Module 1: The Real Number System, Sets, and Directed Numbers

Overview of Number Systems and Daily Applications

• Numbers are essential tools used daily to make informed decisions across a variety of practical contexts, such as:

  • Reading temperatures
  • Measuring recipe ingredients
  • Interpreting bank balances
  • Understanding financial discounts
  • Adjusting recipe quantities

• To work confidently with mathematics, it is foundational to understand the distinct classifications of numbers and how they behave.

• Key Learning Outcomes:

  • Distinguish between different sets of numbers, including integers, whole numbers, rational numbers, irrational numbers, and natural numbers.
  • Identify whether a given set of numbers forms a subset of another set.
  • Locate and manipulate directed numbers on a number line.

• Everyday Situations Involving Numbers:

  • A bus travelling from Grenville to St. George's carrying $15$ passengers.
  • A freezer used to store fish operating at $-18\,^\circ\text{C}$.
  • A recipe for sorrel drink requiring $0.5\,\text{liters}$ of concentrate.
  • The mathematical constant $\pi$ used when calculating the circumference of a circular surface.

• Analytical Reflection Questions:

  • What differences exist among these numbers? (Some represent discrete counts of whole objects, some indicate signed values relative to a zero reference, some are terminating decimals representing continuous liquid measures, and others are non-terminating, non-repeating constants).
  • Can all of these numbers be counted? (No; while passenger count is countable using natural numbers, measurements like temperature, volume, and geometric ratios exist on continuous scales or represent irrational values).
  • Can all of them be written as fractions? (No; while passenger count $15 = \frac{15}{1}$, temperature $-18 = \frac{-18}{1}$, and volume $0.5 = \frac{1}{2}$ can be expressed as fractions of integers, $\pi$ cannot be expressed as a ratio of two integers).

Understanding Sets and Set Theory Notation

• Definition of a Set:

  • A set is a well-defined collection of items usually of the same kind, such as people, letters, or numbers.
  • A set is denoted by a capital letter (e.g., $A$, $B$, $X$) and its elements are enclosed within curly braces $\{\}$.

• Elements of a Set:

  • Each individual item contained within a set is called an element.
  • Example: Let $A$ represent the set of vowels in the English alphabet:     $A = \{a, e, i, o, u\}$
  • The symbol $\in$ denotes 'is an element of'. Therefore:     $a \in A$$e \in A$$i \in A$$o \in A$$u \in A$

• Subsets and Set Inclusion:

  • If $A$ and $B$ are any two sets such that every element of $A$ is contained within $B$, then $A$ is a subset of $B$
  • The symbol $\subset$ denotes 'is contained in' or 'is a subset of'. Written as:     $A \subset B$
  • The symbol $\not\subset$ denotes 'is not contained in' or 'is not a subset of'.

• Demonstration Example of Subsets:

  • Consider the following sets:     $A = \{2, 4, 6\}$$B = \{1, 2, 3, 4, 5, 6, 7\}$$C = \{1, 3, 5, 7\}$
  • Comparison 1: $A \subset B$ because every element in $\{2, 4, 6\}$ is present in $\{1, 2, 3, 4, 5, 6, 7\}$.
  • Comparison 2: $C \subset B$ because every element in $\{1, 3, 5, 7\}$ is present in $\{1, 2, 3, 4, 5, 6, 7\}$.
  • Comparison 3: $A \not\subset C$ because the elements $2$, $4$, and $6$ are not contained in $\{1, 3, 5, 7\}$.

• Exercise #1 Problems and Solutions:

  1. Listing elements of given sets:
  • Set $A = \{\text{even numbers less than } 15\}$$A = \{2, 4, 6, 8, 10, 12, 14\}$
  • Set $X = \{\text{letters used in the word 'mathematics'}\}$$X = \{m, a, t, h, e, i, c, s\}$
  • Set $B = \{\text{the first five letters of the alphabet}\}$$B = \{a, b, c, d, e\}$
  1. Given the sets $A = \{3, 6, 9, 12, 15, 18, 21\}$, $B = \{3, 6, 9\}$, $C = \{9, 12\}$, and $D = \{3, 12, 21\}$:
  • Question a: Which sets are subsets of set $A$?
    • Solution: Sets $B$, $C$, and $D$ are all subsets of set $A$ ($B \subset A$, $C \subset A$, $D \subset A$) because all elements of $B$, $C$, and $D$ are contained within $A$
  • Question b: Is $C \subset B$? Why or why not?
    • Solution: No, $C \not\subset B$ because $12 \in C$ but $12 \notin B$.
  • Question c: Is $B \subset D$? Why or why not?
    • Solution: No, $B \not\subset D$ because the elements $6 \in B$ and $9 \in B$ are not contained in set $D$.

The Real Number System

• The real number system encompasses all numbers that can be located on a continuous number line. The set of real numbers is denoted using the symbol $\mathbb{R}$.

• Natural Numbers ($\mathbb{N}$):

  • The set of natural numbers starts at $1$ and increases infinitely by integer increments.     $\mathbb{N} = \{1, 2, 3, 4, 5, 6, \dots\}$
  • Also termed counting numbers because they are used exclusively for counting objects and ordering sequences.

• Whole Numbers ($\mathbb{W}$):

  • The set of whole numbers includes all natural numbers along with zero ($0$).     $\mathbb{W} = \{0, 1, 2, 3, 4, 5, \dots\}$

• Integers ($\mathbb{Z}$):

  • The set of integers consists of natural numbers, zero, and the negative values of natural numbers.     $\mathbb{Z} = \{\dots, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, \dots\}$
  • Alternatively defined as the complete set of negative and positive whole numbers including zero.
  • Neutrality of Zero: Zero ($0$) is neither positive nor negative; it is classified as neutral.

• Rational Numbers ($\mathbb{Q}$):

  • The set of rational numbers includes natural numbers, whole numbers, integers, and any number that can be expressed as a quotient or fraction of two integers where the denominator is not zero.     $\mathbb{Q} = \left\{\frac{a}{b} : a \in \mathbb{Z}, b \in \mathbb{Z} \text{ and } b \neq 0\right\}$
  • Examples of rational numbers expressed as ratios:     $\frac{3}{25}, \quad \frac{2}{5}, \quad \frac{8}{1}, \quad \frac{-5}{1}, \quad \frac{0}{1}$
  • Decimal Representations of Rational Numbers:
  • Every rational number can be written as a decimal number.
  • Terminating Decimals: Decimals that end after a finite number of digits, such as $0.5$ or $1.684$.
  • Repeating Decimals: Decimals that repeat infinitely in a periodic pattern, such as $\frac{1}{3} = 0.333333\dots$ (or $0.\bar{3}$).

• Irrational Numbers ($\mathbb{I}$ or $\mathbb{Q}'$):

  • The set of numbers that cannot be expressed as a ratio of two integers.
  • Symbolized by $\mathbb{I}$ or $\mathbb{Q}'$
  • When expressed in decimal form, irrational numbers are non-terminating and non-repeating.
  • Examples:     $\sqrt{3}, \quad -\sqrt{7}, \quad \sqrt{\frac{2}{9}}, \quad \pi, \quad \sqrt{2}$

• Real Numbers ($\mathbb{R}$):

  • The set of real numbers is formed by the union of the set of rational numbers and the set of irrational numbers:     $\mathbb{R} = \mathbb{Q} \cup \mathbb{Q}'$

• Hierarchical Structure of the Real Number System:

  • The relations among these number sets form nested subsets:     $\mathbb{N} \subset \mathbb{W} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R}$
  • Irrational numbers $\mathbb{Q}'$ are disjoint from $\mathbb{Q}$, but both $\mathbb{Q} \subset \mathbb{R}$ and $\mathbb{Q}' \subset \mathbb{R}$.

Number Classification and Problem Solutions

• Exercise #2 Problem 1: Categorization Table:

  • Given numbers to evaluate:     $0, \quad 1, \quad 4, \quad -8, \quad 1.5, \quad \frac{2}{3}, \quad -\frac{1}{3}, \quad -0.25, \quad \sqrt{3}, \quad \sqrt{25}, \quad \pi$
  • Comprehensive Breakdown by Set:
  • Natural Numbers ($\mathbb{N}$): $1$, $4$, $\sqrt{25}$ (note: $\sqrt{25} = 5$)
  • Whole Numbers ($\mathbb{W}$): $0$, $1$, $4$, $\sqrt{25}$
  • Integers ($\mathbb{Z}$): $0$, $1$, $4$, $-8$, $\sqrt{25}$
  • Rational Numbers ($\mathbb{Q}$): $0$, $1$, $4$, $-8$, $1.5$, $\frac{2}{3}$, $-\frac{1}{3}$, $-0.25$, $\sqrt{25}$
  • Irrational Numbers ($\mathbb{Q}'$): $\sqrt{3}$, $\pi$

• Exercise #2 Problem 2: Detailed Mathematical Explanations:

  • a. Why is $1.5$ considered a rational number?
  • Explanation: The number $1.5$ is a terminating decimal. It can be written as a fraction of two integers, specifically $\frac{15}{10}$ or in simplest form $\frac{3}{2}$, fitting the exact criteria $\frac{a}{b}$ where $a, b \in \mathbb{Z}$ and $b \neq 0$.
  • b. A student claims that every square root is irrational. Do you agree? Explain using an example.
  • Explanation: Disagree. Not every square root is irrational. The square root of any perfect square evaluates to a rational integer. For example, $\sqrt{25} = 5$ (or $\sqrt{4} = 2$), which is a natural number, whole number, integer, and rational number.
  • c. Explain why $\pi$ is classified as an irrational number.
  • Explanation: The mathematical constant $\pi$ has a decimal expansion ($3.14159265\dots$) that continues infinitely without terminating and without repeating any fixed pattern of digits. Furthermore, it cannot be expressed as an exact ratio of two integers $\frac{a}{b}$.
  • d. Explain why $-8$ belongs to more than one set of numbers.
  • Explanation: The number $-8$ is a negative counting value, making it an integer ($-8 \in \mathbb{Z}$). Because any integer can be expressed as a fraction with a denominator of $1$ (i.e., $\frac{-8}{1}$), $-8$ is also a rational number ($-8 \in \mathbb{Q}$). Consequently, as a rational number, it is also a member of the set of real numbers ($-8 \in \mathbb{R}$).

The Number Line and Directed Numbers

• Definition of Directed Numbers:

  • Directed numbers are integers that combine a specific size (or magnitude) with a specific direction (positive or negative).
  • Positive Numbers: Represent movement in the positive direction, which corresponds to moving to the right on a horizontal number line.
  • Negative Numbers: Represent movement in the negative direction, which corresponds to moving to the left on a horizontal number line.

• Structure of the Number Line:

  • Origin / Midpoint: The number zero ($0$) serves as the origin or starting midpoint.
  • Values situated to the right of zero are positive ($> 0$).
  • Values situated to the left of zero are negative ($< 0$).
  • Number lines provide a visual representation of real numbers by mapping each unique number to a distinct point on a line.

• Rules for Comparing Numbers on a Number Line:

  • When comparing any two numbers on a number line, the larger number is always located to the right of the smaller number.
  • Examples:
  • $5 > 1$, because $5$ lies to the right of $1$
  • $2 > -3$, because $2$ lies to the right of $-3$
  • $-1 > -4$, because $-1$ lies to the right of $-4$

• Addition and Subtraction of Directed Numbers Using a Number Line:

  • To add or subtract directed numbers using a number line:
  1. Locate the initial starting number on the line.
  2. For addition of a positive value, move the specified number of units to the right (positive direction).
  3. For subtraction or adding a negative value, move the specified number of units to the left (negative direction).
  • Worked Example 1: Find the sum of $1$ and $3$ ($1 + 3$):
  • Start at the position of the first number, $1$.
  • Move $3$ units in the positive direction (to the right):       $\text{Position } 1 \xrightarrow{+1} 2 \xrightarrow{+1} 3 \xrightarrow{+1} 4$
  • Conclude that $1 + 3 = 4$.