Precalculus Study Guide: Real Numbers, Intervals, Absolute Value, and Exponents

Real Numbers and Their Classifications

Types of Numbers

The real number system consists of several important subsets, each with unique properties and uses in mathematics.

• Natural Numbers (ℕ): The set of positive whole numbers used for counting: 1, 2, 3, 4, ...

• Whole Numbers: All natural numbers plus zero: 0, 1, 2, 3, ...

• Integers (ℤ): All whole numbers and their negatives: ..., -3, -2, -1, 0, 1, 2, 3, ...

• Rational Numbers (ℚ): Numbers that can be written as a fraction $\frac{a}{b}$, where $a$ and $b$ are integers and $b \neq 0$. Their decimal expansions either terminate or repeat (e.g., $0.5$, $0.333...$, $0.36$).

• Irrational Numbers: Numbers that cannot be written as a fraction of integers. Their decimal expansions are non-terminating and non-repeating (e.g., $\sqrt{2}$, $\pi$, $e$).

• Real Numbers (ℝ): The set of all rational and irrational numbers.

Example: $\sqrt{2} = 1.41421356...$ is irrational because its decimal expansion never repeats or terminates.

Notation:

• $\mathbb{N}$: set of natural numbers

• $\mathbb{Z}$: set of integers

• $\mathbb{Q}$: set of rational numbers

• $\mathbb{R}$: set of real numbers

Visualizing the Real Number System

The Real Number Line

The real number system can be represented as a number line, where each point corresponds to a real number. Positive numbers are to the right of zero, and negative numbers are to the left.

• Example: Points such as $-32$, $-5$, $0$, $1$, $8$, $27$, $64$ can be plotted on the number line.

Intervals and Inequalities

Interval Notation

Intervals are used to describe sets of real numbers between two endpoints. They can be open, closed, or half-open/half-closed.

• Open Interval $(a, b)$: All real numbers between $a$ and $b$, not including $a$ or $b$.

• Closed Interval $[a, b]$: All real numbers between $a$ and $b$, including both $a$ and $b$.

• Half-Open Intervals: $[a, b)$ includes $a$ but not $b$; $(a, b]$ includes $b$ but not $a$.

• Infinite Intervals: $(-\infty, a)$, $(a, \infty)$, $(-\infty, \infty)$, etc.

Example: $(-2, 5)$ represents all real numbers greater than $-2$ and less than $5$ (but not including $-2$ or $5$).

Set-Builder Notation

Set-builder notation describes the elements of a set using a property or rule.

• Example: $\{x \mid 2 < x < 5\}$ means the set of all $x$ such that $2 < x < 5$.

Comparing Interval and Set-Builder Notation

Interval Notation	Set-Builder Notation	Graph
$(a, b)$	$\{x \mid a < x < b\}$	Open circle at $a$ and $b$
$(a, b]$	$\{x \mid a < x \leq b\}$	Open at $a$, closed at $b$
$[a, b)$	$\{x \mid a \leq x < b\}$	Closed at $a$, open at $b$
$[a, b]$	$\{x \mid a \leq x \leq b\}$	Closed circle at $a$ and $b$
$(a, \infty)$	$\{x \mid x > a\}$	Open at $a$, arrow to the right
$(-\infty, b]$	$\{x \mid x \leq b\}$	Arrow to the left, closed at $b$
$(-\infty, \infty)$	$\{x \mid x \text{ is a real number}\}$	Arrow both directions

Using Inequalities and Interval Notation Interchangeably

• Example: The inequality $y \geq \frac{3}{4}$ can be written as $[\frac{3}{4}, \infty)$.

• Always use parentheses for infinity: $(-\infty, a)$ or $(a, \infty)$.

Union and Intersection of Intervals

• Union ($\cup$): Combines all elements from two sets. For example, $[-5, 2) \cup (8, \infty)$ includes all numbers from $-5$ to $2$ (not including $2$) and all numbers greater than $8$.

• Intersection ($\cap$): Includes only elements common to both sets. For example, $[-10, 0) \cap [-2, 6] = [-2, 0)$.

Absolute Value and Distance

Definition of Absolute Value

The absolute value of a number is its distance from zero on the number line, always non-negative.

• For any real number $a$:

$|a| = \begin{cases} a, & \text{if } a \geq 0 \\ -a, & \text{if } a < 0 \end{cases}$

• Example: $|3| = 3$, $|-7| = 7$, $|0| = 0$

Distance Between Two Real Numbers

The distance between two real numbers $a$ and $b$ is given by:

$|a - b|$

• Example: The distance between $-2$ and $4$ is $|-2 - 4| = |-6| = 6$.

• The order of subtraction does not matter: $|a - b| = |b - a|$.

Algebraic Expressions & Rules of Integer Exponents

Algebraic Expressions

An algebraic expression is a mathematical phrase that can include numbers, variables, and operation symbols.

• Example: $6x - 15xy + \frac{3}{2}y^2$ is an algebraic expression.

Properties of Algebraic Operations

• Distributive Property: $a(b + c) = ab + ac$

• Zero-Product Property: If $ab = 0$, then $a = 0$ or $b = 0$

Rules of Integer Exponents

• Product of Powers: $a^m \cdot a^n = a^{m+n}$

• Power of a Power: $(a^m)^n = a^{mn}$

• Power of a Product: $(ab)^n = a^n b^n$

• Quotient of Powers: $\frac{a^m}{a^n} = a^{m-n}$, $a \neq 0$

• Zero Exponent: $a^0 = 1$, $a \neq 0$

• Negative Exponent: $a^{-n} = \frac{1}{a^n}$, $a \neq 0$

Example: $2^3 = 8$, $(-2)^3 = -8$, $3^0 = 1$, $5^{-2} = \frac{1}{25}$

Additional info: The notes also mention that exponents represent repeated multiplication, and that the rules above are essential for simplifying algebraic expressions and solving equations.