mathbb{Z}$ and $n \neq 0 \}$	Can be represented as a fraction
Irrational Numbers	$\mathbb{I}$	$\{h \mid h \notin \mathbb{Q}\}$	Cannot be represented as a fraction
Real Numbers	$\mathbb{R}$	$\mathbb{Q} \cup \mathbb{I}$	Both rational and irrational numbers combined

Fig. 1 - Real Number Set

note

There are various sets that are a subset of another. These include: $\mathbb{N} \subset \mathbb{W} \subset \mathbb{Z} \subset \mathbb{Q}$

Types of Decimals

Fractions when converted into decimals come in two forms...

1. A terminating decimal has a defined terminating point ($ex.\: \frac{15}{8} = 1.875$).
2. A repeating decimal has no terminating point ($ex.\: \frac{4}{11} = 0.36363636... = 0.\bar{36}$). A bar is used to denote that those numbers repeat forever.

Order of Operations

The order is as follows...

1. Paranthesis
2. Exponents
3. Multiplication or Division
4. Addition or Subtraction

Properties of Real Numbers

Property	Definition
Commutative Property of Addition	$a + b = b + a$
Commutative Property of Multiplication	$a \times b = b \times a$
Associative Property of Addition	$a + (b + c) = (a + b) + c$
Associative Property of Multiplication	$a(bc) = (ab)c$
Distributive Property	$a \times (b + c) = a \times b + a \times c$
Identity Property of Addition	$a + 0 = a$
Identity Property of Multiplication	$a \times 1 = a$
Inverse Property of Addition	$a + (-a) = 0$
Inverse Property of Multiplication	$a \times (\frac{1}{a}) = 1$

Algebraic Terms

These are commonly encountered terms in algebra...

1. A constant is a value that does not vary ($ex.\: 5$).
2. A variable represented a value that can vary and is often represented using an alphabetical letter ($ex.\: x$).
3. An algebraic expression is a collection of constants and variables joined together by algebraic operations ($ex.\: x + 5$).
4. An equation is a mathematical statement which indicates that two expressions are equal ($ex.\: 2x + 1 = 7$).
5. A formula is an equation expressing a relationship between constants and variable quantities ($ex.\: A = \pi r^2$ which is the area of a circle).