Definition:

A rational number is any number that can be expressed as a ratio of two integers.

\[ \text{Rational Number} = \frac{p}{q} \]

where \( p \) and \( q \) are integers, and \( q \neq 0 \)

Key Characteristics:

• Can be written as a fraction or ratio

• Decimal form: terminating or repeating

• Includes all integers, whole numbers, natural numbers

• Includes all fractions and mixed numbers

Integers: \( -5, 0, 7, 42 \) (can be written as \( \frac{-5}{1}, \frac{0}{1}, \frac{7}{1}, \frac{42}{1} \))

Fractions: \( \frac{3}{4}, \frac{-7}{8}, \frac{22}{7} \)

Terminating decimals: \( 0.5, 0.75, 3.625, -2.8 \)

→ \( 0.5 = \frac{1}{2} \), \( 0.75 = \frac{3}{4} \), \( 3.625 = \frac{29}{8} \)

Repeating decimals: \( 0.333... = 0.\overline{3}, \; 0.121212... = 0.\overline{12}, \; 2.777... = 2.\overline{7} \)

→ \( 0.\overline{3} = \frac{1}{3} \), \( 0.\overline{12} = \frac{4}{33} \), \( 2.\overline{7} = \frac{25}{9} \)

Perfect square roots: \( \sqrt{4} = 2, \; \sqrt{9} = 3, \; \sqrt{16} = 4, \; \sqrt{25} = 5 \)

Definition:

An irrational number is any real number that CANNOT be expressed as a ratio of two integers.

Key Characteristics:

• Cannot be written as a simple fraction \( \frac{p}{q} \)

• Decimal form: non-terminating AND non-repeating

• Decimal goes on forever without pattern

• Often involves roots, π, e, and other special constants

Non-perfect square roots: \( \sqrt{2}, \sqrt{3}, \sqrt{5}, \sqrt{7}, \sqrt{8}, \sqrt{10} \)

→ \( \sqrt{2} \approx 1.41421356... \) (never ends, never repeats)

Special constants:

• \( \pi \approx 3.14159265358979... \) (ratio of circumference to diameter)

• \( e \approx 2.71828182845904... \) (Euler's number)

• \( \phi \approx 1.61803398874989... \) (golden ratio)

Non-terminating, non-repeating decimals:

• \( 0.101001000100001... \) (pattern but not repeating)

• \( 2.343443444... \) (no repeating block)

Cube roots (non-perfect): \( \sqrt[3]{2}, \sqrt[3]{5}, \sqrt[3]{10} \)

To determine if a number is rational or irrational:

Real Numbers (ℝ) = Rational Numbers (ℚ) ∪ Irrational Numbers

Important Relationships:

ℕ ⊂ 𝕎 ⊂ ℤ ⊂ ℚ ⊂ ℝ

(Natural ⊂ Whole ⊂ Integer ⊂ Rational ⊂ Real)

All natural numbers are rational, but not all rational numbers are natural

A set is closed under an operation if performing that operation on members of the set always produces a result that is also in the set.

✓ Rational numbers are CLOSED under:

• Addition

• Subtraction

• Multiplication

• Division (except by zero)

✗ Irrational numbers are NOT CLOSED under:

• Addition (sometimes rational, sometimes irrational)

• Subtraction (sometimes rational, sometimes irrational)

• Multiplication (sometimes rational, sometimes irrational)

• Division (sometimes rational, sometimes irrational)

Property 1: Rational + Rational = Rational

\( \mathbb{Q} + \mathbb{Q} = \mathbb{Q} \)

The sum of two rational numbers is ALWAYS rational.

Property 2: Rational × Rational = Rational

\( \mathbb{Q} \times \mathbb{Q} = \mathbb{Q} \)

The product of two rational numbers is ALWAYS rational.

Property 3: Rational + Irrational = Irrational

\( \mathbb{Q} + \text{Irrational} = \text{Irrational} \)

The sum of a rational number and an irrational number is ALWAYS irrational.

Property 4: Rational × Irrational = Irrational (if rational ≠ 0)

\( \mathbb{Q} \times \text{Irrational} = \text{Irrational} \) (when \( \mathbb{Q} \neq 0 \))

The product of a non-zero rational number and an irrational number is ALWAYS irrational.

Property 5: Irrational + Irrational = SOMETIMES Rational, SOMETIMES Irrational

\( \text{Irrational} + \text{Irrational} = ? \)

The sum of two irrational numbers can be EITHER rational OR irrational.

Property 6: Irrational × Irrational = SOMETIMES Rational, SOMETIMES Irrational

\( \text{Irrational} \times \text{Irrational} = ? \)

The product of two irrational numbers can be EITHER rational OR irrational.