Real Number Properties

• Real numbers ($\mathbb{R}$) are numbers that can be written in decimal form

Basic Properties of Real Numbers

• Let $a$, $b$, and $c$ be real numbers

Addition Properties

• $a + b$ is a real number (Closure Property)
• $a + b = b + a$ (Commutative Property)
• $(a + b) + c = a + (b + c)$ (Associative Property)
• $a + 0 = a$ (Additive Identity Property)
• $a + (-a) = 0$ (Additive Inverse Property)

Multiplication Properties

• $ab$ is a real number (Closure Property)
• $ab = ba$ (Commutative Property)
• $(ab)c = a(bc)$ (Associative Property)
• $a \cdot 1 = a$ (Multiplicative Identity Property)
• $a \cdot (1/a) = 1$, $a \neq 0$ (Multiplicative Inverse Property)

Distributive Property

• $a(b + c) = ab + ac$

Set of Real Numbers

• The set of real numbers can be divided into rational and irrational numbers

Definition of Rational Numbers

• Rational numbers can be written as $a/b$, where $a$ and $b$ are integers and $b \neq 0$
• Irrational numbers cannot be written in this form

Examples of Rational and Irrational Numbers

• Rational: $5 = \frac{5}{1}$, $\frac{1}{3} = 0.333...$, $0.125 = \frac{1}{8}$, and $0.414213562...$
• Irrational: $\sqrt{2} = 1.414213562...$ and $\pi = 3.141592653...$

Real Number Line

• The real number line is a line where each point represents a real number

Ordering Real Numbers

• If $b - a$ is positive, then $a$ is less than $b$, denoted as $a < b$

Inequalities and the Real Number Line

• $a < b$: $a$ lies to the left of $b$
• $a = b$: $a$ and $b$ are the same point
• $a > b$: $a$ lies to the right of $b$
• $\leq$ and $\geq$ are inequality symbols
• $a \leq b$: $a$ is less than or equal to $b$
• $a \geq b$: $a$ is greater than or equal to $b$
• $a < x < b$: $x$ is between $a$ and $b$

Definition of Absolute Value

• The absolute value of $a$, denoted by $|a|$, is defined as:
  • $|a| = a$, if $a \geq 0$
  • $|a| = -a$, if $a < 0$

Examples of Finding Absolute Value

• $|-4| = -(-4) = 4$
• $|7| = 7$
• $|0| = 0$

Properties of Absolute Value

• Let $a$ and $b$ be real numbers:
  • $|a| \geq 0$
  • $|-a| = |a|$
  • $|ab| = |a||b|$
  • $|a + b| \leq |a| + |b|$ (Triangle Inequality)

Distance Between Two Points on the Real Line

• The distance between $a$ and $b$, denoted by $d(a, b)$, is:
  • $d(a, b) = |b - a| = |a - b|$

Example of Finding the Distance Between Two Points

• The distance between $-3$ and $4$ is $|-3 - 4| = |-7| = 7$

Intervals on the Real Number Line

• The interval $[a, b]$ is the set of all real numbers $x$ such that $a \leq x \leq b$

Definition of Intervals on the Real Number Line

• Notation, Interval Type, Inequality
• $[a, b]$, Closed, $a \leq x \leq b$
• $(a, b)$, Open, $a < x < b$
• $[a, b)$, Half-Open, $a \leq x < b$
• $(a, b]$, Half-Open, $a < x \leq b$
• $(-\infty, a]$, Unbounded, $x \leq a$
• $(-\infty, a)$, Unbounded, $x < a$
• $[b, \infty)$, Unbounded, $x \geq b$
• $(b, \infty)$, Unbounded, $x > b$
• $(-\infty, \infty)$, Entire Real Line

Symbols $\infty$ and $-\infty$

• They do not represent real numbers
• They describe the unboundedness of an interval

Examples of Intervals on the Real Number Line

• $[-1, 2]$: all real numbers $x$ for which $-1 \leq x \leq 2$
• $(-2, \infty)$: all real numbers $x$ for which $x > -2$
• $[-1, 0)$: all real numbers $x$ for which $-1 \leq x < 0$