Example Usage

• Set Membership: $5 \in \mathbb{N}$ (5 is a natural number), but $-3 \notin \mathbb{N}$.
• Number Hierarchy: $\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R} \subset \mathbb{C}$
• Examples: $\frac{2}{3} \in \mathbb{Q}$, $\sqrt{2} \in \mathbb{R}$ but $\sqrt{2} \notin \mathbb{Q}$, $3 + 4i \in \mathbb{C}$

Problem 1: Classify each number: $5$, $-3$, $\frac{2}{3}$, $\sqrt{2}$, $3 + 4i$

Solution:

• $5 \in \mathbb{N}$ (natural number) and $5 \in \mathbb{Z}$ (integer)
• $-3 \in \mathbb{Z}$ (integer) but $-3 \notin \mathbb{N}$ (not natural)
• $\frac{2}{3} \in \mathbb{Q}$ (rational number)
• $\sqrt{2} \in \mathbb{R}$ (real number) but $\sqrt{2} \notin \mathbb{Q}$ (irrational)
• $3 + 4i \in \mathbb{C}$ (complex number)

Answer:

• $5 \in \mathbb{N}, \mathbb{Z}, \mathbb{Q}, \mathbb{R}, \mathbb{C}$
• $-3 \in \mathbb{Z}, \mathbb{Q}, \mathbb{R}, \mathbb{C}$
• $\frac{2}{3} \in \mathbb{Q}, \mathbb{R}, \mathbb{C}$
• $\sqrt{2} \in \mathbb{R}, \mathbb{C}$ but $\sqrt{2} \notin \mathbb{Q}$
• $3 + 4i \in \mathbb{C}$

Problem 2: Show that $\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R} \subset \mathbb{C}$

Solution:

• Every natural number is an integer: $\mathbb{N} \subset \mathbb{Z}$
• Every integer can be written as a fraction: $\mathbb{Z} \subset \mathbb{Q}$
• Rational numbers are real: $\mathbb{Q} \subset \mathbb{R}$
• Real numbers are complex (with imaginary part 0): $\mathbb{R} \subset \mathbb{C}$

Answer: The hierarchy is: $\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R} \subset \mathbb{C}$

Daily Life Applications

Understanding number sets helps you recognize what types of numbers you're working with in everyday situations and choose appropriate operations.

Counting and Whole Numbers

• Natural Numbers ($\mathbb{N}$): Count discrete items. When counting apples, people, or days, you use natural numbers: $1, 2, 3, \ldots \in \mathbb{N}$. You can't have $-3$ apples!
• Integers ($\mathbb{Z}$): Handle positive and negative values. Temperature, bank balances, and elevations use integers. If it's $-5°C$, that's $-5 \in \mathbb{Z}$.

Fractions and Decimals

• Rational Numbers ($\mathbb{Q}$): Work with parts and divisions. When splitting a pizza ($\frac{1}{4}$ per person) or measuring ($2.5$ cups), you use rational numbers. These include all fractions and terminating/repeating decimals.
• Real Numbers ($\mathbb{R}$): Measure continuous quantities. Distances, weights, and most measurements use real numbers. Even irrational numbers like $\pi$ (for circles) or $\sqrt{2}$ (for diagonals) are real.

Practical Examples

• Shopping: Prices like \19.99$are rational numbers ($\in \mathbb{Q}$). You can express this as$\frac{1999}{100}$.
• Cooking: Recipe measurements like $2\frac{1}{2}$ cups are rational numbers. Converting to decimals gives $2.5 \in \mathbb{Q}$.
• Temperature: $-10°F$ is an integer ($-10 \in \mathbb{Z}$), while $98.6°F$ (body temperature) is a rational number ($98.6 \in \mathbb{Q}$).

When Each Set Matters

• Natural Numbers ($\mathbb{N}$): Use for counting items, ages, quantities that can't be negative.
• Integers ($\mathbb{Z}$): Use for temperatures, elevations, financial gains/losses, anything that can be negative.
• Rational Numbers ($\mathbb{Q}$): Use for measurements, prices, percentages, anything that can be expressed as a fraction.
• Real Numbers ($\mathbb{R}$): Use for all physical measurements, distances, areas, volumes.

Problem-Solving Strategy

When working with numbers:

1. Identify the context (counting, measuring, temperature, etc.)
2. Determine the appropriate number set ($\mathbb{N}$ for counting, $\mathbb{Z}$ for negatives, $\mathbb{Q}$ for fractions)
3. Recognize the hierarchy ($\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R}$)
4. Choose operations that make sense for that number set
5. Verify your answer belongs to the expected set

Understanding number sets helps you work with the right type of numbers for each situation!