1. **Problem Statement:** Describe the real number system by recognizing, defining, and distinguishing properties of natural numbers, whole numbers, integers, rational numbers, and irrational numbers. 2. **Step 1: Define each set of numbers** - Natural numbers ($\mathbb{N}$): These are the counting numbers starting from 1, 2, 3, ... - Whole numbers ($\mathbb{W}$): Natural numbers including zero, i.e., 0, 1, 2, 3, ... - Integers ($\mathbb{Z}$): All whole numbers and their negatives, i.e., ..., -3, -2, -1, 0, 1, 2, 3, ... - Rational numbers ($\mathbb{Q}$): Numbers that can be expressed as a fraction $\frac{p}{q}$ where $p$ and $q$ are integers and $q \neq 0$. - Irrational numbers: Numbers that cannot be expressed as a simple fraction, their decimal expansions are non-terminating and non-repeating, e.g., $\sqrt{2}$, $\pi$. 3. **Step 2: Properties and distinctions** - Natural numbers are a subset of whole numbers. - Whole numbers are a subset of integers. - Integers are a subset of rational numbers (since any integer $a$ can be written as $\frac{a}{1}$). - Rational and irrational numbers together form the real numbers ($\mathbb{R}$). 4. **Step 3: Visual diagram description** - Imagine nested sets: Natural numbers inside Whole numbers, inside Integers, inside Rational numbers, and Rational numbers alongside Irrational numbers together forming Real numbers. 5. **Step 4: Summary formula** $$\mathbb{N} \subset \mathbb{W} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R}$$ 6. **Step 5: Important notes** - Every natural number is whole, integer, rational, and real. - Not all real numbers are rational; irrational numbers fill the gaps. This explanation helps understand the hierarchy and properties of the real number system.