The real number system can be divided into several subsets, each with its own defining characteristics. The subsets include natural numbers ($n$), whole numbers ($w$), integers ($z$), rational numbers ($q$), and irrational numbers ($I$). Every real number ($r$) belongs to at least one of these subsets. Below are the subsets of the real number system along with their relationships:

1. Natural Numbers ($n$):

   • This subset includes all positive counting numbers starting from 1, i.e., $\{1, 2, 3, \dots\}$.

2. Whole Numbers ($w$):

   • This subset includes all natural numbers and zero, i.e., $\{0, 1, 2, 3, \dots\}$.

3. Integers ($z$):

   • This subset includes all whole numbers and their negatives, i.e., $\{\dots, -3, -2, -1, 0, 1, 2, 3, \dots\}$.

4. Rational Numbers ($q$):

   • This subset includes all numbers that can be expressed as a fraction $\frac{a}{b}$ where $a$ and $b$ are integers and $b \neq 0$. It includes integers and fractions, as well as terminating and repeating decimals.

5. Irrational Numbers ($I$):

   • This subset includes all real numbers that cannot be expressed as a fraction of two integers. Their decimal expansions are non-terminating and non-repeating, such as $\sqrt{2}$, $\pi$, and $e$.

6. Real Numbers ($r$):

   • This is the set of all possible numbers on the number line, including both rational and irrational numbers.

Relationship Between the Subsets

• $n \subset w \subset z \subset q \subset r$
• $I \subset r$

Example Number and Their Subset Memberships:

1. $3$:

   • Subsets: $n, w, z, q, r$

2. $0$:

   • Subsets: $w, z, q, r$

3. $-5$:

   • Subsets: $z, q, r$

4. $\frac{1}{2}$:

   • Subsets: $q, r$

5. $\sqrt{2}$:

   • Subsets: $I, r$

6. $\pi$:

   • Subsets: $I, r$

Summary

• Natural Numbers ($n$): $\{1, 2, 3, \dots\}$
• Whole Numbers ($w$): $\{0, 1, 2, 3, \dots\}$
• Integers ($z$): $\{\dots, -3, -2, -1, 0, 1, 2, 3, \dots\}$
• Rational Numbers ($q$): Numbers expressible as $\frac{a}{b}$ where $b \neq 0$
• Irrational Numbers ($I$): Numbers that cannot be expressed as fractions
• Real Numbers ($r$): All numbers on the number line

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Do you have any questions or need further explanation on any of these subsets?

Here are five related questions:

1. What is the difference between rational and irrational numbers?
2. How can you determine whether a number is natural, whole, or an integer?
3. What are some examples of irrational numbers other than $\pi$ and $\sqrt{2}$?
4. Can a number be both rational and irrational? Why or why not?
5. How do rational numbers relate to repeating and terminating decimals?

Tip: To identify the subset to which a number belongs, always start with the smallest subset and check upward (e.g., check if it's natural first, then whole, etc.).