Real Numbers

Classification of Numbers

Introduction to the Real Number System [$\mathbb{R}$] 🌍

Welcome to this comprehensive lesson on classifying numbers! Understanding the different sets of numbers is the foundation of all mathematics. We will start with the simplest concepts—counting—and build up to the most complex definitions.

Question: What numbers do we use solely for counting objects?

Answer: $\{1, 2, 3, 4, \dots\}$

1. Foundation: Counting and Natural Numbers [$\mathbb{N}$] 🥕

The numbers we use naturally for counting are the starting point of our number system.

Definition: Natural Numbers [$\mathbb{N}$]

The set of positive non-zero integers used for counting.

$$\mathbb{N} = \{1, 2, 3, 4, 5, \dots\}$$

Introducing Whole Numbers [$\mathbb{W}$]

If we add the concept of "nothing" or "zero" to our counting numbers, we expand the set slightly.

Definition: Whole Numbers [$\mathbb{W}$]

The set of Natural Numbers plus zero.

$$\mathbb{W} = \{0, 1, 2, 3, 4, \dots\}$$

Visually, the relationship is clear: every Natural Number is also a Whole Number. We express this subset relationship as: $\mathbb{N} \subset \mathbb{W}$.

Example 1: Classifying Small Numbers

Consider the number $5$.

Is $5$ in $\mathbb{N}$? Yes, because $5$ is used for counting.
Is $5$ in $\mathbb{W}$? Yes, because $\mathbb{N}$ is a subset of $\mathbb{W}$.

Now consider the number $0$.

Is $0$ in $\mathbb{N}$? No. $\mathbb{N}$ starts at $1$.
Is $0$ in $\mathbb{W}$? Yes. $0$ is the element that defines the difference between $\mathbb{N}$ and $\mathbb{W}$.

Conclusion: $5$ belongs to the narrowest set $\mathbb{N}$. $0$ belongs to the narrowest set $\mathbb{W}$.

2. Expanding to Integers [$\mathbb{Z}$] ❄️

While counting and whole numbers are great for basic arithmetic, they fail when we need to represent concepts below zero, such as temperature, debt, or elevation changes.

Scenario: If a bank account starts with 50 dollars and a user withdraws 75 dollars, the resulting balance is $-25$ dollars. This requires a new set of numbers.

Definition: Integers [$\mathbb{Z}$]

The set of all Whole Numbers and their negative counterparts [the opposite of every Natural Number]. The letter $\mathbb{Z}$ comes from the German word Zahlen, meaning "numbers."

$$\mathbb{Z} = \{\dots, -3, -2, -1, 0, 1, 2, 3, \dots\}$$

The subset relationship continues to grow:

$$\mathbb{N} \subset \mathbb{W} \subset \mathbb{Z}$$

Every Natural Number is an Integer, and every Whole Number is an Integer.

Example 2: Classifying Negative Numbers

Classify the number $-10$ using the narrowest set possible.

Is $-10$ in $\mathbb{N}$? No, it's negative.
Is $-10$ in $\mathbb{W}$? No, it's negative.
Is $-10$ in $\mathbb{Z}$? Yes. It is a negative Whole Number.

Conclusion: $-10$ belongs to the set of Integers [$\mathbb{Z}$].

3. Defining Rational Numbers [$\mathbb{Q}$] ➗

If you divide 1 pizza among 3 people, no integer can represent the size of each slice. This is why we need ratios and fractions.

Definition: Rational Numbers [$\mathbb{Q}$]

The set of numbers that can be expressed as a ratio [fraction] of two integers, where the denominator is not zero. $\mathbb{Q}$ stands for quotient.

$$\mathbb{Q} = \left\{\frac{p}{q} \mid p \in \mathbb{Z}, q \in \mathbb{Z}, q \neq 0\right\}$$

Key Insight: Integers are Rational

Every Integer $p$ can be written as a fraction $p/1$. Therefore, all Integers are Rational Numbers. This extends our relationship:

$$\mathbb{N} \subset \mathbb{W} \subset \mathbb{Z} \subset \mathbb{Q}$$

Decimal Representation of Rational Numbers

When a Rational Number is converted to a decimal, it must exhibit one of two characteristics:

Terminating Decimals: The decimal ends [e.g., $\frac{3}{4} = 0.75$].
Repeating Decimals: The decimal has a pattern that repeats infinitely [e.g., $\frac{1}{3} = 0.333\dots = 0.\overline{3}$].

Example 3: Terminating and Repeating Decimals

Classify the following numbers: $1.5$ and $\frac{2}{11}$.

1. $1.5$: This is a terminating decimal. We can write it as the fraction $\frac{15}{10}$, which simplifies to $\frac{3}{2}$. Since it is a ratio of two integers [$3$ and $2$], it is Rational [$\mathbb{Q}$].

2. $\frac{2}{11}$: When you divide 2 by 11, you get $0.181818\dots$, or $0.\overline{18}$. Since the decimal repeats indefinitely, it is Rational [$\mathbb{Q}$].

4. The Discovery of Irrational Numbers [$\mathbb{I}$ or $\mathbb{Q}'$] 🧮

Around 500 B.C., mathematicians discovered numbers that simply could not be written as a fraction—numbers that are "non-ratio."

Definition: Irrational Numbers [$\mathbb{I}$ or $\mathbb{Q}'$]

The set of numbers whose decimal expansion is non-terminating AND non-repeating. These numbers fall outside of the Rational set.

Irrational numbers are often associated with geometric measures or roots of non-perfect squares.

Famous Examples of Irrational Numbers:

Pi [$\pi$]: Used in circles, $\pi \approx 3.14159265\dots$. Its digits continue forever without any repeating pattern.
The Golden Ratio [$\phi$]: $\frac{1+\sqrt{5}}{2}$.
Non-perfect Roots: $\sqrt{2}, \sqrt{3}, \sqrt{5}$, etc. The square root of any number that is not a perfect square is irrational. For example, $\sqrt{4}=2$ [Rational], but $\sqrt{2} \approx 1.41421356\dots$ [Irrational].

Critical Relationship: Rational [$\mathbb{Q}$] and Irrational [$\mathbb{I}$] numbers are disjoint sets. A number cannot be both Rational and Irrational.

Example 4: Identifying Irrationality

Classify the number $0.1010010001\dots$

This number is non-terminating [it goes on forever] and non-repeating. Although it has a *pattern* [one zero, two zeros, three zeros, etc.], it does not have a *repeating block* of digits [like $\overline{10}$]. Since it does not repeat, it cannot be written as a simple fraction $\frac{p}{q}$.

Conclusion: This number is Irrational [$\mathbb{I}$].

5. Synthesis: The Real Number System [$\mathbb{R}$] 🌟

The set of all Rational and Irrational Numbers together forms the set of Real Numbers.

Definition: Real Numbers [$\mathbb{R}$]

The set composed of all Rational numbers [$\mathbb{Q}$] and all Irrational numbers [$\mathbb{I}$].

$$\mathbb{R} = \mathbb{Q} \cup \mathbb{I}$$

The Real Number System covers every point on the number line.

The Nested Structure of Real Numbers

$$\text{Real Numbers} [\mathbb{R}] = \begin{cases} \text{Rational} [\mathbb{Q}] \begin{cases} \text{Integers} [\mathbb{Z}] \begin{cases} \text{Whole} [\mathbb{W}] \begin{cases} \text{Natural} [\mathbb{N}] \end{cases} \\ \text{Negative Integers} \end{cases} \\ \text{Non-Integer Fractions/Decimals} \end{cases} \\ \text{Irrational} [\mathbb{I}] \end{cases}$$

Classification Workshop: Identifying the Narrowest Set

Number	Type	Justification
$7$	Natural [$\mathbb{N}$]	It is a positive whole number used for counting.
$-4$	Integer [$\mathbb{Z}$]	It is a negative whole number.
$0.6$	Rational [$\mathbb{Q}$]	It is a terminating decimal, $\frac{6}{10}$ or $\frac{3}{5}$.
$\sqrt{25}$	Natural [$\mathbb{N}$]	The value is $5$, which is a counting number.
$\sqrt{13}$	Irrational [$\mathbb{I}$]	$13$ is not a perfect square, resulting in a non-terminating, non-repeating decimal.
$1/3$	Rational [$\mathbb{Q}$]	It is a repeating decimal ($0.\overline{3}$) and a ratio of two integers.

Time for a Quick Check! 🧠

Use the definitions above to classify the following numbers. Select the narrowest set to which the number belongs.

1. To which set does the number $9$ belong most narrowly?

Explanation: $9$ is a positive integer greater than zero, meaning it is a counting number [Natural Number, $\mathbb{N}$].

2. What is the definition of the set of Whole Numbers [$\mathbb{W}$]?

Explanation: Whole Numbers [$\mathbb{W}$] include $0$ and all the Natural Numbers [$\mathbb{N}$], resulting in $\{0, 1, 2, 3, \dots\}$.

3. Classify $-15$ using the narrowest set.

Explanation: $-15$ is a negative counting number, placing it in the set of Integers [$\mathbb{Z}$]. It is not whole or natural because it is negative.

4. Which set notation accurately describes the relationship between Integers [$\mathbb{Z}$] and Rational Numbers [$\mathbb{Q}$]?

Explanation: Every integer $p$ can be written as $p/1$, making it rational. However, rational numbers like $1/2$ are not integers. Therefore, $\mathbb{Z}$ is a subset of $\mathbb{Q}$.

5. Which of the following numbers is an example of an Irrational Number [$\mathbb{I}$]?

Explanation: $7$ is not a perfect square, so $\sqrt{7}$ is irrational [non-terminating, non-repeating]. The other options are terminating or repeating decimals/fractions, making them rational.

6. The decimal expansion of a Rational Number [$\mathbb{Q}$] must be:

Explanation: Rational numbers like $0.5$ terminate, while numbers like $1/3$ repeat [$0.\overline{3}$]. Irrational numbers are the ones that are non-terminating AND non-repeating.

7. To which narrowest set does $0$ belong?

Explanation: $0$ is included in the Whole Numbers [$\mathbb{W}$] but not the Natural Numbers [$\mathbb{N}$]. Since it is an integer, $\mathbb{W}$ is the narrowest applicable set.

8. The number $\pi$ [Pi] is used to calculate the circumference of a circle. Which set does $\pi$ belong to?

Explanation: $\pi$ is the classic example of an irrational number because its decimal expansion [3.14159...] goes on infinitely without repeating.

9. What is the complete set of numbers formed by the union of Rational [$\mathbb{Q}$] and Irrational [$\mathbb{I}$] numbers?

Explanation: The Real Number System [$\mathbb{R}$] is defined as $\mathbb{Q} \cup \mathbb{I}$. These two sets cover every number on the standard number line.

10. How would you classify the number $\frac{5}{8}$?

Explanation: $\frac{5}{8}$ is explicitly written as a ratio of two integers [$5$ and $8$], satisfying the definition of a Rational Number [$\mathbb{Q}$]. In decimal form, it is $0.625$, which terminates.

11. Why is the number $\sqrt{16}$ considered a Natural Number, and not Irrational?

Explanation: The number must be evaluated first. $\sqrt{16} = 4$. Since $4$ is a positive whole number, it belongs to the Natural Numbers [$\mathbb{N}$].

12. Identify the narrowest set for the number $-3.5$.

Explanation: $-3.5$ is a terminating decimal. It can be written as $-\frac{7}{2}$. Since it is a fraction and not a whole number, it is Rational [$\mathbb{Q}$].

13. A number has a decimal representation $0.121221222\dots$. Which set does it belong to?

Explanation: This is a non-repeating, non-terminating decimal. Although it follows a pattern [increasing number of 2s], it does not have a fixed repeating block, meaning it is Irrational [$\mathbb{I}$].

14. What is the primary difference between Natural Numbers [$\mathbb{N}$] and Whole Numbers [$\mathbb{W}$]?

Explanation: $\mathbb{N} = \{1, 2, 3, \dots\}$ and $\mathbb{W} = \{0, 1, 2, 3, \dots\}$. Zero is the single element that distinguishes the Whole Numbers from the Natural Numbers.

15. The expression $\mathbb{Q} \cap \mathbb{I}$ [the intersection of Rational and Irrational numbers] results in:

Explanation: Rational and Irrational numbers are defined as being completely disjoint. A number cannot satisfy both definitions simultaneously, so their intersection is the empty set ($\emptyset$).

16. Which of the following numbers is classified as a Rational Number [$\mathbb{Q}$] but NOT an Integer [$\mathbb{Z}$]?

Explanation: $0.\overline{7}$ is a repeating decimal [equal to $7/9$], making it Rational. Since it is not a whole number or its negative counterpart, it is not an Integer.

17. Why are negative numbers necessary for mathematics?

Explanation: Negative numbers are essential extensions of the number line used to represent values opposite to positive values, such as deficits or movement in the opposite direction from zero.

18. According to the definition of Rational Numbers, what condition must the denominator $q$ satisfy in the fraction $p/q$?

Explanation: The definition requires both $p$ and $q$ to be integers, and division by zero is undefined, so $q$ must not equal $0$.

19. Which number belongs to the set $\mathbb{N}$ but not to the set of negative Integers?

Explanation: Natural Numbers [$\mathbb{N}$] are $\{1, 2, 3, \dots\}$. They are all positive and are therefore not negative integers. $12$ fits this description.

20. True or False: All Real Numbers are Rational.

Explanation: False. Real Numbers [$\mathbb{R}$] include both Rational [$\mathbb{Q}$] and Irrational [$\mathbb{I}$] numbers. Irrational numbers [like $\pi$ or $\sqrt{2}$] are real but not rational.

Lesson Summary 🏆

The Real Number System [$\mathbb{R}$] is a nested hierarchy built upon fundamental definitions:

Natural Numbers [$\mathbb{N}$]: $\{1, 2, 3, \dots\}$ [Counting].
Whole Numbers [$\mathbb{W}$]: $\{0, 1, 2, 3, \dots\}$ [Counting plus zero].
Integers [$\mathbb{Z}$]: $\{\dots, -2, -1, 0, 1, 2, \dots\}$ [Whole Numbers plus negative counterparts].
Rational Numbers [$\mathbb{Q}$]: Numbers that can be written as a fraction $\frac{p}{q}$ [Terminating or repeating decimals]. This set contains all $\mathbb{N}, \mathbb{W}, \text{ and } \mathbb{Z}$.
Irrational Numbers [$\mathbb{I}$]: Numbers that cannot be written as $\frac{p}{q}$ [Non-terminating, non-repeating decimals, like $\pi$].

The final relationship is $\mathbb{N} \subset \mathbb{W} \subset \mathbb{Z} \subset \mathbb{Q}$, and $\mathbb{R} = \mathbb{Q} \cup \mathbb{I}$.

Exit Ticket Requirement: Select the number $7$. Explain how it fits into ALL applicable sets [$\mathbb{R}, \mathbb{Q}, \mathbb{Z}, \mathbb{W}, \mathbb{N}$].

Justification: $7$ is Natural because it is a counting number. Since it is Natural, it is also Whole, Integer, and Rational [as $7/1$]. Since it is Rational, it is also Real. Thus, $7$ belongs to $\mathbb{N}, \mathbb{W}, \mathbb{Z}, \mathbb{Q}, \text{ and } \mathbb{R}$.