Rational Number: A number that can be expressed as a fraction \(\frac{p}{q}\) where \(p\) and \(q\) are integers and \(q \neq 0\)

Formula: \(\text{Rational Number} = \frac{p}{q}\) where \(p, q \in \mathbb{Z}\) and \(q \neq 0\)

✓ Can be expressed as a fraction of two integers

✓ Decimal form either terminates or repeats

✓ Includes all integers, fractions, and finite/repeating decimals

Integers: \(5 = \frac{5}{1}\), \(-3 = \frac{-3}{1}\), \(0 = \frac{0}{1}\)

Fractions: \(\frac{3}{4}\), \(\frac{-7}{2}\), \(\frac{22}{7}\)

Terminating Decimals: \(0.5 = \frac{1}{2}\), \(0.75 = \frac{3}{4}\), \(2.8 = \frac{14}{5}\)

Repeating Decimals: \(0.\overline{3} = \frac{1}{3}\), \(0.\overline{6} = \frac{2}{3}\), \(1.\overline{25} = \frac{41}{33}\)

Irrational Number: A number that CANNOT be expressed as a fraction \(\frac{p}{q}\) where \(p\) and \(q\) are integers

Cannot be written as a ratio of two integers

✓ Cannot be expressed as a fraction of integers

✓ Decimal form is non-terminating and non-repeating

✓ Includes most square roots, π, e, and golden ratio

Square Roots: \(\sqrt{2} \approx 1.414213...\), \(\sqrt{3} \approx 1.732050...\), \(\sqrt{5}\)

Pi: \(\pi \approx 3.141592653...\) (never ends, never repeats)

Euler's Number: \(e \approx 2.718281828...\)

Golden Ratio: \(\phi \approx 1.618033988...\)

Non-repeating Decimals: \(0.10110111011110...\)

Rational: \(\sqrt{n}\) is rational if \(n\) is a perfect square

Irrational: \(\sqrt{n}\) is irrational if \(n\) is NOT a perfect square

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

1. Natural Numbers (ℕ): \(\{1, 2, 3, 4, 5, ...\}\)

Counting numbers starting from 1

2. Whole Numbers (W): \(\{0, 1, 2, 3, 4, 5, ...\}\)

Natural numbers + zero

3. Integers (ℤ): \(\{..., -3, -2, -1, 0, 1, 2, 3, ...\}\)

Whole numbers + negative numbers

4. Rational Numbers (ℚ): \(\frac{p}{q}\) where \(p, q \in \mathbb{Z}, q \neq 0\)

Includes all integers, fractions, terminating/repeating decimals

5. Irrational Numbers: Cannot be expressed as \(\frac{p}{q}\)

Non-terminating, non-repeating decimals

6. Real Numbers (ℝ): Rational + Irrational Numbers

All numbers on the number line

7: Natural, Whole, Integer, Rational, Real

0: Whole, Integer, Rational, Real

-5: Integer, Rational, Real

\(\frac{3}{4}\): Rational, Real

\(\sqrt{2}\): Irrational, Real

\(\pi\): Irrational, Real

Step 1: Can it be written as \(\frac{p}{q}\) where \(p, q\) are integers and \(q \neq 0\)?

→ YES = Rational | NO = Continue to Step 2

Step 2: Is it a decimal?

→ Terminating (ends) = Rational

→ Repeating (pattern repeats) = Rational

→ Non-terminating, non-repeating = Irrational

Step 3: Is it a square root?

→ Perfect square = Rational

→ Not a perfect square = Irrational

✓ 0.75 → Terminates → RATIONAL

✓ 0.333... → Repeats → RATIONAL

✓ \(\sqrt{16}\) → Perfect square (4) → RATIONAL

✗ 0.101001000... → No pattern → IRRATIONAL

✗ \(\sqrt{7}\) → Not perfect square → IRRATIONAL

✗ \(\pi\) → Never ends/repeats → IRRATIONAL

Step 1: Find two consecutive perfect squares that the number lies between

Step 2: Approximate the decimal value

Step 3: Plot the approximate location on the number line

Step 1: \(4 < 7 < 9\) → \(\sqrt{4} < \sqrt{7} < \sqrt{9}\) → \(2 < \sqrt{7} < 3\)

Step 2: \(\sqrt{7} \approx 2.646\)

Step 3: Plot between 2 and 3, closer to 3 (about 65% of the way)

\(\sqrt{2} \approx 1.414\) → Between 1 and 2

\(\sqrt{10} \approx 3.162\) → Between 3 and 4

\(\pi \approx 3.14159\) → Between 3 and 4

\(\sqrt{50} \approx 7.071\) → Between 7 and 8

Step 1: Convert all numbers to decimal form

Step 2: Approximate irrational numbers to same decimal places

Step 3: Compare decimals digit by digit from left to right

Step 4: Order from least to greatest or greatest to least

For Square Roots: \(\sqrt{a} < \sqrt{b}\) if \(a < b\)

Example: \(\sqrt{5} < \sqrt{10}\) because \(5 < 10\)

Mixed Numbers: Convert to decimals for comparison

Use 3-4 decimal places for accuracy

Given: \(\sqrt{10}\), \(\pi\), \(3.14\), \(\frac{22}{7}\)

Step 1 - Convert to Decimals:

• \(\sqrt{10} \approx 3.162\)

• \(\pi \approx 3.142\)

• \(3.14 = 3.140\)

• \(\frac{22}{7} \approx 3.143\)

Answer (Least to Greatest): \(3.14 < \pi < \frac{22}{7} < \sqrt{10}\)

Order: \(\sqrt{5}\), \(2.3\), \(\frac{9}{4}\), \(\sqrt{7}\)

Conversions: \(\sqrt{5} \approx 2.236\), \(\frac{9}{4} = 2.25\), \(\sqrt{7} \approx 2.646\)

Result: \(\sqrt{5} < \frac{9}{4} < 2.3 < \sqrt{7}\)

Method 1: Using Perfect Squares (for square roots)

Find two perfect squares that bracket the number

Example: For \(\sqrt{20}\) → \(16 < 20 < 25\) → \(4 < \sqrt{20} < 5\)

Method 2: Using a Calculator

Use calculator for precise decimal approximation

Method 3: Memorize Common Values

\(\pi \approx 3.14159\), \(e \approx 2.71828\), \(\sqrt{2} \approx 1.414\)

✓ Rational + Rational = Rational

✓ Rational + Irrational = Irrational

✓ Irrational + Irrational = Could be Either

Example: \(\sqrt{2} + (-\sqrt{2}) = 0\) (rational)

Example: \(\sqrt{2} + \sqrt{3}\) (irrational)

✓ Rational × Rational = Rational

✓ Rational (≠ 0) × Irrational = Irrational

✓ Irrational × Irrational = Could be Either

Example: \(\sqrt{2} \times \sqrt{2} = 2\) (rational)

Example: \(\sqrt{2} \times \sqrt{3} = \sqrt{6}\) (irrational)

⚡ Quick Check: Decimals terminate/repeat = Rational | Decimals never end & never repeat = Irrational ⚡

✓ Memorize perfect squares 1-15: Helps identify rational square roots quickly

✓ Check decimal patterns: Look for terminating or repeating patterns

✓ Practice approximation: Learn to estimate irrational numbers between integers

✓ Use a calculator: Verify decimal expansions and approximations

✓ Remember special numbers: π ≈ 3.14159, e ≈ 2.71828, √2 ≈ 1.414

✓ Work backwards: Square a potential answer to verify square roots

✓ Draw number lines: Visual representation helps with ordering

❌ Mistake 1: Thinking all decimals are irrational

✓ Correct: Terminating and repeating decimals are rational

❌ Mistake 2: Assuming \(\sqrt{2}\) can be simplified to a fraction

✓ Correct: \(\sqrt{2}\) is irrational and cannot be expressed as \(\frac{p}{q}\)

❌ Mistake 3: Confusing π with \(\frac{22}{7}\)

✓ Correct: \(\frac{22}{7}\) is just an approximation; π is irrational

❌ Mistake 4: Forgetting that integers are rational

✓ Correct: All integers can be written as \(\frac{n}{1}\), so they're rational

❌ Mistake 5: Assuming irrational + irrational is always irrational

✓ Correct: \(\sqrt{2} + (-\sqrt{2}) = 0\), which is rational