Lesson 1 — Number Sets

🎯 Goal

Understand the different sets of numbers, how they are related, and where they are used in mathematics.

📖 Lesson Description

Numbers are organized into sets based on their properties. Each set is like a “container” for a certain kind of number. Understanding number sets helps us know what rules apply when we calculate, simplify, or solve problems.

🔎 The Number Sets

• Natural Numbers \(\mathbb{N}\): \(1, 2, 3, 4, \dots\)
• Whole Numbers: \(0, 1, 2, 3, 4, \dots\)
• Integers \(\mathbb{Z}\): \(\dots, -3, -2, -1, 0, 1, 2, 3, \dots\)
• Rational Numbers \(\mathbb{Q}\): Fractions or decimals that end or repeat. Example: \(\tfrac{1}{2}, 0.75, -\tfrac{3}{4}\)
• Irrational Numbers: Non-repeating, non-terminating decimals. Example: \(\pi, \sqrt{2}\)
• Real Numbers \(\mathbb{R}\): All rational and irrational numbers (everything on the number line)
• Complex Numbers \(\mathbb{C}\): Numbers with an imaginary part, such as \(3 + 2i\)

📝 Examples

• \(7\) is a natural number, a whole number, an integer, a rational number, and a real number.
• \(-5\) is an integer, a rational number, and a real number.
• \(0.333\ldots = \tfrac{1}{3}\) is a rational number and a real number.
• \(\pi\) is an irrational number and a real number.
• \(4 + 2i\) is a complex number (not real).

✏️ Practice

1. Which sets does the number \(12\) belong to?
2. Is \(0\) a natural number? A whole number? An integer?
3. Classify each: \(-8,\; 2.5,\; \sqrt{5},\; 0.999\ldots\)
4. Give an example of a number that is irrational.
5. Explain why every integer is also a rational number.