Introduction to Number Systems and Types

Broad Classification of Numbers

The world of numbers is vast and interconnected. To bring order and understanding to this diversity, mathematicians classify numbers into various sets based on their shared properties and characteristics. This hierarchical classification helps in studying their behaviour under different operations and relationships.

The most fundamental and overarching classification divides numbers into two major categories: Complex Numbers, which contain Real Numbers and Imaginary Numbers.

Complex Numbers ($\mathbb{C}$)

Complex numbers represent the largest set of numbers typically encountered in many areas of mathematics and engineering. They are numbers that can be expressed in the standard form $a + bi$, where $a$ and $b$ are any real numbers, and $i$ is the imaginary unit, defined by the property $i^2 = -1$.

In a complex number $z = a + bi$, the term $a$ is referred to as the real part of $z$, denoted as $\text{Re}(z) = a$. The term $b$ is referred to as the imaginary part of $z$, denoted as $\text{Im}(z) = b$. Note that the imaginary part is the real coefficient $b$, not $bi$.

The set of all complex numbers is denoted by the symbol $\mathbb{C}$.

Examples of complex numbers include:

• $2 + 3i$ (Here, $a=2, b=3$)
• $-1 - 5i$ (Here, $a=-1, b=-5$)
• $4$ (This is a complex number where $b=0$. It can be written as $4 + 0i$. The real part is 4, imaginary part is 0.)
• $-6i$ (This is a complex number where $a=0$ and $b \neq 0$. It can be written as $0 - 6i$. The real part is 0, imaginary part is -6.)
• $0$ (This is $0 + 0i$. Real part is 0, imaginary part is 0.)
• $\sqrt{2} + \pi i$ (Here, $a=\sqrt{2}, b=\pi$, both are real numbers)

The set of real numbers is a subset of the complex numbers (when the imaginary part is 0). The set of purely imaginary numbers (complex numbers with a real part of 0 and a non-zero imaginary part) is also a subset of complex numbers.

Real Numbers ($\mathbb{R}$)

Real numbers are a fundamental subset of complex numbers where the imaginary part is exactly zero ($b=0$). They are numbers that can be placed on a continuous number line. This set encompasses all numbers used for measurements, quantities, and distances.

The set of real numbers is denoted by the symbol $\mathbb{R}$.

Examples of real numbers include:

• All positive numbers (e.g., $1, 5, 100$)
• All negative numbers (e.g., $-1, -5, -100$)
• Zero ($0$)
• All fractions (e.g., $\frac{1}{2}, -\frac{3}{4}, \frac{22}{7}$)
• All terminating decimals (e.g., $0.25, -3.75$)
• All non-terminating, repeating decimals (e.g., $0.333..., 1.2727...$)
• All irrational numbers (e.g., $\sqrt{2}, \pi, e$)

Real numbers are further subdivided into two distinct and non-overlapping categories: Rational Numbers and Irrational Numbers.

Rational Numbers ($\mathbb{Q}$)

Rational numbers are real numbers that can be expressed as a ratio or fraction of two integers. Specifically, a number $x$ is rational if it can be written in the form $\frac{p}{q}$, where $p$ and $q$ are integers, and $q$ is not equal to zero ($q \neq 0$).

The set of rational numbers is denoted by $\mathbb{Q}$, which originates from the word "quotient".

Key characteristics and examples of rational numbers:

• Integers: Any integer $n$ can be written as $\frac{n}{1}$, so all integers are rational numbers. E.g., $7 = \frac{7}{1}$, $-4 = \frac{-4}{1}$, $0 = \frac{0}{1}$.
• Fractions: Any common fraction $\frac{p}{q}$ where $p, q \in \mathbb{Z}$ and $q \neq 0$. E.g., $\frac{3}{5}, -\frac{8}{3}$.
• Terminating Decimals: Decimals that end after a finite number of digits can always be written as a fraction. E.g., $0.75 = \frac{75}{100} = \frac{3}{4}$, $2.125 = \frac{2125}{1000} = \frac{17}{8}$.
• Non-terminating, Repeating Decimals: Decimals that continue infinitely but with a repeating sequence of digits can also be written as a fraction. E.g., $0.333... = 0.\overline{3} = \frac{1}{3}$. The process for converting repeating decimals to $\frac{p}{q}$ form will be discussed later.

The decimal expansion of a rational number is always either terminating or non-terminating and repeating.

Irrational Numbers ($\mathbb{I}$)

Irrational numbers are real numbers that are not rational. By definition, an irrational number cannot be expressed as a simple fraction $\frac{p}{q}$ where $p$ and $q$ are integers and $q \neq 0$.

The set of irrational numbers is often denoted by $\mathbb{I}$. Since real numbers consist of rational and irrational numbers and they have no overlap, the set of irrational numbers can also be represented as $\mathbb{R} \setminus \mathbb{Q}$ (the set difference between Real numbers and Rational numbers).

Key characteristics and examples of irrational numbers:

• Decimal Expansion: The decimal representation of an irrational number is always non-terminating and non-repeating. The digits after the decimal point go on infinitely without any repeating pattern.
• Roots of Non-perfect Powers: The roots of numbers that are not perfect squares, cubes, etc., are often irrational. E.g., $\sqrt{2}$ (approximately $1.41421356...$), $\sqrt{3}$ (approximately $1.73205081...$), $\sqrt[3]{5}$ (approximately $1.70997595...$).
• Transcendental Numbers: Certain mathematical constants that are not roots of any non-zero polynomial equations with integer coefficients. Famous examples include $\pi$ (Pi, approximately $3.14159265...$) and $e$ (Euler's number, approximately $2.718281828...$).

Irrational numbers fill the gaps on the number line that are not occupied by rational numbers. Together, rational and irrational numbers form the complete set of real numbers.

Imaginary Numbers ($\mathbb{I}_m$ or $i\mathbb{R}$)

Imaginary numbers are a subset of complex numbers where the real part is zero ($a=0$). They are numbers of the form $bi$, where $b$ is a real number and $i$ is the imaginary unit ($i^2 = -1$). When $b=0$, the number is $0i = 0$, which is both real and imaginary. Usually, the term "purely imaginary number" is used when $b \neq 0$.

The set of all imaginary numbers is $\{bi \ | \ b \in \mathbb{R}\}$. The set of purely imaginary numbers is $\{bi \ | \ b \in \mathbb{R}, b \neq 0\}$.

Examples of imaginary numbers:

• $i$ (Here, $b=1$)
• $5i$ (Here, $b=5$)
• $-2.5i$ (Here, $b=-2.5$)
• $\sqrt{-4} = \sqrt{4 \times -1} = \sqrt{4} \times \sqrt{-1} = 2i$
• $\sqrt{-18} = \sqrt{9 \times 2 \times -1} = 3\sqrt{2}i$

A key property is that squaring a purely imaginary number results in a negative real number:

For example, $(7i)^2 = 49 \times (-1) = -49$.

Sub-Classifications of Rational Numbers

Rational numbers themselves can be further divided into simpler sets, forming a hierarchy:

Integers ($\mathbb{Z}$)

Integers are the set of whole numbers and their additive inverses (negatives). They are rational numbers with no fractional component when written in their simplest form. They can be positive, negative, or zero.

The set of integers is denoted by $\mathbb{Z}$, from the German word 'Zahlen' meaning numbers.

$\mathbb{Z} = \{..., -4, -3, -2, -1, 0, 1, 2, 3, 4, ...\}$.

Every integer $n$ can be written as the rational number $\frac{n}{1}$.

Whole Numbers ($\mathbb{W}$)

Whole numbers are a subset of integers that includes zero and all the positive integers. They are the non-negative integers.

The set of whole numbers is denoted by $\mathbb{W}$.

$\mathbb{W} = \{0, 1, 2, 3, 4, 5, ...\}$.

Every whole number is an integer, and thus also a rational and a real number.

Natural Numbers ($\mathbb{N}$ or $\mathbb{Z}^+$)

Natural numbers are the numbers used for counting. They are the positive integers, starting from 1.

The set of natural numbers is most commonly denoted by $\mathbb{N}$. Sometimes, depending on the context or convention being followed, the set of natural numbers might include 0 ($\{0, 1, 2, ...\}$). However, in many mathematical fields, $\mathbb{N}$ starts at 1. Another notation for positive integers is $\mathbb{Z}^+$. We will follow the convention where natural numbers start from 1.

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

Every natural number is a whole number, an integer, a rational number, a real number, and a complex number.

Other specific types of integers (like even, odd, prime, composite) are subsets of integers and will be discussed later.

Summary of Number System Hierarchy

The relationship between these number sets is hierarchical, with each set (except for the most general, Complex numbers) being a subset of the set above it in the hierarchy. This can be shown using the subset symbol $\subset$.

This sequence shows that Natural Numbers are contained within Whole Numbers, Whole Numbers within Integers, Integers within Rational Numbers, Rational Numbers within Real Numbers, and Real Numbers within Complex Numbers.

The set of Real Numbers is composed of two mutually exclusive sets: Rational and Irrational Numbers.

Complex numbers can be defined based on real numbers:

A visual representation often helps to understand this classification:

*(Note: The diagram typically shows the most general set (Complex) at the top, branching or encompassing subsets down to the most specific (Natural), illustrating the 'contains' relationship.)*

Natural Numbers: Definition and Properties

Definition of Natural Numbers

Natural numbers are the most basic set of numbers, intuitively used for counting distinct objects. They are the positive integers, beginning with 1.

The set of natural numbers is universally denoted by the symbol $\mathbb{N}$.

While there are conventions where the set might include 0, in the context of school mathematics, particularly in India, the set of natural numbers $\mathbb{N}$ is commonly defined as starting from 1:

This set includes all positive integers that extend infinitely. Numbers like $1, 2, 10, 500, 100000$ are all natural numbers.

The alternative definition, sometimes denoted as $\mathbb{N}_0$ or $\mathbb{W}$ (Whole Numbers), includes 0:

Unless specifically mentioned otherwise, when we refer to "natural numbers" in these notes, we will be using the definition $\mathbb{N} = \{1, 2, 3, ...\}$.

Properties of Natural Numbers under Basic Operations

Let's examine how the set of natural numbers ($\mathbb{N} = \{1, 2, 3, ...\}$) behaves under the four basic arithmetic operations: addition, subtraction, multiplication, and division. We will look at several important properties.

Let $a, b,$ and $c$ be any three natural numbers.

1. Closure Property

A set is said to be closed under an operation if, when you perform that operation on any two elements from the set, the result is also an element of the same set.

• Closure under Addition:

  For any two natural numbers $a$ and $b$, their sum $a + b$ is always a natural number.

  If $a \in \mathbb{N}$ and $b \in \mathbb{N}$, then $a + b \in \mathbb{N}$.

  (Closure Property of Addition)

  Example: If $a = 5$ and $b = 12$, both are natural numbers. Their sum is $5 + 12 = 17$, which is also a natural number. This holds true for any pair of natural numbers.

  Thus, the set of natural numbers is closed under addition.

• Closure under Subtraction:

  For any two natural numbers $a$ and $b$, their difference $a - b$ is not always a natural number.

  If $a \in \mathbb{N}$ and $b \in \mathbb{N}$, $a - b$ is not necessarily in $\mathbb{N}$.

  Example: If $a = 3$ and $b = 7$, both are natural numbers. Their difference is $3 - 7 = -4$, which is an integer but not a natural number (since natural numbers are positive). Another example: $5 - 5 = 0$, and 0 is not a natural number according to our definition $\mathbb{N} = \{1, 2, 3, ...\}$.

  Thus, the set of natural numbers is not closed under subtraction.

• Closure under Multiplication:

  For any two natural numbers $a$ and $b$, their product $a \times b$ is always a natural number.

  If $a \in \mathbb{N}$ and $b \in \mathbb{N}$, then $a \times b \in \mathbb{N}$.

  (Closure Property of Multiplication)

  Example: If $a = 8$ and $b = 4$, both are natural numbers. Their product is $8 \times 4 = 32$, which is also a natural number. This property holds for any pair of natural numbers.

  Thus, the set of natural numbers is closed under multiplication.

• Closure under Division:

  For any two natural numbers $a$ and $b$, their quotient $a \div b$ is not always a natural number.

  If $a \in \mathbb{N}$ and $b \in \mathbb{N}$, $a \div b$ is not necessarily in $\mathbb{N}$.

  Example: If $a = 10$ and $b = 3$, both are natural numbers. Their quotient is $10 \div 3 = \frac{10}{3}$, which is a rational number but not a natural number. Another example: $5 \div 5 = 1$, which *is* a natural number, but the property must hold for *all* pairs, so one counterexample is enough to show it is not closed.

  Thus, the set of natural numbers is not closed under division.

2. Commutative Property

An operation '*' on a set is commutative if changing the order of the operands does not change the result. Mathematically, $a * b = b * a$ for all elements $a$ and $b$ in the set.

• Commutativity of Addition:

  The order in which two natural numbers are added does not affect the sum.

  If $a \in \mathbb{N}$ and $b \in \mathbb{N}$, then $a + b = b + a$.

  (Commutative Property of Addition)

  Example: $8 + 5 = 13$ and $5 + 8 = 13$. Thus, $8 + 5 = 5 + 8$. This holds for any pair of natural numbers.

  Thus, addition is commutative for natural numbers.

• Commutativity of Subtraction:

  The order in which subtraction is performed does affect the result.

  If $a \in \mathbb{N}$ and $b \in \mathbb{N}$, $a - b \neq b - a$ (unless $a=b=c$).

  Example: $10 - 4 = 6$, but $4 - 10 = -6$. Since $6 \neq -6$, $10 - 4 \neq 4 - 10$.

  Thus, subtraction is not commutative for natural numbers.

• Commutativity of Multiplication:

  The order in which two natural numbers are multiplied does not affect the product.

  If $a \in \mathbb{N}$ and $b \in \mathbb{N}$, then $a \times b = b \times a$.

  (Commutative Property of Multiplication)

  Example: $6 \times 9 = 54$ and $9 \times 6 = 54$. Thus, $6 \times 9 = 9 \times 6$. This property holds for any pair of natural numbers.

  Thus, multiplication is commutative for natural numbers.

• Commutativity of Division:

  The order in which division is performed does affect the result.

  If $a \in \mathbb{N}$ and $b \in \mathbb{N}$, $a \div b \neq b \div a$ (unless $a=b$).

  Example: $12 \div 4 = 3$, but $4 \div 12 = \frac{4}{12} = \frac{1}{3}$. Since $3 \neq \frac{1}{3}$, $12 \div 4 \neq 4 \div 12$.

  Thus, division is not commutative for natural numbers.

3. Associative Property

An operation '*' on a set is associative if the way in which numbers are grouped when performing the operation on three or more numbers does not affect the result. Mathematically, $(a * b) * c = a * (b * c)$ for all elements $a, b,$ and $c$ in the set.

• Associativity of Addition:

  When adding three or more natural numbers, the grouping of numbers does not affect the sum.

  If $a, b, c \in \mathbb{N}$, then $(a + b) + c = a + (b + c)$.

  (Associative Property of Addition)

  Example: Let $a = 2, b = 3, c = 4$.

  Left side: $(2 + 3) + 4 = 5 + 4 = 9$.

  Right side: $2 + (3 + 4) = 2 + 7 = 9$.

  Since $9 = 9$, the property holds. This is true for any natural numbers $a, b, c$.

  Thus, addition is associative for natural numbers.

• Associativity of Subtraction:

  When subtracting three or more natural numbers, the grouping does affect the result.

  If $a, b, c \in \mathbb{N}$, $(a - b) - c \neq a - (b - c)$ (in general).

  Example: Let $a = 10, b = 5, c = 2$.

  Left side: $(10 - 5) - 2 = 5 - 2 = 3$.

  Right side: $10 - (5 - 2) = 10 - 3 = 7$.

  Since $3 \neq 7$, $(10 - 5) - 2 \neq 10 - (5 - 2)$.

  Thus, subtraction is not associative for natural numbers.

• Associativity of Multiplication:

  When multiplying three or more natural numbers, the grouping of numbers does not affect the product.

  If $a, b, c \in \mathbb{N}$, then $(a \times b) \times c = a \times (b \times c)$.

  (Associative Property of Multiplication)

  Example: Let $a = 2, b = 3, c = 4$.

  Left side: $(2 \times 3) \times 4 = 6 \times 4 = 24$.

  Right side: $2 \times (3 \times 4) = 2 \times 12 = 24$.

  Since $24 = 24$, the property holds. This is true for any natural numbers $a, b, c$.

  Thus, multiplication is associative for natural numbers.

• Associativity of Division:

  When dividing three or more natural numbers, the grouping does affect the result.

  If $a, b, c \in \mathbb{N}$, $(a \div b) \div c \neq a \div (b \div c)$ (in general).

  Example: Let $a = 24, b = 6, c = 2$.

  Left side: $(24 \div 6) \div 2 = 4 \div 2 = 2$.

  Right side: $24 \div (6 \div 2) = 24 \div 3 = 8$.

  Since $2 \neq 8$, $(24 \div 6) \div 2 \neq 24 \div (6 \div 2)$.

  Thus, division is not associative for natural numbers.

4. Identity Property (Existence of Identity Element)

An identity element for an operation '*' is an element 'e' such that when combined with any element 'a' from the set using the operation, it leaves 'a' unchanged. Mathematically, $a * e = e * a = a$ for all $a$ in the set, and $e$ must also be in the set.

• Additive Identity:

  The additive identity is the number that, when added to any number, results in the original number. This number is 0 ($a + 0 = 0 + a = a$).

  However, in our definition, the set of natural numbers $\mathbb{N} = \{1, 2, 3, ...\}$ does not include 0.

  The additive identity (0) $\notin \mathbb{N}$.

  Thus, the additive identity does not exist within the set of natural numbers.

• Multiplicative Identity:

  The multiplicative identity is the number that, when multiplied by any number, results in the original number. This number is 1 ($a \times 1 = 1 \times a = a$).

  In our definition, the set of natural numbers $\mathbb{N} = \{1, 2, 3, ...\}$ does include 1.

  For any $a \in \mathbb{N}$, $a \times 1 = 1 \times a = a$.

  (Multiplicative Identity Property)

  Thus, the multiplicative identity exists within the set of natural numbers (it is 1).

5. Inverse Property (Existence of Inverse Element)

For an operation '*' and an identity element 'e', the inverse element of 'a' (denoted $a^{-1}$) is an element in the set such that $a * a^{-1} = a^{-1} * a = e$.

• Additive Inverse:

  For an element $a$, its additive inverse is $-a$, such that $a + (-a) = 0$ (the additive identity). For any natural number $a$ (which is $a \geq 1$), its additive inverse is $-a$ (which is $\leq -1$).

  None of these additive inverses (except for the conceptual inverse of 0, which is 0, but 0 isn't in $\mathbb{N}$) are in the set of natural numbers.

  For $a \in \mathbb{N}$, its additive inverse $(-a)$ $\notin \mathbb{N}$ (for $a > 0$).

  Thus, additive inverses do not exist for elements in the set of natural numbers.

• Multiplicative Inverse:

  For an element $a$, its multiplicative inverse is $\frac{1}{a}$, such that $a \times \frac{1}{a} = 1$ (the multiplicative identity). For any natural number $a > 1$, its multiplicative inverse $\frac{1}{a}$ is a fraction (e.g., the inverse of 2 is $\frac{1}{2}$).

  These fractional inverses are not in the set of natural numbers.

  The only natural number that has a multiplicative inverse which is also a natural number is 1 (its inverse is $\frac{1}{1}=1$).

  For $a \in \mathbb{N}$, its multiplicative inverse $(\frac{1}{a})$ $\notin \mathbb{N}$ (for $a > 1$).

  Thus, multiplicative inverses generally do not exist for elements in the set of natural numbers (except for the number 1 itself).

6. Distributive Property

The distributive property relates two operations, usually multiplication and addition/subtraction. It states that multiplying a number by a sum or difference is the same as multiplying the number by each term in the sum or difference separately and then adding or subtracting the results.

For multiplication over addition:

Since multiplication is commutative for natural numbers ($a \times c = c \times a$ and $b \times c = c \times b$), the left and right distributive properties result in the same outcomes for natural numbers.

Example: Let $a=5, b=3, c=4$.

$a \times (b + c) = 5 \times (3 + 4) = 5 \times 7 = 35$.

$(a \times b) + (a \times c) = (5 \times 3) + (5 \times 4) = 15 + 20 = 35$.

Since $35 = 35$, $a \times (b + c) = (a \times b) + (a \times c)$.

For multiplication over subtraction:

We require $b > c$ here to ensure that $b-c$ is a natural number, making the left side defined within the context of natural numbers operating on natural numbers. Even if $b-c$ results in an integer (when $b \leq c$), the property still holds in larger sets like integers or real numbers.

Example: Let $a=6, b=8, c=2$. (Note that $b > c$)

$a \times (b - c) = 6 \times (8 - 2) = 6 \times 6 = 36$.

$(a \times b) - (a \times c) = (6 \times 8) - (6 \times 2) = 48 - 12 = 36$.

Since $36 = 36$, $a \times (b - c) = (a \times b) - (a \times c)$ when the differences are defined as natural numbers or integers.

Thus, multiplication is distributive over both addition and subtraction for natural numbers.

Summary Table of Properties for Natural Numbers ($\mathbb{N}$)

Understanding these properties is crucial as we move to larger sets of numbers, where some properties that didn't hold for natural numbers might hold.

Whole Numbers: Definition and Properties

Definition of Whole Numbers

Whole numbers are a fundamental set of numbers formed by combining the set of natural numbers with the number zero. If we follow the convention that natural numbers $\mathbb{N} = \{1, 2, 3, 4, ...\}$, then the set of whole numbers, denoted by $\mathbb{W}$, is:

This means the set of whole numbers includes zero and all the positive integers. In other words, whole numbers are the set of non-negative integers.

Based on this definition, the set of natural numbers is a proper subset of the set of whole numbers:

Examples of whole numbers include $0, 1, 10, 250, 1000000$.

It is important to note that every natural number is a whole number, but the converse is not true, as $0$ is a whole number but not a natural number (under our adopted convention for $\mathbb{N}$).

Properties of Whole Numbers under Basic Operations

Let $a, b,$ and $c$ be any three whole numbers. Let's examine the properties of whole numbers under the basic arithmetic operations: addition, subtraction, multiplication, and division.

1. Closure Property

A set is closed under an operation if performing the operation on any two elements from the set always produces a result that is also within the same set.

• Closure under Addition:

  For any two whole numbers $a$ and $b$, their sum $a + b$ is always a whole number.

  If $a \in \mathbb{W}$ and $b \in \mathbb{W}$, then $a + b \in \mathbb{W}$.

  (Closure Property of Addition)

  Example 1. Check if the sum of two whole numbers 8 and 0 is a whole number.

  Answer:

  Given whole numbers are $a = 8$ and $b = 0$. Both $8 \in \mathbb{W}$ and $0 \in \mathbb{W}$.

  Their sum is $a + b = 8 + 0 = 8$.

  The result $8$ is a whole number ($8 \in \mathbb{W}$). This confirms the closure property for this specific case.

  This property holds true for any pair of whole numbers, including sums involving 0.

  Thus, the set of whole numbers is closed under addition.

• Closure under Subtraction:

  For any two whole numbers $a$ and $b$, their difference $a - b$ is not always a whole number.

  If $a \in \mathbb{W}$ and $b \in \mathbb{W}$, $a - b$ is not necessarily in $\mathbb{W}$.

  Example 1. Show that whole numbers are not closed under subtraction.

  Answer:

  Let $a = 3$ and $b = 5$. Both $a$ and $b$ are whole numbers ($3 \in \mathbb{W}$, $5 \in \mathbb{W}$).

  Consider their difference: $a - b = 3 - 5 = -2$.

  The result, $-2$, is an integer but it is not a whole number (since whole numbers must be non-negative, starting from 0).

  Since the result of subtracting two whole numbers is not always a whole number, the set of whole numbers is not closed under subtraction.

• Closure under Multiplication:

  For any two whole numbers $a$ and $b$, their product $a \times b$ is always a whole number.

  If $a \in \mathbb{W}$ and $b \in \mathbb{W}$, then $a \times b \in \mathbb{W}$.

  (Closure Property of Multiplication)

  Example 1. Verify the closure property of multiplication for whole numbers using 7 and 0.

  Answer:

  Given whole numbers are $a = 7$ and $b = 0$. Both $7 \in \mathbb{W}$ and $0 \in \mathbb{W}$.

  Their product is $a \times b = 7 \times 0 = 0$.

  The result $0$ is a whole number ($0 \in \mathbb{W}$). This holds true for any pair of whole numbers.

  Thus, the set of whole numbers is closed under multiplication.

• Closure under Division:

  For any two whole numbers $a$ and $b$, their quotient $a \div b$ is not always a whole number.

  If $a \in \mathbb{W}$ and $b \in \mathbb{W}$, $a \div b$ is not necessarily in $\mathbb{W}$.

  Example 1. Give an example to show that division is not closed for whole numbers.

  Answer:

  Let $a = 5$ and $b = 2$. Both $a$ and $b$ are whole numbers ($5 \in \mathbb{W}$, $2 \in \mathbb{W}$).

  Consider their quotient: $a \div b = 5 \div 2 = \frac{5}{2}$.

  The result, $\frac{5}{2}$ or $2.5$, is a rational number but it is not a whole number.

  Also, note that division by zero ($a \div 0$ where $a \in \mathbb{W}, a \neq 0$) is undefined, and $0 \div 0$ is indeterminate. The result of division is not guaranteed to be a whole number.

  Thus, the set of whole numbers is not closed under division.

2. Commutative Property

An operation '*' is commutative if changing the order of the operands does not affect the result ($a * b = b * a$).

• Commutativity of Addition:

  The order in which two whole numbers are added does not affect the sum.

  If $a \in \mathbb{W}$ and $b \in \mathbb{W}$, then $a + b = b + a$.

  (Commutative Property of Addition)

  Example 1. Check if addition is commutative for the whole numbers 0 and 9.

  Answer:

  Let $a=0$ and $b=9$. Both $0 \in \mathbb{W}$ and $9 \in \mathbb{W}$.

  Consider $a+b$: $0 + 9 = 9$.

  Consider $b+a$: $9 + 0 = 9$.

  Since $0 + 9 = 9 + 0$, the property holds for this pair. This holds for any pair of whole numbers.

  Thus, addition is commutative for whole numbers.

• Commutativity of Subtraction:

  Subtraction of whole numbers is not commutative.

  If $a \in \mathbb{W}$ and $b \in \mathbb{W}$, $a - b \neq b - a$ (unless $a=b$).

  Example 1. Demonstrate that subtraction is not commutative for whole numbers using 15 and 6.

  Answer:

  Let $a = 15$ and $b = 6$. Both $15 \in \mathbb{W}$ and $6 \in \mathbb{W}$.

  Consider $a-b$: $15 - 6 = 9$.

  Consider $b-a$: $6 - 15 = -9$.

  Since $9 \neq -9$, $15 - 6 \neq 6 - 15$.

  Thus, subtraction is not commutative for whole numbers.

• Commutativity of Multiplication:

  The order in which two whole numbers are multiplied does not affect the product.

  If $a \in \mathbb{W}$ and $b \in \mathbb{W}$, then $a \times b = b \times a$.

  (Commutative Property of Multiplication)

  Example 1. Verify the commutative property of multiplication for the whole numbers 4 and 7.

  Answer:

  Let $a=4$ and $b=7$. Both $4 \in \mathbb{W}$ and $7 \in \mathbb{W}$.

  Consider $a \times b$: $4 \times 7 = 28$.

  Consider $b \times a$: $7 \times 4 = 28$.

  Since $4 \times 7 = 7 \times 4$, the property holds for this pair. This is true for any pair of whole numbers.

  Thus, multiplication is commutative for whole numbers.

• Commutativity of Division:

  Division of whole numbers is not commutative.

  If $a \in \mathbb{W}$ and $b \in \mathbb{W}$, $a \div b \neq b \div a$ (unless $a=b$, or one is 1 and the other is 1, excluding division by 0).

  Example 1. Show that division is not commutative for whole numbers using 10 and 2.

  Answer:

  Let $a = 10$ and $b = 2$. Both $10 \in \mathbb{W}$ and $2 \in \mathbb{W}$.

  Consider $a \div b$: $10 \div 2 = 5$.

  Consider $b \div a$: $2 \div 10 = \frac{2}{10} = \frac{1}{5}$.

  Since $5 \neq \frac{1}{5}$, $10 \div 2 \neq 2 \div 10$.

  Thus, division is not commutative for whole numbers.

3. Associative Property

An operation '*' is associative if the grouping of numbers does not affect the result when performing the operation on three or more elements $[(a * b) * c = a * (b * c)]$.

• Associativity of Addition:

  When adding three or more whole numbers, the way they are grouped does not affect the sum.

  If $a, b, c \in \mathbb{W}$, then $(a + b) + c = a + (b + c)$.

  (Associative Property of Addition)

  Example 1. Verify the associative property of addition for whole numbers 5, 0, and 3.

  Answer:

  Let $a = 5, b = 0, c = 3$. All are whole numbers.

  Consider $(a + b) + c$: $(5 + 0) + 3 = 5 + 3 = 8$.

  Consider $a + (b + c)$: $5 + (0 + 3) = 5 + 3 = 8$.

  Since $(5 + 0) + 3 = 5 + (0 + 3)$, the property holds for this example. It holds for any three whole numbers.

  Thus, addition is associative for whole numbers.

• Associativity of Subtraction:

  Subtraction of whole numbers is not associative.

  If $a, b, c \in \mathbb{W}$, $(a - b) - c \neq a - (b - c)$ (in general).

  Example 1. Show that subtraction is not associative for whole numbers using 15, 5, and 2.

  Answer:

  Let $a = 15, b = 5, c = 2$. All are whole numbers.

  Consider $(a - b) - c$: $(15 - 5) - 2 = 10 - 2 = 8$.

  Consider $a - (b - c)$: $15 - (5 - 2) = 15 - 3 = 12$.

  Since $8 \neq 12$, $(15 - 5) - 2 \neq 15 - (5 - 2)$.

  Thus, subtraction is not associative for whole numbers.

• Associativity of Multiplication:

  When multiplying three or more whole numbers, the grouping of numbers does not affect the product.

  If $a, b, c \in \mathbb{W}$, then $(a \times b) \times c = a \times (b \times c)$.

  (Associative Property of Multiplication)

  Example 1. Verify the associative property of multiplication for whole numbers 2, 3, and 0.

  Answer:

  Let $a = 2, b = 3, c = 0$. All are whole numbers.

  Consider $(a \times b) \times c$: $(2 \times 3) \times 0 = 6 \times 0 = 0$.

  Consider $a \times (b \times c)$: $2 \times (3 \times 0) = 2 \times 0 = 0$.

  Since $(2 \times 3) \times 0 = 2 \times (3 \times 0)$, the property holds for this example. It holds for any three whole numbers.

  Thus, multiplication is associative for whole numbers.

• Associativity of Division:

  Division of whole numbers is not associative.

  If $a, b, c \in \mathbb{W}$, $(a \div b) \div c \neq a \div (b \div c)$ (in general, and avoiding division by zero).

  Example 1. Show that division is not associative for whole numbers using 36, 6, and 3.

  Answer:

  Let $a = 36, b = 6, c = 3$. All are whole numbers.

  Consider $(a \div b) \div c$: $(36 \div 6) \div 3 = 6 \div 3 = 2$.

  Consider $a \div (b \div c)$: $36 \div (6 \div 3) = 36 \div 2 = 18$.

  Since $2 \neq 18$, $(36 \div 6) \div 3 \neq 36 \div (6 \div 3)$.

  Thus, division is not associative for whole numbers.

4. Identity Property (Existence of Identity Element)

An identity element for an operation '*' is an element 'e' in the set such that for any element 'a' in the set, $a * e = e * a = a$.

• Additive Identity:

  The additive identity is the number that, when added to any number, results in the original number. This number is 0 ($a + 0 = 0 + a = a$).

  Since 0 is an element of the set of whole numbers ($\mathbb{W} = \{0, 1, 2, ...\}$), the additive identity exists in $\mathbb{W}$.

  For any $a \in \mathbb{W}$, $a + 0 = 0 + a = a$.

  (Additive Identity Property)

  Example 1. Identify the additive identity in whole numbers and verify it with a number.

  Answer:

  The additive identity is $0$. For any whole number, say $15 \in \mathbb{W}$:

  $15 + 0 = 15$ and $0 + 15 = 15$.

  Since $0$ is in $\mathbb{W}$, the additive identity exists in the set of whole numbers and is $0$.

• Multiplicative Identity:

  The multiplicative identity is the number that, when multiplied by any number, results in the original number. This number is 1 ($a \times 1 = 1 \times a = a$).

  Since 1 is an element of the set of whole numbers ($\mathbb{W} = \{0, 1, 2, ...\}$), the multiplicative identity exists in $\mathbb{W}$.

  For any $a \in \mathbb{W}$, $a \times 1 = 1 \times a = a$.

  (Multiplicative Identity Property)

  Example 1. Identify the multiplicative identity in whole numbers and verify it with a number.

  Answer:

  The multiplicative identity is $1$. For any whole number, say $23 \in \mathbb{W}$:

  $23 \times 1 = 23$ and $1 \times 23 = 23$.

  Since $1$ is in $\mathbb{W}$, the multiplicative identity exists in the set of whole numbers and is $1$.

5. Inverse Property (Existence of Inverse Element)

For an operation '*' and an identity element 'e', the inverse element of 'a' ($a^{-1}$) is an element in the set such that $a * a^{-1} = a^{-1} * a = e$.

• Additive Inverse:

  The additive inverse of a whole number $a$ is $-a$, such that $a + (-a) = 0$ (the additive identity).

  For any non-zero whole number $a$ (i.e., $a \geq 1$), its additive inverse $-a$ is a negative integer (e.g., the inverse of 5 is -5).

  Negative integers are not in the set of whole numbers $\mathbb{W}$.

  Only the number 0 has an additive inverse within $\mathbb{W}$, which is $0$ itself ($0 + 0 = 0$).

  For $a \in \mathbb{W}$ ($a > 0$), its additive inverse $(-a)$ $\notin \mathbb{W}$.

  Example 1. Does the number 7 have an additive inverse in the set of whole numbers?

  Answer:

  The additive inverse of 7 is the number $x$ such that $7 + x = 0$. Solving for $x$, we get $x = -7$.

  The number $-7$ is a negative integer and is not included in the set of whole numbers $\mathbb{W} = \{0, 1, 2, ...\}$.

  Therefore, 7 does not have an additive inverse within the set of whole numbers.

  Thus, additive inverses generally do not exist for elements in the set of whole numbers (except for 0).

• Multiplicative Inverse:

  The multiplicative inverse of a whole number $a$ is $\frac{1}{a}$, such that $a \times \frac{1}{a} = 1$ (the multiplicative identity).

  For any whole number $a > 1$, its multiplicative inverse $\frac{1}{a}$ is a fraction or decimal which is not a whole number (e.g., the inverse of 4 is $\frac{1}{4}$ or 0.25).

  The number 0 does not have a multiplicative inverse, as division by zero is undefined (there is no number $x$ such that $0 \times x = 1$).

  Only the number 1 has a multiplicative inverse within $\mathbb{W}$, which is $1$ itself ($1 \times \frac{1}{1} = 1$).

  For $a \in \mathbb{W}$ ($a > 1$), its multiplicative inverse $(\frac{1}{a})$ $\notin \mathbb{W}$.

  Example 1. Does the number 5 have a multiplicative inverse in the set of whole numbers?

  Answer:

  The multiplicative inverse of 5 is the number $x$ such that $5 \times x = 1$. Solving for $x$, we get $x = \frac{1}{5}$ or $0.2$.

  The number $\frac{1}{5}$ is a fraction and is not included in the set of whole numbers $\mathbb{W} = \{0, 1, 2, ...\}$.

  Therefore, 5 does not have a multiplicative inverse within the set of whole numbers.

  Thus, multiplicative inverses generally do not exist for elements in the set of whole numbers (except for 1).

6. Distributive Property

The distributive property connects multiplication with addition or subtraction. For whole numbers, multiplication distributes over both addition and subtraction.

• Multiplication over Addition:

  Multiplying a whole number by the sum of two other whole numbers is equal to the sum of the products of the first number with each of the other two numbers.

  If $a, b, c \in \mathbb{W}$, then $a \times (b + c) = (a \times b) + (a \times c)$.

  (Distributive Property over Addition)

  Example 1. Verify the distributive property of multiplication over addition for $a=6, b=4, c=5$.

  Answer:

  Let $a=6, b=4, c=5$. All are whole numbers.

  Left side: $a \times (b + c) = 6 \times (4 + 5) = 6 \times 9 = 54$.

  Right side: $(a \times b) + (a \times c) = (6 \times 4) + (6 \times 5) = 24 + 30 = 54$.

  Since $54 = 54$, $6 \times (4 + 5) = (6 \times 4) + (6 \times 5)$.

  The property holds for these whole numbers.

• Multiplication over Subtraction:

  Multiplying a whole number by the difference of two other whole numbers (where the subtraction results in a whole number) is equal to the difference of the products.

  If $a, b, c \in \mathbb{W}$ and $b \ge c$, then $a \times (b - c) = (a \times b) - (a \times c)$.

  (Distributive Property over Subtraction)

  Example 1. Verify the distributive property of multiplication over subtraction for $a=8, b=10, c=3$.

  Answer:

  Let $a=8, b=10, c=3$. All are whole numbers, and $b \ge c$ (10 > 3).

  Left side: $a \times (b - c) = 8 \times (10 - 3) = 8 \times 7 = 56$.

  Right side: $(a \times b) - (a \times c) = (8 \times 10) - (8 \times 3) = 80 - 24 = 56$.

  Since $56 = 56$, $8 \times (10 - 3) = (8 \times 10) - (8 \times 3)$.

  The property holds for these whole numbers.

Successor and Predecessor

For any whole number $n$, its successor is the number that comes immediately after it, which is $n+1$. The successor of any whole number is always a whole number.

For any whole number $n$ greater than 0 (i.e., $n \in \{1, 2, 3, ...\}$), its predecessor is the number that comes immediately before it, which is $n-1$.

The number 0 is a whole number, but it does not have a predecessor within the set of whole numbers, because $0 - 1 = -1$, which is an integer but not a whole number.

Examples:

• The successor of 5 is $5+1=6$. Both 5 and 6 are whole numbers.
• The predecessor of 5 is $5-1=4$. Both 5 and 4 are whole numbers.
• The successor of 0 is $0+1=1$. Both 0 and 1 are whole numbers.
• The predecessor of 0 is $0-1=-1$. 0 is a whole number, but -1 is not a whole number. Hence, 0 has no predecessor in $\mathbb{W}$.

Summary Table of Properties for Whole Numbers ($\mathbb{W}$)

Comparing these properties with those of natural numbers, we see that including zero in the set of whole numbers provides an additive identity, which was absent in the set of natural numbers.

Integers: Definition and Properties

Definition of Integers

Integers are a larger set of numbers compared to whole numbers. They are formed by including all the whole numbers and their negative counterparts. The set of integers extends infinitely in both positive and negative directions from zero.

The set of integers is denoted by the symbol $\mathbb{Z}$. This symbol comes from the German word 'Zahlen', which means 'numbers'.

The set of integers $\mathbb{Z}$ can be explicitly written as:

From the definition, it's clear that all whole numbers are integers, and since natural numbers are a subset of whole numbers, all natural numbers are also integers. This expands our hierarchy of number sets:

Examples of integers include $-100, -50, -1, 0, 1, 10, 75, 1000$.

Integers are typically classified into three categories:

• Positive Integers ($\mathbb{Z}^+$): These are the integers greater than zero. $\mathbb{Z}^+ = \{1, 2, 3, ...\}$, which is the same as the set of Natural Numbers ($\mathbb{N}$).
• Negative Integers ($\mathbb{Z}^-$): These are the integers less than zero. $\mathbb{Z}^- = \{..., -3, -2, -1\}$.
• Zero ($0$): The integer $0$ is neither positive nor negative. It serves as the dividing point between positive and negative integers on the number line.

So, the set of integers can also be expressed as the union of negative integers, zero, and positive integers:

Properties of Integers under Basic Operations

Let $a, b,$ and $c$ be any three integers. Let's examine how the set of integers ($\mathbb{Z}$) behaves under the four basic arithmetic operations: addition, subtraction, multiplication, and division.

1. Closure Property

A set is closed under an operation if performing the operation on any two elements from the set always produces a result that is also an element of the same set.

• Closure under Addition:

  For any two integers $a$ and $b$, their sum $a + b$ is always an integer.

  If $a \in \mathbb{Z}$ and $b \in \mathbb{Z}$, then $a + b \in \mathbb{Z}$.

  (Closure Property of Addition)

  Example 1. Check if the sum of two integers -5 and 12 is an integer.

  Answer:

  Given integers are $a = -5$ and $b = 12$. Both $-5 \in \mathbb{Z}$ and $12 \in \mathbb{Z}$.

  Their sum is $a + b = -5 + 12 = 7$.

  The result $7$ is an integer ($7 \in \mathbb{Z}$). This confirms the closure property for this specific case.

  Example 2. Check if the sum of two integers -8 and -3 is an integer.

  Answer:

  Given integers are $a = -8$ and $b = -3$. Both $-8 \in \mathbb{Z}$ and $-3 \in \mathbb{Z}$.

  Their sum is $a + b = -8 + (-3) = -11$.

  The result $-11$ is an integer ($-11 \in \mathbb{Z}$).

  This property holds true for any pair of integers, including combinations of positive, negative, and zero.

  Thus, the set of integers is closed under addition.

• Closure under Subtraction:

  For any two integers $a$ and $b$, their difference $a - b$ is always an integer.

  If $a \in \mathbb{Z}$ and $b \in \mathbb{Z}$, then $a - b \in \mathbb{Z}$.

  (Closure Property of Subtraction)

  Example 1. Check if the difference of two integers 4 and 9 is an integer.

  Answer:

  Given integers are $a = 4$ and $b = 9$. Both $4 \in \mathbb{Z}$ and $9 \in \mathbb{Z}$.

  Their difference is $a - b = 4 - 9 = -5$.

  The result, $-5$, is an integer ($-5 \in \mathbb{Z}$).

  Example 2. Check if the difference of two integers -6 and -2 is an integer.

  Answer:

  Given integers are $a = -6$ and $b = -2$. Both $-6 \in \mathbb{Z}$ and $-2 \in \mathbb{Z}$.

  Their difference is $a - b = -6 - (-2) = -6 + 2 = -4$.

  The result, $-4$, is an integer ($-4 \in \mathbb{Z}$).

  This property holds because the set of integers includes both positive and negative numbers and zero. Any difference between two integers will fall into one of these categories.

  Thus, the set of integers is closed under subtraction.

• Closure under Multiplication:

  For any two integers $a$ and $b$, their product $a \times b$ is always an integer.

  If $a \in \mathbb{Z}$ and $b \in \mathbb{Z}$, then $a \times b \in \mathbb{Z}$.

  (Closure Property of Multiplication)

  Example 1. Check if the product of two integers -7 and 4 is an integer.

  Answer:

  Given integers are $a = -7$ and $b = 4$. Both $-7 \in \mathbb{Z}$ and $4 \in \mathbb{Z}$.

  Their product is $a \times b = (-7) \times 4 = -28$.

  The result $-28$ is an integer ($-28 \in \mathbb{Z}$).

  Example 2. Check if the product of two integers -9 and -5 is an integer.

  Answer:

  Given integers are $a = -9$ and $b = -5$. Both $-9 \in \mathbb{Z}$ and $-5 \in \mathbb{Z}$.

  Their product is $a \times b = (-9) \times (-5) = 45$.

  The result $45$ is an integer ($45 \in \mathbb{Z}$).

  This property holds true for any pair of integers.

  Thus, the set of integers is closed under multiplication.

• Closure under Division:

  For any two integers $a$ and $b$, their quotient $a \div b$ is not always an integer. Division by zero ($b=0$) is undefined.

  If $a \in \mathbb{Z}$ and $b \in \mathbb{Z}$ ($b \neq 0$), $a \div b$ is not necessarily in $\mathbb{Z}$.

  Example 1. Give an example to show that division is not closed for integers.

  Answer:

  Let $a = 8$ and $b = 3$. Both $a$ and $b$ are integers ($8 \in \mathbb{Z}$, $3 \in \mathbb{Z}$, $b \neq 0$).

  Consider their quotient: $a \div b = 8 \div 3 = \frac{8}{3}$.

  The result, $\frac{8}{3}$, is a rational number but it is not an integer.

  Therefore, the set of integers is not closed under division.

2. Commutative Property

An operation '*' is commutative if changing the order of the operands does not change the result ($a * b = b * a$).

• Commutativity of Addition:

  The order in which two integers are added does not affect the sum.

  If $a \in \mathbb{Z}$ and $b \in \mathbb{Z}$, then $a + b = b + a$.

  (Commutative Property of Addition)

  Example 1. Check if addition is commutative for the integers -7 and 5.

  Answer:

  Let $a=-7$ and $b=5$. Both $-7 \in \mathbb{Z}$ and $5 \in \mathbb{Z}$.

  Consider $a+b$: $-7 + 5 = -2$.

  Consider $b+a$: $5 + (-7) = 5 - 7 = -2$.

  Since $-7 + 5 = 5 + (-7)$, the property holds for this pair. This holds for any pair of integers.

  Thus, addition is commutative for integers.

• Commutativity of Subtraction:

  Subtraction of integers is not commutative.

  If $a \in \mathbb{Z}$ and $b \in \mathbb{Z}$, $a - b \neq b - a$ (unless $a=b$).

  Example 1. Demonstrate that subtraction is not commutative for integers using 10 and -4.

  Answer:

  Let $a = 10$ and $b = -4$. Both $10 \in \mathbb{Z}$ and $-4 \in \mathbb{Z}$.

  Consider $a-b$: $10 - (-4) = 10 + 4 = 14$.

  Consider $b-a$: $-4 - 10 = -14$.

  Since $14 \neq -14$, $10 - (-4) \neq -4 - 10$.

  Thus, subtraction is not commutative for integers.

• Commutativity of Multiplication:

  The order in which two integers are multiplied does not affect the product.

  If $a \in \mathbb{Z}$ and $b \in \mathbb{Z}$, then $a \times b = b \times a$.

  (Commutative Property of Multiplication)

  Example 1. Verify the commutative property of multiplication for the integers -6 and 8.

  Answer:

  Let $a=-6$ and $b=8$. Both $-6 \in \mathbb{Z}$ and $8 \in \mathbb{Z}$.

  Consider $a \times b$: $(-6) \times 8 = -48$.

  Consider $b \times a$: $8 \times (-6) = -48$.

  Since $(-6) \times 8 = 8 \times (-6)$, the property holds for this pair. This is true for any pair of integers.

  Thus, multiplication is commutative for integers.

• Commutativity of Division:

  Division of integers is not commutative.

  If $a \in \mathbb{Z}$ and $b \in \mathbb{Z}$ ($b \neq 0, a \neq 0$), $a \div b \neq b \div a$ (unless $|a|=|b|$).

  Example 1. Show that division is not commutative for integers using 12 and -3.

  Answer:

  Let $a = 12$ and $b = -3$. Both $12 \in \mathbb{Z}$ and $-3 \in \mathbb{Z}$ ($b \neq 0$).

  Consider $a \div b$: $12 \div (-3) = -4$.

  Consider $b \div a$: $(-3) \div 12 = \frac{-3}{12} = -\frac{1}{4}$.

  Since $-4 \neq -\frac{1}{4}$, $12 \div (-3) \neq (-3) \div 12$.

  Thus, division is not commutative for integers.

3. Associative Property

An operation '*' on a set is associative if the way in which numbers are grouped when performing the operation on three or more numbers does not affect the result [$(a * b) * c = a * (b * c)$].

• Associativity of Addition:

  When adding three or more integers, the way they are grouped does not affect the sum.

  If $a, b, c \in \mathbb{Z}$, then $(a + b) + c = a + (b + c)$.

  (Associative Property of Addition)

  Example 1. Verify the associative property of addition for integers -2, 5, and -3.

  Answer:

  Let $a = -2, b = 5, c = -3$. All are integers.

  Consider $(a + b) + c$: $(-2 + 5) + (-3) = 3 + (-3) = 0$.

  Consider $a + (b + c)$: $-2 + (5 + (-3)) = -2 + (5 - 3) = -2 + 2 = 0$.

  Since $( -2 + 5 ) + ( -3 ) = -2 + ( 5 + ( -3 ) ) = 0$, the property holds for this example. It holds for any three integers.

  Thus, addition is associative for integers.

• Associativity of Subtraction:

  Subtraction of integers is not associative.

  If $a, b, c \in \mathbb{Z}$, $(a - b) - c \neq a - (b - c)$ (in general).

  Example 1. Show that subtraction is not associative for integers using 10, 5, and -2.

  Answer:

  Let $a = 10, b = 5, c = -2$. All are integers.

  Consider $(a - b) - c$: $(10 - 5) - (-2) = 5 - (-2) = 5 + 2 = 7$.

  Consider $a - (b - c)$: $10 - (5 - (-2)) = 10 - (5 + 2) = 10 - 7 = 3$.

  Since $7 \neq 3$, $(10 - 5) - (-2) \neq 10 - (5 - (-2))$.

  Thus, subtraction is not associative for integers.

• Associativity of Multiplication:

  When multiplying three or more integers, the grouping of numbers does not affect the product.

  If $a, b, c \in \mathbb{Z}$, then $(a \times b) \times c = a \times (b \times c)$.

  (Associative Property of Multiplication)

  Example 1. Verify the associative property of multiplication for integers -3, 2, and -4.

  Answer:

  Let $a = -3, b = 2, c = -4$. All are integers.

  Consider $(a \times b) \times c$: $(-3 \times 2) \times (-4) = (-6) \times (-4) = 24$.

  Consider $a \times (b \times c)$: $-3 \times (2 \times (-4)) = -3 \times (-8) = 24$.

  Since $(-3 \times 2) \times (-4) = -3 \times (2 \times -4) = 24$, the property holds for this example. It holds for any three integers.

  Thus, multiplication is associative for integers.

• Associativity of Division:

  Division of integers is not associative.

  If $a, b, c \in \mathbb{Z}$, $(a \div b) \div c \neq a \div (b \div c)$ (in general, and avoiding division by zero).

  Example 1. Show that division is not associative for integers using 48, -8, and 2.

  Answer:

  Let $a = 48, b = -8, c = 2$. All are integers, and $b \neq 0, c \neq 0$.

  Consider $(a \div b) \div c$: $(48 \div -8) \div 2 = (-6) \div 2 = -3$.

  Consider $a \div (b \div c)$: $48 \div (-8 \div 2) = 48 \div (-4) = -12$.

  Since $-3 \neq -12$, $(48 \div -8) \div 2 \neq 48 \div (-8 \div 2)$.

  Thus, division is not associative for integers.

4. Identity Property (Existence of Identity Element)

An identity element for an operation '*' is an element 'e' in the set such that for any element 'a' in the set, $a * e = e * a = a$.

• Additive Identity:

  The additive identity is the number that, when added to any number, results in the original number. This number is 0 ($a + 0 = 0 + a = a$).

  Since 0 is an element of the set of integers ($\mathbb{Z} = \{..., -1, 0, 1, ...\}$), the additive identity exists in $\mathbb{Z}$.

  For any $a \in \mathbb{Z}$, $a + 0 = 0 + a = a$.

  (Additive Identity Property)

  Example 1. Identify the additive identity in integers and verify it with a negative integer.

  Answer:

  The additive identity is $0$. For any integer, say $-18 \in \mathbb{Z}$:

  $-18 + 0 = -18$ and $0 + (-18) = -18$.

  Since $0$ is in $\mathbb{Z}$, the additive identity exists in the set of integers and is $0$.

• Multiplicative Identity:

  The multiplicative identity is the number that, when multiplied by any number, results in the original number. This number is 1 ($a \times 1 = 1 \times a = a$).

  Since 1 is an element of the set of integers ($\mathbb{Z} = \{..., -1, 0, 1, ...\}$), the multiplicative identity exists in $\mathbb{Z}$.

  For any $a \in \mathbb{Z}$, $a \times 1 = 1 \times a = a$.

  (Multiplicative Identity Property)

  Example 1. Identify the multiplicative identity in integers and verify it with an integer.

  Answer:

  The multiplicative identity is $1$. For any integer, say $-5 \in \mathbb{Z}$:

  $-5 \times 1 = -5$ and $1 \times (-5) = -5$.

  Since $1$ is in $\mathbb{Z}$, the multiplicative identity exists in the set of integers and is $1$.

5. Inverse Property (Existence of Inverse Element)

For an operation '*' and an identity element 'e', the inverse element of 'a' ($a^{-1}$) is an element in the set such that $a * a^{-1} = a^{-1} * a = e$.

• Additive Inverse:

  For every integer $a$, its additive inverse is the integer $-a$, such that $a + (-a) = 0$ (the additive identity).

  Since the set of integers includes both positive and negative counterparts for every non-zero integer, and 0 for itself, the additive inverse of any integer is also an integer.

  For any $a \in \mathbb{Z}$, there exists $-a \in \mathbb{Z}$ such that $a + (-a) = (-a) + a = 0$.

  (Additive Inverse Property)

  Example 1. Find the additive inverse of 9 and -11 in the set of integers.

  Answer:

  For $a = 9$, the additive inverse is $-a = -9$. Since $9 + (-9) = 0$ and $-9 \in \mathbb{Z}$, the additive inverse of 9 is -9.

  For $a = -11$, the additive inverse is $-a = -(-11) = 11$. Since $-11 + 11 = 0$ and $11 \in \mathbb{Z}$, the additive inverse of -11 is 11.

  For $a = 0$, the additive inverse is $-0 = 0$. Since $0 + 0 = 0$ and $0 \in \mathbb{Z}$, the additive inverse of 0 is 0.

  Additive inverses exist for all integers within the set of integers.

• Multiplicative Inverse:

  The multiplicative inverse of a non-zero integer $a$ is $\frac{1}{a}$, such that $a \times \frac{1}{a} = 1$ (the multiplicative identity).

  For an integer $a$, its multiplicative inverse $\frac{1}{a}$ is an integer only if $a$ is 1 or -1.

  If $a \in \mathbb{Z}$ and $a \neq 0$, its multiplicative inverse is $\frac{1}{a}$.

  For $a \in \mathbb{Z}$ ($a \neq 0$), its multiplicative inverse $(\frac{1}{a})$ $\notin \mathbb{Z}$ (except for $a=1$ and $a=-1$).

  Example 1. Find the multiplicative inverse of 5 and -1 in the set of integers, if they exist.

  Answer:

  For $a = 5$, the multiplicative inverse is $\frac{1}{5}$. $\frac{1}{5}$ is a rational number but not an integer. So, 5 does not have a multiplicative inverse in $\mathbb{Z}$.

  For $a = -1$, the multiplicative inverse is $\frac{1}{-1} = -1$. Since $-1$ is an integer ($ -1 \in \mathbb{Z}$), the multiplicative inverse of -1 is -1.

  For $a=1$, the multiplicative inverse is $\frac{1}{1}=1$. Since $1$ is an integer ($1 \in \mathbb{Z}$), the multiplicative inverse of 1 is 1.

  The integer 0 does not have a multiplicative inverse.

  Multiplicative inverses generally do not exist for integers, except for the special cases of 1 and -1.

6. Distributive Property

The distributive property connects multiplication with addition or subtraction. For integers, multiplication distributes over both addition and subtraction.

• Multiplication over Addition:

  Multiplying an integer by the sum of two other integers is equal to the sum of the products of the first integer with each of the other two integers.

  If $a, b, c \in \mathbb{Z}$, then $a \times (b + c) = (a \times b) + (a \times c)$.

  (Distributive Property over Addition)

  Example 1. Verify the distributive property of multiplication over addition for $a=-2, b=3, c=-5$.

  Answer:

  Let $a=-2, b=3, c=-5$. All are integers.

  Left side: $a \times (b + c) = -2 \times (3 + (-5)) = -2 \times (3 - 5) = -2 \times (-2) = 4$.

  Right side: $(a \times b) + (a \times c) = (-2 \times 3) + (-2 \times -5) = (-6) + (10) = 4$.

  Since $4 = 4$, $-2 \times (3 + (-5)) = (-2 \times 3) + (-2 \times -5)$.

  The property holds for these integers.

• Multiplication over Subtraction:

  Multiplying an integer by the difference of two other integers is equal to the difference of the products.

  If $a, b, c \in \mathbb{Z}$, then $a \times (b - c) = (a \times b) - (a \times c)$.

  (Distributive Property over Subtraction)

  Example 1. Verify the distributive property of multiplication over subtraction for $a=4, b=7, c=-3$.

  Answer:

  Let $a=4, b=7, c=-3$. All are integers.

  Left side: $a \times (b - c) = 4 \times (7 - (-3)) = 4 \times (7 + 3) = 4 \times 10 = 40$.

  Right side: $(a \times b) - (a \times c) = (4 \times 7) - (4 \times -3) = 28 - (-12) = 28 + 12 = 40$.

  Since $40 = 40$, $4 \times (7 - (-3)) = (4 \times 7) - (4 \times -3)$.

  The property holds for these integers.

Summary Table of Properties for Integers ($\mathbb{Z}$)

Compared to whole numbers, the set of integers is closed under subtraction and provides an additive inverse for every element within the set. This makes integers a more robust set for solving equations involving addition and subtraction.

Rational Numbers: Definition, Classification, and Properties

Definition of Rational Numbers

Rational numbers are a set of numbers that include all integers, fractions, and terminating or repeating decimals. Formally, a rational number is defined as any number that can be expressed as a ratio or fraction of two integers.

The set of rational numbers is denoted by the symbol $\mathbb{Q}$, which stands for Quotient.

The formal definition of the set of rational numbers is:

In the expression $\frac{p}{q}$, $p$ is called the numerator and $q$ is called the denominator. The condition $q \neq 0$ is crucial because division by zero is undefined.

Examples of rational numbers include:

• Simple fractions: $\frac{1}{2}, \frac{3}{4}, -\frac{5}{6}, \frac{22}{7}$
• Integers: Any integer $n$ can be written in the form $\frac{p}{q}$ by taking $p=n$ and $q=1$. For example, $5 = \frac{5}{1}$, $-8 = \frac{-8}{1}$, $0 = \frac{0}{1}$. Thus, all integers are rational numbers.
• Terminating decimals: $0.5 = \frac{5}{10} = \frac{1}{2}$, $2.75 = \frac{275}{100} = \frac{11}{4}$, $-0.125 = \frac{-125}{1000} = -\frac{1}{8}$.
• Non-terminating repeating decimals: $0.333... = 0.\overline{3} = \frac{1}{3}$, $1.272727... = 1.\overline{27} = \frac{126}{99} = \frac{14}{11}$.

Since all integers can be expressed as a ratio of an integer to 1, the set of integers is a subset of the set of rational numbers. This extends our number system hierarchy:

Decimal Representation of Rational Numbers

A key characteristic of rational numbers is the nature of their decimal expansion. When a rational number $\frac{p}{q}$ is converted into decimal form, the result is always one of two types:

• Terminating Decimal Expansion:

  These are decimals that end after a finite number of digits. This occurs when the division of $p$ by $q$ results in a remainder of zero after a finite number of steps.

  Examples:

  • $\frac{1}{4} = 0.25$
  • $\frac{3}{8} = 0.375$
  • $\frac{7}{20} = 0.35$

  A rational number $\frac{p}{q}$, where $p$ and $q$ are integers with no common factors other than 1 (i.e., in simplest form), will have a terminating decimal expansion if and only if the prime factorization of the denominator, $q$, contains only the prime numbers 2 and/or 5. For example, $4 = 2^2$, $8 = 2^3$, $20 = 2^2 \times 5$.

• Non-terminating Repeating (Recurring) Decimal Expansion:

  These are decimals that continue infinitely, but with a specific block of digits that repeats indefinitely. This occurs when the division of $p$ by $q$ never results in a remainder of zero, but the remainders start repeating, causing the digits in the quotient to repeat.

  The repeating part is indicated by a bar ($\overline{\phantom{aa}}$) over the repeating block of digits.

  Examples:

  • $\frac{1}{3} = 0.333... = 0.\overline{3}$
  • $\frac{2}{7} = 0.285714285714... = 0.\overline{285714}$
  • $\frac{1}{9} = 0.111... = 0.\overline{1}$
  • $\frac{5}{11} = 0.454545... = 0.\overline{45}$

  A rational number $\frac{p}{q}$, in simplest form, will have a non-terminating repeating decimal expansion if the prime factorization of the denominator, $q$, contains any prime factor other than 2 or 5. For example, $3 = 3^1$, $7 = 7^1$, $9 = 3^2$, $11 = 11^1$.

This property is a defining characteristic: Any number that can be written as $\frac{p}{q}$ has a decimal expansion that is either terminating or repeating. Conversely, any number with a terminating or repeating decimal expansion can be written as $\frac{p}{q}$, proving it is rational.

Converting Repeating Decimals to $\frac{p}{q}$ Form

Since every non-terminating repeating decimal is a rational number, it can be expressed in the form $\frac{p}{q}$. We can use an algebraic method to perform this conversion.

Answer:

Answer:

Answer:

Properties of Rational Numbers under Basic Operations

The set of rational numbers ($\mathbb{Q}$) possesses many important algebraic properties under the basic arithmetic operations. These properties make $\mathbb{Q}$ a field in abstract algebra.

Let $a, b,$ and $c$ be any three rational numbers.

1. Closure Property

The set of rational numbers is closed under addition, subtraction, multiplication, and division (with the exception of division by zero).

• Closure under Addition:

  The sum of any two rational numbers is always a rational number.

  If $a \in \mathbb{Q}$ and $b \in \mathbb{Q}$, then $a + b \in \mathbb{Q}$.

  (Closure Property of Addition)

  Explanation: Let $a = \frac{p}{q}$ and $b = \frac{r}{s}$, where $p, q, r, s \in \mathbb{Z}$ and $q \neq 0, s \neq 0$.

  $a + b = \frac{p}{q} + \frac{r}{s} = \frac{ps + qr}{qs}$

  Since $p, q, r, s$ are integers, $ps$, $qr$, and $qs$ are also integers (closure under multiplication for integers). Also, $qs \neq 0$ because $q \neq 0$ and $s \neq 0$. Therefore, $\frac{ps + qr}{qs}$ is in the form of an integer ratio over a non-zero integer, which is the definition of a rational number.

  Example 1. Check if the sum of $\frac{1}{2}$ and $\frac{3}{4}$ is rational.

  Answer:

  $\frac{1}{2} + \frac{3}{4} = \frac{1 \times 2}{2 \times 2} + \frac{3}{4} = \frac{2}{4} + \frac{3}{4} = \frac{2+3}{4} = \frac{5}{4}$.

  $\frac{5}{4}$ is in the form $\frac{p}{q}$ where $p=5, q=4$, both integers, $q \neq 0$. Thus, $\frac{5}{4}$ is a rational number. The set of rational numbers is closed under addition.

• Closure under Subtraction:

  The difference between any two rational numbers is always a rational number.

  If $a \in \mathbb{Q}$ and $b \in \mathbb{Q}$, then $a - b \in \mathbb{Q}$.

  (Closure Property of Subtraction)

  Explanation: Let $a = \frac{p}{q}$ and $b = \frac{r}{s}$.

  $a - b = \frac{p}{q} - \frac{r}{s} = \frac{ps - qr}{qs}$

  Similar to addition, $ps - qr$ is an integer (closure under multiplication and subtraction for integers), and $qs \neq 0$. Thus, the result is a rational number.

  Example 1. Check if the difference between $\frac{2}{3}$ and $\frac{1}{5}$ is rational.

  Answer:

  $\frac{2}{3} - \frac{1}{5} = \frac{2 \times 5}{3 \times 5} - \frac{1 \times 3}{5 \times 3} = \frac{10}{15} - \frac{3}{15} = \frac{10-3}{15} = \frac{7}{15}$.

  $\frac{7}{15}$ is in the form $\frac{p}{q}$ where $p=7, q=15$, both integers, $q \neq 0$. Thus, $\frac{7}{15}$ is a rational number. The set of rational numbers is closed under subtraction.

• Closure under Multiplication:

  The product of any two rational numbers is always a rational number.

  If $a \in \mathbb{Q}$ and $b \in \mathbb{Q}$, then $a \times b \in \mathbb{Q}$.

  (Closure Property of Multiplication)

  Explanation: Let $a = \frac{p}{q}$ and $b = \frac{r}{s}$.

  $a \times b = \frac{p}{q} \times \frac{r}{s} = \frac{p \times r}{q \times s} = \frac{pr}{qs}$

  Since $p, q, r, s$ are integers, $pr$ and $qs$ are integers. Since $q \neq 0$ and $s \neq 0$, $qs \neq 0$. Thus, the result $\frac{pr}{qs}$ is a rational number.

  Example 1. Check if the product of $-\frac{3}{5}$ and $\frac{10}{9}$ is rational.

  Answer:

  $-\frac{3}{5} \times \frac{10}{9} = \frac{-3 \times 10}{5 \times 9} = \frac{-30}{45}$.

  Simplify the fraction by dividing numerator and denominator by 15:

  $\frac{-30}{45} = \frac{\cancel{-30}^{-2}}{\cancel{45}_3} = -\frac{2}{3}$

  $-\frac{2}{3}$ is in the form $\frac{p}{q}$ where $p=-2, q=3$, both integers, $q \neq 0$. Thus, $-\frac{2}{3}$ is a rational number. The set of rational numbers is closed under multiplication.

• Closure under Division:

  The quotient of a rational number $a$ divided by a non-zero rational number $b$ is always a rational number.

  If $a \in \mathbb{Q}$ and $b \in \mathbb{Q}$ ($b \neq 0$), then $a \div b \in \mathbb{Q}$.

  (Closure Property of Division for non-zero divisor)

  Explanation: Let $a = \frac{p}{q}$ and $b = \frac{r}{s}$, where $p, q, r, s \in \mathbb{Z}$ and $q \neq 0, s \neq 0$. Since $b \neq 0$, we also must have $r \neq 0$.

  $a \div b = \frac{p}{q} \div \frac{r}{s} = \frac{p}{q} \times \frac{s}{r} = \frac{ps}{qr}$

  Since $p, q, r, s$ are integers, $ps$ and $qr$ are integers. Since $q \neq 0$ and $r \neq 0$, $qr \neq 0$. Thus, the result $\frac{ps}{qr}$ is a rational number.

  Example 1. Check if the quotient of $\frac{4}{5}$ divided by $-\frac{2}{15}$ is rational.

  Answer:

  $\frac{4}{5} \div (-\frac{2}{15}) = \frac{4}{5} \times (\text{reciprocal of } -\frac{2}{15}) = \frac{4}{5} \times (-\frac{15}{2})$.

  Now multiply the fractions:

  $\frac{4}{5} \times (-\frac{15}{2}) = \frac{4 \times (-15)}{5 \times 2} = \frac{-60}{10}$

  Simplify the fraction:

  $\frac{-60}{10} = -6$

  The result, $-6$, is an integer. Since all integers are rational numbers, $-6 \in \mathbb{Q}$.

  Thus, the set of rational numbers is closed under division by any non-zero rational number.

2. Commutative Property

Addition and multiplication of rational numbers are commutative.

• Commutativity of Addition:

  The order of adding two rational numbers does not affect the sum.

  If $a \in \mathbb{Q}$ and $b \in \mathbb{Q}$, then $a + b = b + a$.

  (Commutative Property of Addition)

  Explanation: Based on the commutative property of addition for integers, which holds for the numerators $ps$ and $qr$ in the sum $\frac{ps + qr}{qs}$.

  Example 1. Check if addition is commutative for $\frac{1}{3}$ and $-\frac{2}{5}$.

  Answer:

  $\frac{1}{3} + (-\frac{2}{5}) = \frac{1 \times 5 + (-2) \times 3}{3 \times 5} = \frac{5 - 6}{15} = \frac{-1}{15}$.

  $-\frac{2}{5} + \frac{1}{3} = \frac{(-2) \times 3 + 1 \times 5}{5 \times 3} = \frac{-6 + 5}{15} = \frac{-1}{15}$.

  Since both results are equal, $\frac{1}{3} + (-\frac{2}{5}) = -\frac{2}{5} + \frac{1}{3}$. Addition is commutative for rational numbers.

• Commutativity of Multiplication:

  The order of multiplying two rational numbers does not affect the product.

  If $a \in \mathbb{Q}$ and $b \in \mathbb{Q}$, then $a \times b = b \times a$.

  (Commutative Property of Multiplication)

  Explanation: Based on the commutative property of multiplication for integers, which holds for the numerators $pr$ and $rp$ and denominators $qs$ and $sq$ in the product $\frac{pr}{qs}$.

  Example 1. Check if multiplication is commutative for $\frac{3}{7}$ and $\frac{5}{4}$.

  Answer:

  $\frac{3}{7} \times \frac{5}{4} = \frac{3 \times 5}{7 \times 4} = \frac{15}{28}$.

  $\frac{5}{4} \times \frac{3}{7} = \frac{5 \times 3}{4 \times 7} = \frac{15}{28}$.

  Since both results are equal, $\frac{3}{7} \times \frac{5}{4} = \frac{5}{4} \times \frac{3}{7}$. Multiplication is commutative for rational numbers.

• Subtraction and Division: Not commutative (similar to integers and whole numbers).

3. Associative Property

Addition and multiplication of rational numbers are associative.

• Associativity of Addition:

  When adding three or more rational numbers, the grouping does not affect the sum.

  If $a, b, c \in \mathbb{Q}$, then $(a + b) + c = a + (b + c)$.

  (Associative Property of Addition)

  Explanation: This property extends from the associative property of addition for integers.

  Example 1. Verify associative property for addition for $\frac{1}{2}, \frac{1}{3}, \frac{1}{4}$.

  Answer:

  Let $a = \frac{1}{2}, b = \frac{1}{3}, c = \frac{1}{4}$.

  $(a + b) + c = (\frac{1}{2} + \frac{1}{3}) + \frac{1}{4} = (\frac{3+2}{6}) + \frac{1}{4} = \frac{5}{6} + \frac{1}{4} = \frac{10+3}{12} = \frac{13}{12}$.

  $a + (b + c) = \frac{1}{2} + (\frac{1}{3} + \frac{1}{4}) = \frac{1}{2} + (\frac{4+3}{12}) = \frac{1}{2} + \frac{7}{12} = \frac{6+7}{12} = \frac{13}{12}$.

  Since both results are $\frac{13}{12}$, $(\frac{1}{2} + \frac{1}{3}) + \frac{1}{4} = \frac{1}{2} + (\frac{1}{3} + \frac{1}{4})$. Addition is associative for rational numbers.

• Associativity of Multiplication:

  When multiplying three or more rational numbers, the grouping does not affect the product.

  If $a, b, c \in \mathbb{Q}$, then $(a \times b) \times c = a \times (b \times c)$.

  (Associative Property of Multiplication)

  Example 1. Verify associative property for multiplication for $\frac{2}{3}, \frac{1}{2}, \frac{3}{4}$.

  Answer:

  Let $a = \frac{2}{3}, b = \frac{1}{2}, c = \frac{3}{4}$.

  $(a \times b) \times c = (\frac{2}{3} \times \frac{1}{2}) \times \frac{3}{4} = (\frac{2 \times 1}{3 \times 2}) \times \frac{3}{4} = \frac{2}{6} \times \frac{3}{4} = \frac{\cancel{1}}{\cancel{3}} \times \frac{\cancel{3}}{\cancel{4}_2} = \frac{1}{1 \times 2} = \frac{1}{2}$. (Or $\frac{2}{6} \times \frac{3}{4} = \frac{6}{24} = \frac{1}{4}$... Let's re-calculate the simplification)

  $\frac{2}{6} \times \frac{3}{4} = \frac{\cancel{2}^1}{\cancel{6}_3} \times \frac{\cancel{3}^1}{4} = \frac{1 \times 1}{3 \times 4} = \frac{1}{4}$.

  $a \times (b \times c) = \frac{2}{3} \times (\frac{1}{2} \times \frac{3}{4}) = \frac{2}{3} \times (\frac{1 \times 3}{2 \times 4}) = \frac{2}{3} \times \frac{3}{8}$.

  Multiply and simplify:

  $\frac{2}{3} \times \frac{3}{8} = \frac{\cancel{2}^1}{\cancel{3}^1} \times \frac{\cancel{3}^1}{\cancel{8}_4} = \frac{1 \times 1}{1 \times 4} = \frac{1}{4}$.

  Since both results are $\frac{1}{4}$, $(\frac{2}{3} \times \frac{1}{2}) \times \frac{3}{4} = \frac{2}{3} \times (\frac{1}{2} \times \frac{3}{4})$. Multiplication is associative for rational numbers.

• Subtraction and Division: Not associative (similar to integers and whole numbers).

4. Identity Property (Existence of Identity Element)

Identity elements exist for both addition and multiplication in the set of rational numbers.

• Additive Identity:

  The additive identity is the number 0. For any rational number $a$, $a + 0 = 0 + a = a$.

  Since $0 = \frac{0}{1}$ is a rational number ($p=0 \in \mathbb{Z}, q=1 \in \mathbb{Z}, q \neq 0$), the additive identity exists in $\mathbb{Q}$.

  For any $a \in \mathbb{Q}$, $a + 0 = 0 + a = a$.

  (Additive Identity Property)

  Example 1. Verify the additive identity property for $\frac{5}{8}$.

  Answer:

  $\frac{5}{8} + 0 = \frac{5}{8} + \frac{0}{8} = \frac{5+0}{8} = \frac{5}{8}$.

  $0 + \frac{5}{8} = \frac{0}{8} + \frac{5}{8} = \frac{0+5}{8} = \frac{5}{8}$.

  Since $\frac{5}{8} + 0 = 0 + \frac{5}{8} = \frac{5}{8}$, the additive identity property holds for $\frac{5}{8}$.

• Multiplicative Identity:

  The multiplicative identity is the number 1. For any rational number $a$, $a \times 1 = 1 \times a = a$.

  Since $1 = \frac{1}{1}$ is a rational number ($p=1 \in \mathbb{Z}, q=1 \in \mathbb{Z}, q \neq 0$), the multiplicative identity exists in $\mathbb{Q}$.

  For any $a \in \mathbb{Q}$, $a \times 1 = 1 \times a = a$.

  (Multiplicative Identity Property)

  Example 1. Verify the multiplicative identity property for $-\frac{7}{9}$.

  Answer:

  $-\frac{7}{9} \times 1 = -\frac{7}{9} \times \frac{1}{1} = \frac{-7 \times 1}{9 \times 1} = -\frac{7}{9}$.

  $1 \times (-\frac{7}{9}) = \frac{1}{1} \times (-\frac{7}{9}) = \frac{1 \times -7}{1 \times 9} = -\frac{7}{9}$.

  Since $-\frac{7}{9} \times 1 = 1 \times (-\frac{7}{9}) = -\frac{7}{9}$, the multiplicative identity property holds for $-\frac{7}{9}$.

5. Inverse Property (Existence of Inverse Element)

Inverse elements exist for addition for all rational numbers, and for multiplication for all non-zero rational numbers.

• Additive Inverse:

  For every rational number $a$, there exists an additive inverse, denoted by $-a$, such that $a + (-a) = 0$ (the additive identity).

  If $a = \frac{p}{q}$ is a rational number, its additive inverse is $-a = -\frac{p}{q} = \frac{-p}{q}$. Since $p \in \mathbb{Z}$, $-p \in \mathbb{Z}$, and $q \in \mathbb{Z}, q \neq 0$, $\frac{-p}{q}$ is also a rational number.

  For any $a \in \mathbb{Q}$, there exists $-a \in \mathbb{Q}$ such that $a + (-a) = (-a) + a = 0$.

  (Additive Inverse Property)

  Example 1. Find the additive inverse of $\frac{3}{5}$ and $-\frac{2}{7}$ in the set of rational numbers.

  Answer:

  For $a = \frac{3}{5}$, the additive inverse is $-a = -\frac{3}{5}$. Since $-\frac{3}{5}$ is a rational number, the additive inverse exists in $\mathbb{Q}$. Check: $\frac{3}{5} + (-\frac{3}{5}) = \frac{3-3}{5} = \frac{0}{5} = 0$.

  For $a = -\frac{2}{7}$, the additive inverse is $-a = -(-\frac{2}{7}) = \frac{2}{7}$. Since $\frac{2}{7}$ is a rational number, the additive inverse exists in $\mathbb{Q}$. Check: $-\frac{2}{7} + \frac{2}{7} = \frac{-2+2}{7} = \frac{0}{7} = 0$.

  For $a = 0$, the additive inverse is $-0 = 0$. Since $0 \in \mathbb{Q}$, the additive inverse of 0 is 0.

  Additive inverses exist for all rational numbers within the set of rational numbers.

• Multiplicative Inverse (Reciprocal):

  For every non-zero rational number $a$, there exists a multiplicative inverse, denoted by $a^{-1}$ or $\frac{1}{a}$, such that $a \times a^{-1} = a^{-1} \times a = 1$ (the multiplicative identity).

  If $a = \frac{p}{q}$ is a non-zero rational number ($p \neq 0, q \neq 0$), its multiplicative inverse is $\frac{1}{a} = \frac{1}{\frac{p}{q}} = \frac{q}{p}$. Since $q \in \mathbb{Z}$, $p \in \mathbb{Z}$, and $p \neq 0$ (because $a \neq 0$), $\frac{q}{p}$ is also a rational number.

  For any $a \in \mathbb{Q}$ ($a \neq 0$), there exists $a^{-1} = \frac{1}{a} \in \mathbb{Q}$ such that $a \times \frac{1}{a} = \frac{1}{a} \times a = 1$.

  (Multiplicative Inverse Property)

  Example 1. Find the multiplicative inverse of $\frac{4}{7}$ and $-5$ in the set of rational numbers, if they exist.

  Answer:

  For $a = \frac{4}{7}$, the multiplicative inverse is $\frac{1}{a} = \frac{1}{\frac{4}{7}} = \frac{7}{4}$. Since $\frac{7}{4}$ is a rational number, the multiplicative inverse exists in $\mathbb{Q}$. Check: $\frac{4}{7} \times \frac{7}{4} = \frac{4 \times 7}{7 \times 4} = \frac{28}{28} = 1$.

  For $a = -5$. First, write -5 as a rational number: $-5 = \frac{-5}{1}$. The multiplicative inverse is $\frac{1}{a} = \frac{1}{-5} = -\frac{1}{5}$. Since $-\frac{1}{5}$ is a rational number, the multiplicative inverse exists in $\mathbb{Q}$. Check: $-5 \times (-\frac{1}{5}) = \frac{-5}{1} \times \frac{-1}{5} = \frac{(-5) \times (-1)}{1 \times 5} = \frac{5}{5} = 1$.

  The number 0 does not have a multiplicative inverse because $\frac{1}{0}$ is undefined. The multiplicative inverse exists for all rational numbers except 0.

6. Distributive Property

Multiplication of rational numbers distributes over addition and subtraction.

• Multiplication over Addition:

  If $a, b, c \in \mathbb{Q}$, then $a \times (b + c) = (a \times b) + (a \times c)$.

  (Distributive Property over Addition)

  Example 1. Verify the distributive property for $a=\frac{1}{2}, b=\frac{2}{3}, c=-\frac{1}{4}$.

  Answer:

  Left side: $a \times (b + c) = \frac{1}{2} \times (\frac{2}{3} + (-\frac{1}{4})) = \frac{1}{2} \times (\frac{2}{3} - \frac{1}{4})$.

  Find the sum in the parenthesis: $\frac{2}{3} - \frac{1}{4} = \frac{2 \times 4 - 1 \times 3}{12} = \frac{8 - 3}{12} = \frac{5}{12}$.

  Multiply by $a$: $\frac{1}{2} \times \frac{5}{12} = \frac{1 \times 5}{2 \times 12} = \frac{5}{24}$.

  Right side: $(a \times b) + (a \times c) = (\frac{1}{2} \times \frac{2}{3}) + (\frac{1}{2} \times (-\frac{1}{4}))$.

  Calculate the products: $\frac{1}{2} \times \frac{2}{3} = \frac{1 \times 2}{2 \times 3} = \frac{2}{6} = \frac{1}{3}$.

  $\frac{1}{2} \times (-\frac{1}{4}) = \frac{1 \times (-1)}{2 \times 4} = \frac{-1}{8} = -\frac{1}{8}$.

  Add the products: $\frac{1}{3} + (-\frac{1}{8}) = \frac{1}{3} - \frac{1}{8} = \frac{1 \times 8 - 1 \times 3}{24} = \frac{8 - 3}{24} = \frac{5}{24}$.

  Since both sides equal $\frac{5}{24}$, the distributive property holds: $\frac{1}{2} \times (\frac{2}{3} + (-\frac{1}{4})) = (\frac{1}{2} \times \frac{2}{3}) + (\frac{1}{2} \times (-\frac{1}{4}))$.

• Multiplication over Subtraction:

  If $a, b, c \in \mathbb{Q}$, then $a \times (b - c) = (a \times b) - (a \times c)$.

  (Distributive Property over Subtraction)

  Example 1. Verify the distributive property for $a=-\frac{1}{3}, b=\frac{3}{4}, c=\frac{1}{2}$.

  Answer:

  Left side: $a \times (b - c) = -\frac{1}{3} \times (\frac{3}{4} - \frac{1}{2})$.

  Find the difference in the parenthesis: $\frac{3}{4} - \frac{1}{2} = \frac{3}{4} - \frac{1 \times 2}{2 \times 2} = \frac{3}{4} - \frac{2}{4} = \frac{3-2}{4} = \frac{1}{4}$.

  Multiply by $a$: $-\frac{1}{3} \times \frac{1}{4} = \frac{-1 \times 1}{3 \times 4} = -\frac{1}{12}$.

  Right side: $(a \times b) - (a \times c) = (-\frac{1}{3} \times \frac{3}{4}) - (-\frac{1}{3} \times \frac{1}{2})$.

  Calculate the products: $-\frac{1}{3} \times \frac{3}{4} = \frac{-1 \times 3}{3 \times 4} = \frac{-3}{12} = -\frac{1}{4}$.

  $-\frac{1}{3} \times \frac{1}{2} = \frac{-1 \times 1}{3 \times 2} = \frac{-1}{6}$.

  Subtract the products: $-\frac{1}{4} - (-\frac{1}{6}) = -\frac{1}{4} + \frac{1}{6} = \frac{-1 \times 6 + 1 \times 4}{24} = \frac{-6 + 4}{24} = \frac{-2}{24} = -\frac{1}{12}$.

  Since both sides equal $-\frac{1}{12}$, the distributive property holds: $-\frac{1}{3} \times (\frac{3}{4} - \frac{1}{2}) = (-\frac{1}{3} \times \frac{3}{4}) - (-\frac{1}{3} \times \frac{1}{2})$.

7. Density Property

The set of rational numbers is dense on the number line. This means that between any two distinct rational numbers, no matter how close they are, there exists infinitely many other rational numbers.

More formally, for any two distinct rational numbers $a$ and $b$ with $a < b$, there exists at least one rational number $c$ such that $a < c < b$. Since there is one, we can repeat the process between $a$ and $c$, and $c$ and $b$, and so on, finding infinitely many rational numbers.

A simple way to find a rational number between two distinct rational numbers $a$ and $b$ is to calculate their average: $\frac{a+b}{2}$.

Proof that $\frac{a+b}{2}$ is rational: If $a, b \in \mathbb{Q}$, then $a+b \in \mathbb{Q}$ (closure under addition) and $2 \in \mathbb{Q}$ ($2=\frac{2}{1}$). Since $2 \neq 0$, $\frac{a+b}{2}$ is the quotient of two rational numbers (with the divisor being non-zero), which is also a rational number (closure under division).

Proof that $a < \frac{a+b}{2} < b$:

Given $a < b$.

Add $a$ to both sides of the inequality $a < b$:

Divide both sides by 2 (since 2 is positive, the inequality direction doesn't change):

Similarly, add $b$ to both sides of the inequality $a < b$:

Divide both sides by 2:

Combining (1) and (2), we get $a < \frac{a+b}{2} < b$. This shows that the average is a rational number lying between $a$ and $b$.

To find more rational numbers between $a$ and $b$, we can find the average of $a$ and $\frac{a+b}{2}$, and the average of $\frac{a+b}{2}$ and $b$, and continue this process indefinitely.

Answer:

Answer:

Summary Table of Properties for Rational Numbers ($\mathbb{Q}$)

The set of rational numbers is closed under all four basic arithmetic operations (except division by zero) and satisfies the main algebraic properties, making it a very useful and structured number system.

Irrational Numbers: Definition, Classification, and Properties

Definition of Irrational Numbers

Irrational numbers are real numbers that stand in contrast to rational numbers. By definition, an irrational number is any real number that cannot be expressed as a simple fraction $\frac{p}{q}$, where $p$ and $q$ are integers and $q$ is a non-zero integer ($q \neq 0$).

The set of irrational numbers is commonly denoted by $\mathbb{I}$. Since the set of real numbers $\mathbb{R}$ is the union of rational numbers $\mathbb{Q}$ and irrational numbers $\mathbb{I}$, and these sets are disjoint, we can also define the set of irrational numbers as $\mathbb{R} \setminus \mathbb{Q}$ (the set of real numbers excluding the rational numbers).

The most defining characteristic of an irrational number lies in its decimal representation:

The decimal expansion of an irrational number is always non-terminating and non-repeating.

This means that the digits after the decimal point go on infinitely without ever forming a repeating block or pattern.

Examples of irrational numbers:

• $\sqrt{2} \approx 1.41421356237...$
• $\sqrt{3} \approx 1.73205080756...$
• $\pi \approx 3.14159265358...$ (Pi)
• $e \approx 2.71828182845...$ (Euler's number)
• Numbers constructed with explicit non-repeating patterns, e.g., $0.10110111011110...$

The relationship between rational ($\mathbb{Q}$) and irrational ($\mathbb{I}$) numbers is that they are mutually exclusive subsets of the real numbers ($\mathbb{R}$).

Sources and Examples of Irrational Numbers

Irrational numbers arise from various mathematical contexts. Some common sources include:

• Roots of non-perfect powers:

  The $n$-th root of an integer $k$ is irrational if $k$ is not the $n$-th power of an integer. For example, $\sqrt{k}$ is irrational if $k$ is not a perfect square ($1, 4, 9, 16, ...$). Similarly, $\sqrt[3]{k}$ is irrational if $k$ is not a perfect cube ($1, 8, 27, 64, ...$), and so on.

  Examples:

  • $\sqrt{2}$ (Square root of 2 - 2 is not a perfect square)
  • $\sqrt{3}$ (Square root of 3 - 3 is not a perfect square)
  • $\sqrt{5}$ (Square root of 5 - 5 is not a perfect square)
  • $\sqrt[3]{2}$ (Cube root of 2 - 2 is not a perfect cube)
  • $\sqrt[4]{7}$ (Fourth root of 7 - 7 is not a perfect fourth power)

• Certain mathematical constants:

  Some fundamental constants that appear in various areas of mathematics and science are irrational.

  • Pi ($\pi$): The ratio of a circle's circumference to its diameter. It is approximately $3.14159265...$. $\pi$ is not only irrational but also a transcendental number, meaning it is not a root of any non-zero polynomial equation with integer coefficients. All transcendental numbers are irrational.
  • Euler's number ($e$): The base of the natural logarithm. It is approximately $2.718281828...$. Like $\pi$, $e$ is also a transcendental number and therefore irrational.
  • The golden ratio ($\phi$) is approximately $1.618...$, which is equal to $\frac{1+\sqrt{5}}{2}$. Since it involves the irrational number $\sqrt{5}$ in a way that the sum/quotient does not simplify to a rational number, $\phi$ is irrational.

• Numbers constructed with explicit non-repeating decimal patterns:

  It is possible to construct decimal numbers that are non-terminating and non-repeating by following a pattern that guarantees no block of digits will repeat. For example:

  • $0.12112111211112...$ (one 2, then one 1, one 2, then two 1s, one 2, then three 1s, etc.)
  • $0.01001000100001...$ (one 1, one 0, then one 1, two 0s, one 1, three 0s, etc.)

  Since these decimals do not repeat, they cannot be expressed in $\frac{p}{q}$ form and are thus irrational.

Proof of Irrationality: The Case of $\sqrt{2}$

The proof that $\sqrt{2}$ is an irrational number is a classic example of mathematical reasoning using proof by contradiction (also known as reductio ad absurdum). It's a cornerstone result demonstrating the existence of numbers beyond the rational set.

Proof:

Properties of Irrational Numbers under Basic Operations

Unlike rational numbers, the set of irrational numbers is not closed under any of the four basic arithmetic operations (addition, subtraction, multiplication, and division). Performing operations involving irrational numbers can result in either a rational number or another irrational number.

Let $a$ be a rational number ($a \in \mathbb{Q}$) and $b, c$ be irrational numbers ($b, c \in \mathbb{I}$). Assume $a \neq 0$, $b \neq 0$, $c \neq 0$, and for division, the divisor is non-zero.

• Sum and Difference:

  • Rational $\pm$ Irrational = Irrational

    The sum or difference of a rational number and an irrational number is always irrational.

    Example: $5 + \sqrt{3}$. If we assume $5 + \sqrt{3}$ is rational (say $r$), then $\sqrt{3} = r - 5$. Since $r$ and $5$ are rational, their difference $r-5$ is rational (closure of subtraction for rationals). This would imply $\sqrt{3}$ is rational, which is false. Therefore, the assumption is wrong, and $5 + \sqrt{3}$ is irrational.

    Similarly, $2 - \sqrt{5}$ is irrational, $0 + \pi = \pi$ is irrational, $-3.5 - e$ is irrational.

  • Irrational $\pm$ Irrational: Can be Rational or Irrational.

    The sum or difference of two irrational numbers can be either rational or irrational.

    Examples (Rational Result):

    • $\sqrt{2} + (-\sqrt{2}) = 0$ (rational)
    • $(2 + \sqrt{3}) - \sqrt{3} = 2$ (rational)
    • $(5 - \sqrt{7}) + (2 + \sqrt{7}) = 7$ (rational)

    Examples (Irrational Result):

    • $\sqrt{2} + \sqrt{3}$ (irrational)
    • $\sqrt{2} + \sqrt{2} = 2\sqrt{2}$ (irrational, as $2$ is non-zero rational and $\sqrt{2}$ is irrational)
    • $\sqrt{8} - \sqrt{2} = 2\sqrt{2} - \sqrt{2} = \sqrt{2}$ (irrational)

• Product and Quotient:

  • Non-zero Rational $\times$ Irrational = Irrational

    The product of a non-zero rational number and an irrational number is always irrational.

    Example: $3 \times \sqrt{5} = 3\sqrt{5}$. If we assume $3\sqrt{5}$ is rational (say $r$), then $\sqrt{5} = \frac{r}{3}$. Since $r$ is rational and $3$ is a non-zero rational, their quotient $\frac{r}{3}$ is rational (closure of division for rationals). This would imply $\sqrt{5}$ is rational, which is false. Therefore, the assumption is wrong, and $3\sqrt{5}$ is irrational.

    Similarly, $-2 \times \pi = -2\pi$ is irrational, $\frac{1}{4} \times \sqrt{7} = \frac{\sqrt{7}}{4}$ is irrational.

  • Non-zero Rational $\div$ Irrational = Irrational (Proof is similar to multiplication, e.g., if $\frac{a}{b} = r$, then $b = \frac{a}{r}$).
  • Irrational $\times$ Irrational: Can be Rational or Irrational.

    The product of two irrational numbers can be either rational or irrational.

    Examples (Rational Result):

    • $\sqrt{2} \times \sqrt{2} = (\sqrt{2})^2 = 2$ (rational)
    • $\sqrt{3} \times \sqrt{12} = \sqrt{3 \times 12} = \sqrt{36} = 6$ (rational)
    • $(1+\sqrt{2}) \times (1-\sqrt{2}) = 1^2 - (\sqrt{2})^2 = 1 - 2 = -1$ (rational)

    Examples (Irrational Result):

    • $\sqrt{2} \times \sqrt{3} = \sqrt{6}$ (irrational)
    • $\sqrt{5} \times \sqrt{7} = \sqrt{35}$ (irrational)
    • $\pi \times \sqrt{2}$ (irrational)
  • Irrational $\div$ Irrational: Can be Rational or Irrational.

    The quotient of two irrational numbers can be either rational or irrational (provided the divisor is non-zero).

    Examples (Rational Result):

    • $\sqrt{8} \div \sqrt{2} = \frac{\sqrt{8}}{\sqrt{2}} = \sqrt{\frac{8}{2}} = \sqrt{4} = 2$ (rational)
    • $\sqrt{27} \div \sqrt{3} = \sqrt{\frac{27}{3}} = \sqrt{9} = 3$ (rational)
    • $\pi \div \pi = 1$ (rational, provided $\pi \neq 0$, which is true)

    Examples (Irrational Result):

    • $\sqrt{6} \div \sqrt{2} = \sqrt{\frac{6}{2}} = \sqrt{3}$ (irrational)
    • $\sqrt{10} \div \sqrt{3} = \sqrt{\frac{10}{3}}$ (irrational)
    • $\pi \div \sqrt{2} = \frac{\pi}{\sqrt{2}}$ (irrational)

This lack of closure means that when you perform an operation on two irrational numbers, you cannot assume the result will also be irrational; you need to evaluate the specific case.

Density Property

Similar to rational numbers, irrational numbers are also dense on the number line. This means that between any two distinct irrational numbers, there are infinitely many other irrational numbers.

Even more significantly, the real number line is "filled" by both rational and irrational numbers. Between any two distinct real numbers (whether both rational, both irrational, or one of each), there is always at least one rational number and at least one irrational number, and consequently, infinitely many of both.

Example: Between the two rational numbers 0.1 and 0.2, there exists the irrational number $0.1010010001...$ and the rational number 0.15.

Between the two irrational numbers $\sqrt{2} \approx 1.414$ and $\sqrt{3} \approx 1.732$, there exist rational numbers like 1.5 and 1.6, and irrational numbers like $\sqrt{2.5}$ and $\sqrt{2} + 0.1$.

This density property highlights that both sets of numbers are "everywhere" on the real number line, interleaved infinitely amongst each other.

Real Numbers: Definition and Relationship between Number Systems

Definition of Real Numbers

The set of Real Numbers, denoted by $\mathbb{R}$, encompasses all the numbers that can be represented by points on a continuous line called the number line. It is formed by combining the set of rational numbers and the set of irrational numbers.

Formally, the set of real numbers is the union of the set of rational numbers ($\mathbb{Q}$) and the set of irrational numbers ($\mathbb{I}$).

Every real number is either rational or irrational, but not both. The rational and irrational number sets are mutually exclusive ($\mathbb{Q} \cap \mathbb{I} = \emptyset$).

Examples of real numbers are virtually any number you encounter: $7, -3, 0, \frac{1}{2}, -4.75, \sqrt{5}, \pi, e$. The set of real numbers covers all possibilities for distance, magnitude, and quantity along a single dimension.

The key characteristic of real numbers is that they completely fill the number line. There is a one-to-one correspondence between every real number and every point on the infinite number line.

Relationship and Hierarchy of Number Systems within Real Numbers

We have seen how smaller sets of numbers are contained within larger sets. All the number sets discussed so far (Natural, Whole, Integers, Rational, Irrational) are subsets of the real numbers. Let's summarize their relationships within $\mathbb{R}$:

• Natural Numbers ($\mathbb{N}$): These are the positive counting numbers: $\{1, 2, 3, 4, ...\}$. They are a subset of Whole Numbers.
• Whole Numbers ($\mathbb{W}$): These are the natural numbers including zero: $\{0, 1, 2, 3, ...\}$. They are a subset of Integers.
• Integers ($\mathbb{Z}$): These include all whole numbers and their negative counterparts: $\{..., -3, -2, -1, 0, 1, 2, 3, ...\}$. They are a subset of Rational Numbers.
• Rational Numbers ($\mathbb{Q}$): These are numbers that can be expressed as a fraction of two integers $\frac{p}{q}$ ($q \neq 0$). Their decimal expansions are either terminating or non-terminating repeating. They are a subset of Real Numbers.
• Irrational Numbers ($\mathbb{I}$): These are real numbers that cannot be expressed as a fraction $\frac{p}{q}$. Their decimal expansions are non-terminating and non-repeating. They are also a subset of Real Numbers, and distinct from Rational Numbers.

This hierarchy, moving from the most restrictive to the most inclusive set (within the real number system), is represented by successive subset relationships:

And the set of real numbers is composed of the rational and irrational numbers, which are disjoint sets:

This structure shows that every natural number is a whole number, every whole number is an integer, every integer is a rational number, and every rational number is a real number. Also, every irrational number is a real number.

The broader classification places Real Numbers as a subset of Complex Numbers, where the imaginary part is zero: $\mathbb{R} \subset \mathbb{C}$.

Properties of Real Numbers under Basic Operations

The set of real numbers $\mathbb{R}$, together with the operations of addition (+) and multiplication ($\times$), satisfies a comprehensive set of properties that form the basis of algebra and analysis. These properties make the real numbers a field.

Let $a, b, c$ be any three real numbers.

1. Closure Property

The set of real numbers is closed under addition, subtraction, multiplication, and division (except by zero).

• Closure under Addition: The sum of any two real numbers is a real number.

  If $a \in \mathbb{R}$ and $b \in \mathbb{R}$, then $a + b \in \mathbb{R}$.

  Example: $\sqrt{2} + 3$ is irrational, which is real. $5 + (-8) = -3$ is rational, which is real. $\pi + \pi = 2\pi$ is irrational, which is real.

• Closure under Subtraction: The difference of any two real numbers is a real number.

  If $a \in \mathbb{R}$ and $b \in \mathbb{R}$, then $a - b \in \mathbb{R}$.

  Example: $\sqrt{5} - \sqrt{5} = 0$ is rational, which is real. $10 - \pi$ is irrational, which is real.

• Closure under Multiplication: The product of any two real numbers is a real number.

  If $a \in \mathbb{R}$ and $b \in \mathbb{R}$, then $a \times b \in \mathbb{R}$.

  Example: $\sqrt{3} \times \sqrt{3} = 3$ is rational, which is real. $4 \times \pi = 4\pi$ is irrational, which is real.

• Closure under Division: The quotient of a real number $a$ by a non-zero real number $b$ is a real number.

  If $a \in \mathbb{R}$ and $b \in \mathbb{R}$ ($b \neq 0$), then $a \div b \in \mathbb{R}$.

  Example: $6 \div \sqrt{2} = \frac{6}{\sqrt{2}} = \frac{6\sqrt{2}}{2} = 3\sqrt{2}$ is irrational, which is real. $\sqrt{18} \div \sqrt{2} = \sqrt{9} = 3$ is rational, which is real.

2. Commutative Property

The order of operands does not affect the result for addition and multiplication.

• Commutativity of Addition: $a + b = b + a$.

  For $a, b \in \mathbb{R}$, $a + b = b + a$.

  Example: $\sqrt{5} + 7 = 7 + \sqrt{5}$.

• Commutativity of Multiplication: $a \times b = b \times a$.

  For $a, b \in \mathbb{R}$, $a \times b = b \times a$.

  Example: $\pi \times \sqrt{2} = \sqrt{2} \times \pi$.

3. Associative Property

The grouping of operands does not affect the result for addition and multiplication.

• Associativity of Addition: $(a + b) + c = a + (b + c)$.

  For $a, b, c \in \mathbb{R}$, $(a + b) + c = a + (b + c)$.

  Example: $(2 + \sqrt{3}) + \pi = 2 + (\sqrt{3} + \pi)$.

• Associativity of Multiplication: $(a \times b) \times c = a \times (b \times c)$.

  For $a, b, c \in \mathbb{R}$, $(a \times b) \times c = a \times (b \times c)$.

  Example: $(5 \times \sqrt{2}) \times \sqrt{3} = 5 \times (\sqrt{2} \times \sqrt{3}) = 5\sqrt{6}$.

4. Identity Property (Existence of Identity Element)

Identity elements exist for both addition and multiplication within the real numbers.

• Additive Identity: There exists a unique real number, 0, such that for any real number $a$, $a + 0 = 0 + a = a$.

  For any $a \in \mathbb{R}$, $a + 0 = 0 + a = a$.

  Since $0 \in \mathbb{Q}$ and $\mathbb{Q} \subset \mathbb{R}$, $0$ is a real number.

• Multiplicative Identity: There exists a unique real number, 1, such that for any real number $a$, $a \times 1 = 1 \times a = a$.

  For any $a \in \mathbb{R}$, $a \times 1 = 1 \times a = a$.

  Since $1 \in \mathbb{Q}$ and $\mathbb{Q} \subset \mathbb{R}$, $1$ is a real number.

5. Inverse Property (Existence of Inverse Element)

Inverse elements exist for addition for all real numbers, and for multiplication for all non-zero real numbers.

• Additive Inverse: For every real number $a$, there exists a unique real number, denoted by $-a$, such that $a + (-a) = (-a) + a = 0$.

  For any $a \in \mathbb{R}$, there exists $-a \in \mathbb{R}$ such that $a + (-a) = (-a) + a = 0$.

  Example: The additive inverse of $5$ is $-5$. The additive inverse of $-\sqrt{7}$ is $\sqrt{7}$. The additive inverse of $\pi$ is $-\pi$. Since if $a$ is real, $-a$ is also real, additive inverses exist for all real numbers within $\mathbb{R}$.

• Multiplicative Inverse (Reciprocal): For every non-zero real number $a$, there exists a unique real number, denoted by $a^{-1}$ or $\frac{1}{a}$, such that $a \times \frac{1}{a} = \frac{1}{a} \times a = 1$.

  For any $a \in \mathbb{R}, a \neq 0$, there exists $a^{-1} = \frac{1}{a} \in \mathbb{R}$ such that $a \times \frac{1}{a} = \frac{1}{a} \times a = 1$.

  Example: The multiplicative inverse of $4$ is $\frac{1}{4}$. The multiplicative inverse of $-\frac{2}{3}$ is $-\frac{3}{2}$. The multiplicative inverse of $\sqrt{5}$ is $\frac{1}{\sqrt{5}} = \frac{\sqrt{5}}{5}$. The multiplicative inverse of $\pi$ is $\frac{1}{\pi}$. Since if $a$ is a non-zero real, $\frac{1}{a}$ is also a non-zero real, multiplicative inverses exist for all non-zero real numbers within $\mathbb{R}$.

6. Distributive Property

Multiplication distributes over addition and subtraction in the set of real numbers.

• Multiplication over Addition:

  If $a, b, c \in \mathbb{R}$, then $a \times (b + c) = (a \times b) + (a \times c)$.

  Example: $3 \times (5 + \sqrt{2}) = 3 \times 5 + 3 \times \sqrt{2} = 15 + 3\sqrt{2}$.

• Multiplication over Subtraction:

  If $a, b, c \in \mathbb{R}$, then $a \times (b - c) = (a \times b) - (a \times c)$.

  Example: $\sqrt{7} \times (10 - \sqrt{3}) = \sqrt{7} \times 10 - \sqrt{7} \times \sqrt{3} = 10\sqrt{7} - \sqrt{21}$.

Order Properties of Real Numbers

The real numbers are an ordered set. This means that for any two distinct real numbers, one is always greater than the other. This property allows us to arrange real numbers on the number line in increasing order.

For any two real numbers $a$ and $b$, exactly one of the following relations is true:

• $a < b$ ($a$ is less than $b$)
• $a = b$ ($a$ is equal to $b$)
• $a > b$ ($a$ is greater than $b$)

This property is crucial for comparing and ordering real numbers, solving inequalities, and understanding the structure of the number line.

Completeness Property of Real Numbers

The Completeness Property is a fundamental characteristic that distinguishes the set of real numbers from the set of rational numbers. While rational numbers are dense (meaning between any two rationals there's another rational), there are "gaps" on the number line where irrational numbers lie ($\sqrt{2}, \pi$, etc.). The real number line has no such gaps; it is continuous.

There are several equivalent ways to state the completeness property, often encountered in higher mathematics (calculus and analysis). Some common formulations include:

• Least Upper Bound Property (Supremum Property): Every non-empty set of real numbers that is bounded above has a least upper bound (supremum) in $\mathbb{R}$. Similarly, every non-empty set of real numbers that is bounded below has a greatest lower bound (infimum) in $\mathbb{R}$. For example, the set $\{x \in \mathbb{Q} \ | \ x^2 < 2\}$ is bounded above by 1.5, 2, etc., but its least upper bound is $\sqrt{2}$, which is not in $\mathbb{Q}$. The completeness property ensures that $\sqrt{2}$ exists in $\mathbb{R}$.
• Nested Interval Theorem: For a sequence of nested closed intervals $[a_1, b_1] \supset [a_2, b_2] \supset [a_3, b_3] \supset ...$ such that the length of the intervals approaches zero as $n \to \infty$, there exists exactly one point common to all intervals. This common point is guaranteed to be a real number.
• Cauchy Convergence Criterion: A sequence of real numbers converges to a real number if and only if it is a Cauchy sequence (a sequence whose terms get arbitrarily close to each other).

This property ensures that the real number line is continuous and contains the limits of convergent sequences and the bounds of bounded sets, which is essential for calculus and advanced mathematics.

Summary Table of Properties for Real Numbers ($\mathbb{R}$)

The real numbers form the foundation for much of mathematics and science, providing a continuum essential for concepts like limits, derivatives, and integrals.

Other Number Types (Prime, Composite, etc.)

In addition to the broad classifications of numbers based on their structure (like rational or irrational), we can also categorize numbers based on their properties related to factors, divisors, and specific patterns. These classifications often apply to subsets of integers, most commonly the positive integers, which are the natural numbers.

Even and Odd Numbers

This classification applies to the set of integers ($\mathbb{Z}$).

• Even Numbers: An integer is called an even number if it is exactly divisible by 2. In other words, an even number is any integer that can be written in the form $2k$, where $k$ is an integer.

  $\text{Set of Even Integers} = \{ x \in \mathbb{Z} \ | \ x = 2k, \text{ for some } k \in \mathbb{Z} \}$

  Examples: ..., $-6, -4, -2, 0, 2, 4, 6, 8, ...$

  Note that 0 is considered an even number because it fits the definition ($0 = 2 \times 0$, and 0 is an integer).

• Odd Numbers: An integer is called an odd number if it is not exactly divisible by 2. An odd number is an integer that leaves a remainder of 1 when divided by 2. Odd numbers can be written in the form $2k+1$, where $k$ is an integer.

  $\text{Set of Odd Integers} = \{ x \in \mathbb{Z} \ | \ x = 2k+1, \text{ for some } k \in \mathbb{Z} \}$

  Examples: ..., $-7, -5, -3, -1, 1, 3, 5, 7, ...$

Every integer belongs to exactly one of these two categories: it is either even or odd.

Properties related to Even and Odd Numbers:

• Even $\pm$ Even = Even (e.g., $4 + 6 = 10$, $-2 - 8 = -10$)
• Odd $\pm$ Odd = Even (e.g., $3 + 5 = 8$, $9 - 7 = 2$)
• Even $\pm$ Odd = Odd (e.g., $4 + 3 = 7$, $10 - 5 = 5$)
• Even $\times$ Even = Even (e.g., $4 \times 6 = 24$)
• Odd $\times$ Odd = Odd (e.g., $3 \times 5 = 15$)
• Even $\times$ Odd = Even (e.g., $4 \times 3 = 12$)

Prime and Composite Numbers

This classification specifically applies to natural numbers greater than 1 ($\mathbb{N} \setminus \{1\}$).

• Prime Numbers: A prime number is a natural number greater than 1 that has exactly two distinct positive divisors: 1 and the number itself.

  The divisors of a prime number $p$ are only 1 and $p$.

  Examples: 2, 3, 5, 7, 11, 13, 17, 19, 23, ...

  Important points about prime numbers:

  • The number 1 is not considered a prime number. It has only one positive divisor (itself).
  • The number 2 is the smallest prime number. It is also the only even prime number. All other even numbers greater than 2 have 2 as a divisor in addition to 1 and themselves, so they have at least three divisors.
  • Prime numbers are the building blocks of the natural numbers through multiplication.

• Composite Numbers: A composite number is a natural number greater than 1 that is not prime. In other words, a composite number is a natural number greater than 1 that has more than two positive divisors.

  A composite number can be expressed as a product of two smaller natural numbers (greater than 1).

  Examples: 4 (divisors: 1, 2, 4), 6 (divisors: 1, 2, 3, 6), 8 (divisors: 1, 2, 4, 8), 9 (divisors: 1, 3, 9), 10 (divisors: 1, 2, 5, 10), 12, 14, 15, 16, ...

The number 1 is neither prime nor composite. It belongs to its own category among natural numbers.

So, the set of natural numbers $\mathbb{N} = \{1, 2, 3, 4, ...\}$ can be partitioned into three sets: $\{1\}$, the set of prime numbers, and the set of composite numbers.

Fundamental Theorem of Arithmetic

Also known as the Unique Factorization Theorem, this fundamental theorem of number theory states that every integer greater than 1 can be uniquely represented as a product of prime numbers, ignoring the order of the factors.

Statement: Every composite number can be expressed (factorised) as a product of primes, and this factorisation is unique, apart from the order in which the prime factors occur.

For example, the number 12 can be factorized as $2 \times 2 \times 3$, or $2^2 \times 3$. No other combination of prime numbers will multiply to give 12. The order might change ($2 \times 3 \times 2$), but the set of prime factors and their powers will always be $\{2, 2, 3\}$ or $\{2^2, 3^1\}$.

Answer:

Co-prime Numbers (or Relatively Prime Numbers)

Two integers $a$ and $b$ are said to be co-prime or relatively prime if their only common positive divisor is 1. This is equivalent to saying that their greatest common divisor (GCD) is 1.

Note that $a$ and $b$ do not have to be prime numbers themselves for them to be co-prime. The definition applies to any two integers (though often discussed for positive integers). By convention, 0 is co-prime only to 1 and -1. We usually consider non-zero integers for this concept.

Examples:

• Are 4 and 9 co-prime?

  Divisors of 4 are $\{1, 2, 4\}$. Divisors of 9 are $\{1, 3, 9\}$. The only common positive divisor is 1. $\text{GCD}(4, 9) = 1$. Yes, 4 and 9 are co-prime.

• Are 10 and 15 co-prime?

  Divisors of 10 are $\{1, 2, 5, 10\}$. Divisors of 15 are $\{1, 3, 5, 15\}$. Common positive divisors are $\{1, 5\}$. The greatest common divisor is 5. $\text{GCD}(10, 15) = 5 \neq 1$. No, 10 and 15 are not co-prime.

• Are 7 and 13 co-prime?

  7 is prime, its divisors are $\{1, 7\}$. 13 is prime, its divisors are $\{1, 13\}$. The only common positive divisor is 1. $\text{GCD}(7, 13) = 1$. Yes, 7 and 13 are co-prime. Two distinct prime numbers are always co-prime.

• Are 1 and 8 co-prime?

  Divisors of 1 are $\{1\}$. Divisors of 8 are $\{1, 2, 4, 8\}$. The only common positive divisor is 1. $\text{GCD}(1, 8) = 1$. Yes, 1 and 8 are co-prime. The number 1 is co-prime with every integer.

The concept of co-prime numbers is important in various areas, including fractions (simplifying to lowest terms), modular arithmetic, and cryptography.

Perfect Numbers

A perfect number is a positive integer that is equal to the sum of its proper positive divisors.

Proper divisors of a number are all its positive divisors, excluding the number itself.

Example 1: Is 6 a perfect number?

The positive divisors of 6 are 1, 2, 3, and 6.

The proper positive divisors of 6 are the divisors excluding 6, which are 1, 2, and 3.

Sum of proper divisors $= 1 + 2 + 3 = 6$.

Since the sum of the proper divisors is equal to the number itself (6 = 6), 6 is a perfect number.

Example 2: Is 28 a perfect number?

The positive divisors of 28 are 1, 2, 4, 7, 14, and 28.

The proper positive divisors of 28 are 1, 2, 4, 7, and 14.

Sum of proper divisors $= 1 + 2 + 4 + 7 + 14 = 28$.

Since the sum of the proper divisors is equal to the number itself (28 = 28), 28 is a perfect number.

The next perfect number is 496, and after that 8128. These numbers become increasingly rare.

There is a strong connection between perfect numbers and a special type of prime number called Mersenne primes (prime numbers of the form $2^p - 1$, where $p$ is also a prime number). Euclid proved that if $2^p - 1$ is a Mersenne prime, then the number $2^{p-1}(2^p - 1)$ is an even perfect number.

Other Specific Number Types

Number theory is rich with various classifications of numbers based on specific properties, relationships, or geometric arrangements. Some other notable types include:

• Square Numbers (or Perfect Squares): These are numbers obtained by squaring an integer (multiplying an integer by itself). They can be arranged to form a square pattern of dots.

  $\text{Set of Square Numbers} = \{ n^2 \ | \ n \in \mathbb{Z} \}$

  Examples: $1^2=1, 2^2=4, 3^2=9, 4^2=16, 5^2=25, ...$ Also include squares of negative integers and zero: $(-1)^2=1, (-2)^2=4, 0^2=0$. Conventionally, when listing square numbers, we usually mean positive square numbers: $1, 4, 9, 16, 25, ...$

• Cube Numbers (or Perfect Cubes): These are numbers obtained by cubing an integer (multiplying an integer by itself three times).

  $\text{Set of Cube Numbers} = \{ n^3 \ | \ n \in \mathbb{Z} \}$

  Examples: $1^3=1, 2^3=8, 3^3=27, 4^3=64, ...$ Also include cubes of negative integers and zero: $(-1)^3=-1, (-2)^3=-8, 0^3=0$. The positive cube numbers are: $1, 8, 27, 64, 125, ...$

• Triangular Numbers: These are numbers that can be represented as a triangular arrangement of dots. They are formed by the sum of consecutive natural numbers starting from 1.

  $\text{The } n\text{-th triangular number} = T_n = 1 + 2 + 3 + ... + n = \frac{n(n+1)}{2}$

  Examples: $T_1 = 1$, $T_2 = 1+2=3$, $T_3 = 1+2+3=6$, $T_4 = 1+2+3+4=10$, $T_5 = 1+2+3+4+5=15, ...$ The sequence is 1, 3, 6, 10, 15, 21, 28, ...

• Fibonacci Numbers: These form a sequence starting with 0 and 1, where each subsequent number is the sum of the two preceding ones. It's often denoted by $F_n$.

  Definition: $F_0 = 0, F_1 = 1, F_n = F_{n-1} + F_{n-2}$ for $n > 1$.

  Sequence: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ...

• Transcendental Numbers: These are real numbers that are not algebraic numbers. An algebraic number is a root of a non-zero polynomial equation with integer coefficients. Transcendental numbers cannot be roots of such polynomials.

  Examples: The most famous transcendental numbers are $\pi$ and $e$. Proving a number is transcendental is generally very difficult.

These are just a few examples of the many fascinating types of numbers studied in number theory and other branches of mathematics. Each type has unique properties and applications.