Chapter 2 Whole Numbers (Concepts)

Introduction of Whole Numbers

In our study of numbers, we first learn about Natural Numbers. These are the fundamental numbers we use for counting things in the real world, such as apples, books, or people. The set of natural numbers, often denoted by $\mathbb{N}$, begins with 1 and continues infinitely.

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

However, natural numbers are not sufficient to represent every situation. For example, if you have 3 apples and you eat all 3, how many apples are left? The answer is "none" or "nothing". To represent this concept of "nothingness" mathematically, we need the number zero (0).

When we take the entire collection of natural numbers and include the number zero, we form a new, more complete set of numbers. This set is called the set of Whole Numbers.

The set of whole numbers is commonly represented by the symbol $\mathbb{W}$.

$\mathbb{W} = \{0, 1, 2, 3, 4, 5, 6, ...\}$

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Relationship between Natural Numbers and Whole Numbers

The definitions of natural and whole numbers show a very clear and simple relationship. The set of whole numbers is essentially the set of natural numbers with just one extra element: zero.

• Every natural number is also a whole number. (e.g., 7 is a natural number and also a whole number).
• Zero (0) is a whole number, but it is not a natural number.

This means that the set of natural numbers is a subset of (is contained within) the set of whole numbers.

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Smallest and Largest Whole Number

Smallest Whole Number

If we look at the set of whole numbers, $\mathbb{W} = \{0, 1, 2, 3, ...\}$, the sequence begins with the number 0. There is no whole number smaller than 0.

Therefore, the smallest whole number is $\mathbf{0}$.

Largest Whole Number

The set of whole numbers, like the natural numbers, is infinite. This means it goes on forever without end. For any whole number you can think of, no matter how large, we can always find a larger one simply by adding 1 to it.

For any whole number $n$, the number $n+1$ is also a whole number and is greater than $n$.

Therefore, there is no largest whole number.

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Predecessor and Successor

For any number in a sequence, the number that comes immediately before it is its predecessor, and the number that comes immediately after it is its successor.

• The successor of a whole number is found by adding 1 to it.
  Successor of $n = n + 1$.
  For example, the successor of $15$ is $15+1=16$. The successor of $0$ is $0+1=1$. Every single whole number has a successor.
• The predecessor of a whole number (except 0) is found by subtracting 1 from it.
  Predecessor of $n = n - 1$ (for $n > 0$).
  For example, the predecessor of $15$ is $15-1=14$. The predecessor of $1$ is $1-1=0$.

The special case of Zero: The number 0 has a successor (which is 1), but it does not have a predecessor within the set of whole numbers. The number before 0 is -1, which is an integer but not a whole number. This is a key difference from natural numbers, where 1 has no predecessor.

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Representation of Whole Numbers on a Number Line

A number line is a straight line with points marked at equal intervals. It is a visual way to represent numbers. To represent whole numbers on a number line:

1. Draw a straight horizontal line.
2. Mark a point on the far left and label it 0. This is the starting point for whole numbers.
3. Mark points at equal distances to the right of 0.
4. Label these points with the numbers 1, 2, 3, 4, and so on.
5. Place an arrow at the right end of the line to show that the numbers continue infinitely and there is no largest whole number.

The number line helps us visualize the order of numbers. Any number on the right is greater than any number on its left. For example, since 5 is to the right of 2, we can say $5 > 2$.

Fundamental Operations on Whole Numbers

In mathematics, there are four main operations that we perform on numbers: Addition, Subtraction, Multiplication, and Division. These are often called the fundamental operations. We can perform these operations on whole numbers. Understanding how these operations work with whole numbers is essential.

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Addition of Whole Numbers

Addition is the process of combining two or more numbers. When we add whole numbers, the result is called the sum. Addition of whole numbers can be thought of as combining collections of objects.

Answer:

Properties of Addition of Whole Numbers:

Addition of whole numbers has some important properties:

• Closure Property:

  When you add any two whole numbers, the result is always a whole number. We say that whole numbers are closed under addition.

  Example: Take two whole numbers, $2$ and $3$. Their sum is $2 + 3 = 5$. Since $5$ is also a whole number, the closure property holds. If $a \in \mathbb{W}$ and $b \in \mathbb{W}$, then $a+b \in \mathbb{W}$.

• Commutative Property:

  The order in which you add two whole numbers does not change the sum. You can swap the numbers, and the result will be the same. We say that addition of whole numbers is commutative.

  For any two whole numbers $a$ and $b$, $a + b = b + a$.

  Example: $5 + 7 = 12$. Also, $7 + 5 = 12$. So, $5 + 7 = 7 + 5$.

• Associative Property:

  When you add three or more whole numbers, the way you group them (using brackets) does not change the final sum. We say that addition of whole numbers is associative.

  For any three whole numbers $a, b,$ and $c$, $(a + b) + c = a + (b + c)$.

  Example: Let's add $2, 3,$ and $4$.
  Grouping 1: $(2 + 3) + 4 = 5 + 4 = 9$.
  Grouping 2: $2 + (3 + 4) = 2 + 7 = 9$.
  Since the results are the same, $(2 + 3) + 4 = 2 + (3 + 4)$.

• Additive Identity (Zero Property):

  When you add zero ($0$) to any whole number, the sum is the whole number itself. Zero is called the additive identity for whole numbers.

  For any whole number $a$, $a + 0 = a$ and $0 + a = a$.

  Example: $10 + 0 = 10$. $0 + 55 = 55$.

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Property	Rule	Example
Closure Property	The sum of any two whole numbers is always a whole number.	$8 + 9 = 17$. Since 17 is a whole number, the set is closed.
Commutative Property	The order of adding two numbers does not change the sum. $a + b = b + a$	$5 + 12 = 17$ and $12 + 5 = 17$. So, $5 + 12 = 12 + 5$.
Associative Property	The grouping of numbers does not change the sum when adding three or more numbers. $(a + b) + c = a + (b + c)$	$(2 + 3) + 4 = 5 + 4 = 9$ $2 + (3 + 4) = 2 + 7 = 9$
Additive Identity	Adding zero (0) to any whole number leaves it unchanged. Zero is the additive identity. $a + 0 = a$	$45 + 0 = 45$.

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Subtraction of Whole Numbers

Subtraction is the inverse (opposite) operation of addition. It means taking away one number from another or finding the difference between two numbers. The result of subtraction is called the difference.

Answer:

Properties of Subtraction of Whole Numbers:

Subtraction of whole numbers has different properties compared to addition:

• Closure Property:

  When you subtract one whole number from another, the result is not always a whole number. For example, $3 - 5$ is not a whole number (it's $-2$). Therefore, whole numbers are not closed under subtraction.

• Commutative Property:

  The order in which you subtract numbers matters. $a - b$ is generally not equal to $b - a$. Subtraction of whole numbers is not commutative.

  Example: $5 - 3 = 2$, but $3 - 5$ is not a whole number. Even for cases like $5-5=0$ and $5-5=0$, taking other examples shows it's not generally true, e.g., $5-3 \ne 3-5$.

• Associative Property:

  The way you group numbers for subtraction affects the result. $(a - b) - c$ is generally not equal to $a - (b - c)$. Subtraction of whole numbers is not associative.

  Example: Let's subtract using $8, 4,$ and $2$.
  Grouping 1: $(8 - 4) - 2 = 4 - 2 = 2$.
  Grouping 2: $8 - (4 - 2) = 8 - 2 = 6$.
  Since $2 \neq 6$, $(8 - 4) - 2 \neq 8 - (4 - 2)$.

• Identity Property:

  Subtracting zero from a whole number gives the number itself ($a - 0 = a$). However, $0 - a$ is not equal to $a$ (unless $a=0$) and is generally not a whole number if $a > 0$. So, there is no additive identity for subtraction in whole numbers in the same way as for addition.

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Multiplication of Whole Numbers

Multiplication can be thought of as repeated addition. For example, $3 \times 5$ means adding $5$ three times ($5 + 5 + 5$). The result of multiplication is called the product.

Answer:

Properties of Multiplication of Whole Numbers:

Multiplication of whole numbers also has some important properties:

• Closure Property:

  When you multiply any two whole numbers, the result is always a whole number. Whole numbers are closed under multiplication.

  Example: Take $4 \in \mathbb{W}$ and $6 \in \mathbb{W}$. Their product is $4 \times 6 = 24$. Since $24$ is a whole number, closure holds. If $a \in \mathbb{W}$ and $b \in \mathbb{W}$, then $a \times b \in \mathbb{W}$.

• Commutative Property:

  The order in which you multiply two whole numbers does not change the product. Multiplication of whole numbers is commutative.

  For any two whole numbers $a$ and $b$, $a \times b = b \times a$.

  Example: $5 \times 7 = 35$. Also, $7 \times 5 = 35$. So, $5 \times 7 = 7 \times 5$.

• Associative Property:

  When you multiply three or more whole numbers, the way you group them (using brackets) does not change the final product. Multiplication of whole numbers is associative.

  For any three whole numbers $a, b,$ and $c$, $(a \times b) \times c = a \times (b \times c)$.

  Example: Let's multiply $2, 3,$ and $4$.
  Grouping 1: $(2 \times 3) \times 4 = 6 \times 4 = 24$.
  Grouping 2: $2 \times (3 \times 4) = 2 \times 12 = 24$.
  Since the results are the same, $(2 \times 3) \times 4 = 2 \times (3 \times 4)$.

• Multiplicative Identity (Property of One):

  When you multiply any whole number by one ($1$), the product is the whole number itself. One is called the multiplicative identity for whole numbers.

  For any whole number $a$, $a \times 1 = a$ and $1 \times a = a$.

  Example: $15 \times 1 = 15$. $1 \times 99 = 99$.

• Multiplication by Zero:

  When you multiply any whole number by zero ($0$), the product is always zero.

  For any whole number $a$, $a \times 0 = 0$ and $0 \times a = 0$.

  Example: $25 \times 0 = 0$. $0 \times 1234 = 0$.

• Distributive Property of Multiplication over Addition:

  This property relates multiplication and addition. If you multiply a number by the sum of two other numbers, the result is the same as multiplying the number by each of the other numbers separately and then adding the products.

  For any whole numbers $a, b,$ and $c$, $a \times (b + c) = (a \times b) + (a \times c)$.

  Example: Let's calculate $5 \times (2 + 3)$ using the property.
  Left side: $5 \times (2 + 3) = 5 \times 5 = 25$.
  Right side: $(5 \times 2) + (5 \times 3) = 10 + 15 = 25$.
  Since $25 = 25$, the property holds: $5 \times (2 + 3) = (5 \times 2) + (5 \times 3)$.

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Property	Rule	Example
Closure Property	The product of any two whole numbers is always a whole number.	$7 \times 8 = 56$. Since 56 is a whole number, the set is closed.
Commutative Property	The order of multiplying two numbers does not change the product. $a \times b = b \times a$	$4 \times 9 = 36$ and $9 \times 4 = 36$. So, $4 \times 9 = 9 \times 4$.
Associative Property	The grouping of numbers does not change the product. $(a \times b) \times c = a \times (b \times c)$	$(2 \times 3) \times 5 = 6 \times 5 = 30$ $2 \times (3 \times 5) = 2 \times 15 = 30$
Multiplicative Identity	Multiplying any whole number by one (1) leaves it unchanged. One is the multiplicative identity. $a \times 1 = a$	$72 \times 1 = 72$.
Multiplication by Zero	Multiplying any whole number by zero (0) results in zero.	$39 \times 0 = 0$.
Distributive Property	Multiplication distributes over addition. $a \times (b + c) = (a \times b) + (a \times c)$	$5 \times (10+2) = 5 \times 12 = 60$ $(5 \times 10)+(5 \times 2) = 50+10=60$

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Division of Whole Numbers

Division is the inverse (opposite) operation of multiplication. It is the process of splitting a number into equal parts or groups. In a division problem like $a \div b = c$:

• $a$ is the Dividend (the total number being divided).
• $b$ is the Divisor (the number by which the dividend is divided, or the number of equal parts).
• $c$ is the Quotient (the result of the division, or the number in each part).

Sometimes, when we divide, the dividend is not perfectly divisible by the divisor. In such cases, there is an amount left over, which is called the Remainder.

The relationship between the Dividend, Divisor, Quotient, and Remainder is given by the Division Algorithm:

Dividend $= ($ Divisor $\times$ Quotient $) +$ Remainder

where the Remainder must always be a whole number greater than or equal to $0$ and strictly less than the Divisor.

Remainder $\ge 0$ and Remainder $<$ Divisor.

Answer:

Properties of Division of Whole Numbers:

Division of whole numbers has certain constraints and specific properties:

• Closure Property:

  When you divide one whole number by another whole number, the result is not always a whole number. For example, $7 \div 2 = 3.5$, which is not a whole number. Therefore, whole numbers are not closed under division.

• Division by Zero:

  Division of any non-zero whole number by zero ($0$) is undefined. This is a fundamental rule in mathematics because it leads to a logical contradiction and has no meaningful answer.

  To understand why, let's remember the relationship between multiplication and division. If we say $5 \div 0 = q$ (where q is some quotient), this must mean that the inverse is true: $q \times 0 = 5$.

  However, we know that any number multiplied by 0 is always 0. Since no number 'q' exists that can make this statement true, the operation is impossible.

  Attempting to show this with the long division structure also reveals the problem:

  $\begin{array}{r} ? \phantom{)} \\ 0{\overline{\smash{\big)}\,5\phantom{)}}} \end{array}$

  We ask: "What number can we put in the quotient (?) such that $? \times 0$ is equal to or just less than 5?" No matter what number we choose for the quotient (1, 10, 500...), the product will always be 0.

  $\begin{array}{r} \text{any number} \\ 0{\overline{\smash{\big)}\,5\phantom{)}}} \\ \underline{-~\phantom{(}0\phantom{)}} \\ 5\phantom{)} \end{array}$

  This leaves a remainder of 5. A crucial rule of division is that the remainder must be less than the divisor. Here, the remainder is 5 and the divisor is 0. The condition $5 < 0$ is false. Since the process breaks the rules of division, it is undefined.

• Division of Zero by a Non-zero Number:

  Dividing zero ($0$) by any whole number other than zero always results in zero.

  For any whole number $a \neq 0$, $0 \div a = 0$.

  Example: Find $0 \div 10$.

  This asks "what number multiplied by 10 gives 0?". The only answer is 0. Using long division:

  $\begin{array}{r} 0 \phantom{)} \\ 10{\overline{\smash{\big)}\,0\phantom{)}}} \\ \underline{-~\phantom{(}0\phantom{)}} \\ 0\phantom{)} \end{array}$

  The quotient is 0 and the remainder is 0.

  (If you have 0 sweets to share among 10 friends, each friend gets 0 sweets.)

• Division by One:

  Dividing any whole number by one ($1$) gives the number itself.

  For any whole number $a$, $a \div 1 = a$.

  Example: Find $20 \div 1$.

  Using long division:

  $\begin{array}{r} 20 \phantom{)} \\ 1{\overline{\smash{\big)}\,20\phantom{)}}} \\ \underline{-~\phantom{(}2\phantom{)}} \\ 00\phantom{)} \\ \underline{-~\phantom{(}0\phantom{)}} \\ 0\phantom{)} \end{array}$

  The quotient is 20 and the remainder is 0.

  (If you have 20 chocolates to give to 1 person, that person gets all 20 chocolates.)

Patterns in Whole Numbers

Whole numbers can be arranged in various ways to reveal interesting and beautiful patterns. One of the simplest ways to visualize numbers and their properties is by arranging dots in basic geometric shapes like lines, rectangles, squares, and triangles. These are sometimes called "figurate numbers".

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Arranging Numbers as Lines

Every whole number greater than 1 can be represented as a line of dots. This is the most basic representation, where the number of dots simply corresponds to the value of the number.

Number 3:
$$\begin{matrix} \bullet & \bullet & \bullet \end{matrix}$$

Number 5:
$$\begin{matrix} \bullet & \bullet & \bullet & \bullet & \bullet \end{matrix}$$

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Arranging Numbers as Rectangles

Some numbers can be arranged to form a rectangle with more than one row and more than one column. The ability to form such a rectangle is a key indicator of the number's factors.

Numbers that can be arranged as a rectangle (with at least two rows and two columns) are called composite numbers. This is because a composite number can be expressed as a product of two smaller whole numbers (other than 1), which can represent the number of rows and columns.

Number 6: The number 6 is composite. It can be shown as a rectangle of 2 rows and 3 columns, or 3 rows and 2 columns.

Numbers that can only be arranged as a single line (like 2, 3, 5, 7, etc.) are called prime numbers. They cannot form a rectangle with both dimensions greater than 1.

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Arranging Numbers as Squares

Some numbers can be arranged in the shape of a perfect square, where the number of rows is equal to the number of columns. These numbers are called square numbers.

A square number is the result of multiplying a whole number by itself (e.g., $n \times n$ or $n^2$).

Number 4 ($2 \times 2$):
$$\begin{matrix} \bullet & \bullet \\ \bullet & \bullet \end{matrix}$$

Number 9 ($3 \times 3$):
$$\begin{matrix} \bullet & \bullet & \bullet \\ \bullet & \bullet & \bullet \\ \bullet & \bullet & \bullet \end{matrix}$$

The sequence of square numbers is $1, 4, 9, 16, 25, 36, ...$

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Arranging Numbers as Triangles

Some numbers can be arranged as equilateral triangles, where each new row has one more dot than the row above it. These are called triangular numbers.

Triangular numbers are formed by the sum of consecutive natural numbers starting from 1.

Number 3 ($1 + 2$):
$$\begin{matrix} \bullet \\ \bullet & \bullet \end{matrix}$$

Number 6 ($1 + 2 + 3$):
$$\begin{matrix} \bullet \\ \bullet & \bullet \\ \bullet & \bullet & \bullet \end{matrix}$$

Number 10 ($1 + 2 + 3 + 4$):
$$\begin{matrix} \bullet \\ \bullet & \bullet \\ \bullet & \bullet & \bullet \\ \bullet & \bullet & \bullet & \bullet \end{matrix}$$

The sequence of triangular numbers is $1, 3, 6, 10, 15, 21, ...$

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Observing Patterns

By looking at these shapes, we can discover some fascinating relationships between different types of numbers.

Combining Triangular Numbers to make Square Numbers

A remarkable pattern emerges when we add two consecutive triangular numbers.

For example, let's take the 2nd triangular number (3) and the 3rd triangular number (6).

$3 + 6 = 9$, which is the square number $3^2$.

We can visualize this by taking the dot patterns for 3 and 6 and fitting them together. If we flip one of the triangles, it fits perfectly with the other to form a square.

This holds true for any pair of consecutive triangular numbers:

• $1 + 3 = 4 = 2^2$
• $3 + 6 = 9 = 3^2$
• $6 + 10 = 16 = 4^2$
• $10 + 15 = 25 = 5^2$

This shows a deep connection between these two types of patterned numbers.