Fundamentals - Theory & Concepts

Fundamentals

Algebra serves as the language of mathematics, providing a structured way to represent and solve problems using symbols and rules. Before tackling complex equations, it is essential to master the foundational principles of arithmetic, the properties of real numbers, and the logic behind manipulating expressions. This section builds the groundwork for all subsequent algebraic topics.

Sets of Numbers

The real number system consists of several subsets. Understanding these sets helps in identifying the domain and nature of solutions.

Number Systems

Natural Numbers ( $\mathbb{N}$ ): Counting numbers {1, 2, 3, $\dots$ }.
Whole Numbers ( $\mathbb{W}$ ): Natural numbers plus zero {0, 1, 2, 3, $\dots$ }.
Integers ( $\mathbb{Z}$ ): Whole numbers and their negatives { $\dots, -2, -1, 0, 1, 2, \dots$ }.
Rational Numbers ( $\mathbb{Q}$ ): Numbers that can be expressed as a fraction $\frac{p}{q}$ where $p, q$ are integers and $q \neq 0$ . This includes terminating ( $0.5$ ) and repeating ( $0.333\dots$ ) decimals.
Irrational Numbers ( $\mathbb{I}$ ): Numbers that cannot be written as a simple fraction. Their decimal expansions are non-terminating and non-repeating (e.g., $\pi, \sqrt{2}, e$ ).
Real Numbers ( $\mathbb{R}$ ): The set of all rational and irrational numbers.
Complex Numbers ( $\mathbb{C}$ ): Numbers of the form $a + bi$ , where $a, b \in \mathbb{R}$ and $i = \sqrt{-1}$ .

Prime Factorization, GCD, and LCM

Understanding the building blocks of numbers is crucial for simplifying fractions and solving problems involving repeating events.

Number Properties

Prime Factorization: Expressing a composite number as the product of its prime factors (e.g., $12 = 2^2 \cdot 3$ ).
Greatest Common Divisor (GCD): The largest positive integer that divides two or more numbers without a remainder. Useful for reducing fractions.
Least Common Multiple (LCM): The smallest positive integer that is a multiple of two or more numbers. Essential for finding common denominators.

Order of Operations (PEMDAS)

To ensure everyone calculates the same result for a given mathematical expression, we follow a strict order of operations, often remembered by the acronym PEMDAS:

PEMDAS Rules

Parentheses: Perform operations inside the innermost grouping symbols like brackets [], braces {}, and fraction bars first.
Exponents: Evaluate powers and square roots.
Multiplication and Division: These are inverse operations with equal priority. Perform them from left to right.
Addition and Subtraction: These are also inverse operations with equal priority. Perform them from left to right.

Caution

Multiplication does not always come before division. If you see

10 \div 2 \times 5

, you calculate from left to right:

10 \div 2 = 5

, then

5 \times 5 = 25

Properties of Real Numbers

Visualize the commutative property of addition and the definition of distance between points using the interactive number line below. Adjust the values of

a

and

b

to see how vector addition works and how

|a - b|

represents the distance between them.

Properties of Real Numbers on the Number Line

Value of

a

Value of

b

-2

Commutative Property of Addition:

a + b = b + a

3 + (-2) = -2 + (3)

1 = 1

Visualize $a + b$ and $b + a$ reaching the same sum.

These axioms form the rules of the game for algebra. For any real numbers

a

b

, and

c

Algebraic Axioms

Commutative Property: Changing the order does not change the result. Addition: $a + b = b + a$ , Multiplication: $ab = ba$
Associative Property: Changing the grouping does not change the result. Addition: $(a + b) + c = a + (b + c)$ , Multiplication: $(ab)c = a(bc)$
Distributive Property: Multiplication distributes over addition/subtraction. $a(b + c) = ab + ac$
Identity Property: Additive Identity: $a + 0 = a$ , Multiplicative Identity: $a \cdot 1 = a$
Inverse Property: Additive Inverse: $a + (-a) = 0$ , Multiplicative Inverse (Reciprocal): $a \cdot \frac{1}{a} = 1$ (where $a \neq 0$ )

Properties of Equality

These properties justify the steps we take when solving equations:

Equality Axioms

Reflexive Property: $a = a$ (A value always equals itself).
Symmetric Property: If $a = b$ , then $b = a$ .
Transitive Property: If $a = b$ and $b = c$ , then $a = c$ .
Substitution Property: If $a = b$ , then $b$ can be substituted for $a$ in any expression without changing its value.

Absolute Value

Geometrically, the absolute value of a number

x

, denoted

|x|

, is its distance from zero on the number line. Since distance cannot be negative, absolute value is always non-negative.

Absolute Value Definition

The piecewise definition of absolute value.

|x| = \begin{cases} x & \text{if } x \ge 0 \\\\ -x & \text{if } x \lt 0 \end{cases}

Variables

Symbol	Description	Unit
$\|x\|$	The absolute value of x	-
$x$	A real number	-

Key Properties

$|x| \ge 0$
$|-x| = |x|$
Product Rule: $|xy| = |x||y|$
Quotient Rule: $|\frac{x}{y}| = \frac{|x|}{|y|}$ (if $y \neq 0$ )
Triangle Inequality: $|x + y| \le |x| + |y|$

Inequalities

Inequalities compare the relative size of two values. They are solved similarly to equations, with one critical difference.

The Golden Rule of Inequalities

Whenever you multiply or divide both sides of an inequality by a negative number, you must reverse the direction of the inequality sign.

Interval Notation Guide

Notation Rules

Open Interval: $(a, b)$ corresponds to $a \lt x \lt b$ . Use parentheses when endpoints are excluded.
Closed Interval: $[a, b]$ corresponds to $a \le x \le b$ . Use brackets when endpoints are included.
Half-Open Interval: $[a, b)$ or $(a, b]$ mixes inclusion and exclusion.
Infinity: Always use parentheses with $\infty$ or $-\infty$ (e.g., $[a, \infty)$ ).

Interactive Visualizer

Visualize different inequalities and their corresponding interval notations on a number line.

Inequality Visualizer

Value:2

Inequality

x \ge 2

Interval

[2, \infty)

Key Takeaways

Order Matters: Always follow PEMDAS. Multiplication and Division are tied; Addition and Subtraction are tied. Solve ties from left to right.
Sign Flipping: The most common error in inequalities is forgetting to flip the sign when multiplying or dividing by a negative number.
Absolute Value: Represents distance. $|x| = a$ typically splits into two equations: $x = a$ and $x = -a$ .
Properties: The Distributive Property ( $a(b+c) = ab+ac$ ) is a key tool for simplifying expressions and solving linear equations.
Sets of Numbers: Knowing the difference between integers, rational, and real numbers helps in defining the domain of functions.

Quiz Me

NextFundamentals - Examples & Applications