3.2: Whole and Natural Numbers

Understanding Whole and Natural Numbers

Mathematics is all around us, from the time we spend budgeting our expenses to the measurements we make in cooking. A strong foundation in basic number concepts, such as whole and natural numbers, is essential for more complex mathematical thinking. This section will refresh your understanding of these fundamental concepts, explore their properties, and demonstrate their real-world applications.

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What Are Natural Numbers?

Definition: Natural numbers, (we use the symbol \(\mathbb{N}\) to represent the natural numbers), are the set of positive numbers used for counting and ordering. They start at 1 and continue infinitely. This set of numbers is the most basic form of numbers that we use daily. Natural Numbers: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, …

Visual Representation:

Imagine a number line that begins at 1 and continues to the right indefinitely. This number line represents natural numbers. They are sequential and used to count objects, rank items, and perform basic arithmetic.

Examples of using Natural Numbers:

• Counting Items: You have 5 apples.
• Ordering: Your book is number 7 on the list.

Important Note:

Natural numbers do not include zero or negative numbers. They are always positive integers starting from 1.

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What Are Whole Numbers?

Definition: Whole numbers, (we use the symbol \(\mathbb{N}_{0}\) or \(\mathbb{W}\) to represent whole numbers), include all natural numbers plus zero. This set represents the numbers we use for counting as well as indicating the absence of quantity. Whole Numbers: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, …

Visual Representation:

Consider a number line that starts at 0 and continues to the right indefinitely. This line represents whole numbers, showing that they include zero and extend infinitely.

Examples of Whole Numbers:

• Count of Objects: You have 3 books on the table.
• Zero Scenario: There are 0 apples in the basket.

Important Note:

Whole numbers include zero, which signifies a null quantity, while natural numbers do not include zero.

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Comparing Natural and Whole Numbers

Key Differences:

Feature	Natural Numbers	Whole Numbers
Includes Zero	No	Yes
Starting Point	1	0
Examples	1, 2, 3, 4, 5	0, 1, 2, 3, 4, 5
Application	Counting, Ordering	Counting, Ordering, Indicating absence

Exercise \(\PageIndex{1}\)

Determine which of the following numbers are natural numbers and which are whole numbers:

\(7, 0, -3, 12, \frac{3}{4}\)

Answer

Natural Numbers: \(7, 12\); Whole Numbers: \(0, 7, 12\)

Exercise \(\PageIndex{2}\)

Determine which of the following numbers are natural numbers and which are whole numbers:

\(\frac{1}{3}, -15, -\frac{4}{9}, 0.1, 20, 1,783, -4.65\)

Answer

Natural Numbers: \(20, 1,783\); Whole Numbers: \(20, 1,783\)

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Comparison of Properties

Whole numbers and natural numbers are foundational concepts in mathematics. Here’s a comparison of their properties to help you understand their differences and similarities:

Property	Natural Numbers	Whole Numbers
Starting Point	1	0
Zero	Not included	Included
Infinite	Yes	Yes
Used For	Counting and ordering	Counting, ordering, and representing quantities (including none)
Includes Negative Numbers	No	No
Includes Fractions/Decimals	No	No

What does it mean to be 'Closed'

A set is closed under an operation if applying that operation to any elements of the set results in another element that is also in the set. This means if you apply certain operations on natural numbers the result will also be a natural number. Likewise, if you perform certain operations on whole numbers the result will also be a whole number.

Property	Natural Numbers	Whole Numbers
Addition	Closed under addition	Closed under addition
Subtraction	Not closed	Not closed
Multiplication	Closed under multiplication	Closed under multiplication
Division	Not closed	Not closed
Commutative Property	Yes (for addition and multiplication)	Yes (for addition and multiplication)
Associative Property	Yes (for addition and multiplication)	Yes (for addition and multiplication)
Distributive Property	Yes	Yes

The Closure Property

The closure property for natural numbers means that if you take any two natural numbers and perform a specific mathematical operation on them, the result will always be a natural number. This only happens with addition and multiplication for natural numbers. (This also applies to whole numbers.)

Closure Property of Addition

Addition: Adding two natural or whole numbers results in a larger natural or whole number.

Example for Natural Numbers: \(3 + 4 = 7\)

Example for Whole Numbers: \(0 + 6 = 6\)

Closure Property of Multiplication

Multiplication: Multiplying two natural or whole numbers always results in a natural or whole number.

Example for Natural Numbers: \(3 \times 4 = 12\), the result will always be a larger natural number than the numbers you are multiplying

Example for Whole Numbers: \(0 \times 12 = 0\), the result will always be zero or a larger natural number than the numbers you are multiplying

The Natural Numbers are NOT closed on subtraction or division

You might find some cases where the result is a natural or whole number but to be 'closed' it must result in a natural or whole number in all cases.

Subtraction: Subtracting a natural number from a natural number may result in a number that is not always natural.

Example for Natural Numbers: \(10 - 7 = 3\), is a natural number BUT \(3 - 12 = -9\), is not a natural number, therefore, natural numbers are not closed on subtraction

Example for Whole Numbers: \(13 - 0 = 13\), which is in the set of whole numbers BUT \(0 - 8 = -8\), is not in the set of whole numbers, therefore, whole numbers are not closed on subtraction

Division: Dividing two natural or whole numbers can result in a fraction or decimal.

Example for Natural Numbers: \(9 \div 12 = 0.75\), is not a natural number

Example for Whole Numbers: \(6 \div 0 = undefined\), not even a number

The Commutative Property

The commutative property states that the order in which two numbers are added or multiplied does not change the result.

Commutative Property of Addition: \(a + b = b + a\)

Example: \(7 + 13 = 13 + 7\), both equal \(20\)

Commutative Property of Multiplication: \(a \times b = b \times a\)

Example: \( 5 \times 6 = 6 \times 5\). both equal \(30\)

The Natural Numbers are NOT commutative on subtraction or division

Subtraction: \(a - b \neq b - a\)

Example: \(5 - 3 \neq 3 - 5\)

Division: \( a \div b \neq b \div a\)

Example: \(10 \div 2 \neq 2 \div 10\)

Exercise \(\PageIndex{3}\)

Show how using the commutative property can be helpful with adding this list of numbers

\(3 + 15 + 4 + 17 + 20 + 19 + 5 + 1 + 16\)

Answer

Since the commutative property allows us to reorder numbers, we could reorder these in such a way that each pair of numbers adds to 20 then it would be easier to add several 20's together

\(3 + 15 + 4 + 17 + 20 + 19 + 5 + 1 + 16 = 3 + 17 + 15 + 5 + 4 + 16 + 20 + 19 + 1\)

So, \(3 + 17 + 15 + 5 + 4 + 16 + 20 + 19 + 1 = 20 + 20 + 20 + 20 + 20 = 100\)

Exercise \(\PageIndex{4}\)

Show how using the commutative property can be helpful with adding this list of numbers.

\(200 + 4 + 600 + 3 + 5 + 100 + 6\)

Answer

Since the commutative property allows us to reorder numbers, we could reorder these in such a way that the smaller numbers are first and the larger numbers are last

\(200 + 4 + 600 + 3 + 5 + 100 + 6 = 3 + 4 + 5 + 6 + 200 + 600 + 100\)

So, \(3 + 4 + 5 + 6 + 200 + 600 + 100 = 7 + 11 + 800 + 100 = 18 + 900 = 918\)

The Associative Property

The associative property is a key mathematical concept that shows how the grouping of numbers in addition and multiplication does not affect the result of these operations.

Associative Property of Addition: \(a + (b + c) = (a + b) + c\)

Example:
\(2 + (5 + 9) = (2 + 5) + 9\)
\(2 + 14 = 7 + 9\)
\(16 = 16\)

Associative Property of Multiplication: \(a \times (b \times c) = (a \times b) \times c\)

Example:
\(4 \times (6 \times 2) = (4 \times 6) \times 2\)
\(4 \times 12 = 24 \times 2\)
\(48 = 48\)

Associative Property does not apply to subtraction and division

Example:
\(2 - (5 - 9) = (2 - 5) - 9\)
\(2 - (-4) = (-3) - 9\)
\(2 + 4 = -3 - 9\)
\(6 \neq -12\)

Example:
\(2 \div (10 \div 3) = (2 \div 10) \div 3\)
\(2 \div \frac{10}{3} = \frac{2}{10} \div 3\)
\(2 \times \frac{3}{10}= \frac{2}{10} \times \frac{1}{3}\)
\(\frac{6}{10} \neq \frac{2}{30}\)
\(\frac{3}{5} \neq \frac{1}{15}\)

Exercise \(\PageIndex{5}\)

Using the associative property show an equivalent expression for \(3 \times (4 \times 11)\).

Answer

Since the associative property (for multiplication) is \(a \times (b \times c) = (a \times b) \times c\), let \(a = 3\), \(b = 4\), and \(c = 11\), then the equivalent expression for \(3 \times (4 \times 11)\) is \((3 \times 4) \times 11\).

Proof:
\(3 \times (4 \times 11) = 3 \times 44 = 132\)
\((3 \times 4) \times 11 = 12 \times 11 = 132\)

Exercise \(\PageIndex{6}\)

Using the associative property show an equivalent expression for \((20 + 94) + 356\).

Answer

Since the associative property (for addition) is \(a + (b + c) = (a + b) + c\), let \(a = 20\), \(b = 94\), and \(c = 356\), then the equivalent expression for \((20 + 94) + 356\) is \20 + (94 + 356)\).

Proof:
\((20 + 94) + 356 = 114 + 356 = 470\)
\(20 + (94 + 356) = 20 + 450 = 470\)

The Distributive Property

The distributive property states that for any numbers \(a\), \(b\), and \(c\):

\(a \times (b + c) = (a \times b) + (a \times c)\)

This means that when you multiply a number by a sum, you can distribute the multiplication to each term in the sum and then add the results.

Similarly, it applies to subtraction:

\(a \times (b - c) = (a \times b) - (a \times c)\)

This means that when you multiply a number by a difference, you can distribute the multiplication to each term in the difference and then subtract the results.

Exercise \(\PageIndex{7}\)

Using the distribution property show an equivalent expression for \(6 \times (7 + 3)\).

Answer

Since the distributive property is \(a \times (b + c) = (a \times b) + (a \times c)\), let \(a = 6\), \(b = 7\), and \(c = 3\), then the equivalent expression for \(6 \times (7 + 3)\) is \((6 \times 7) + (6 \times 3)\)

Proof:
\(6 \times (7 + 3) = 6 \times 10 = 60\)
\((6 \times 7) + (6 \times 3) = 42 + 18 = 60\)

Exercise \(\PageIndex{8}\)

Using the distribution property show an equivalent expression for \((10 \times 15) + (10 \times 9)\)

Answer

Since the distributive property is \((a \times b) + (a \times c) = a \times (b + c) \), let \(a = 10\), \(b = 15\), and \(c = 9\), then the equivalent expression for \((10 \times 15) + (10 \times 9)\) is \(10 \times (15 + 9)\)

Proof:
\((10 \times 15) + (10 \times 9) = 150 + 90 = 240\)
\(10 \times (15 + 9) = 10 \times 24 = 240\)

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Why is it Helpful to Know About Operations Being 'Closed'

Understanding that natural numbers are closed under addition and multiplication is fundamental for both practical arithmetic and advanced mathematical concepts. This closure property simplifies problem-solving because you can confidently perform these operations knowing the results will always be natural numbers. For example, adding 3 and 4 gives you 7, a natural number, and multiplying 2 and 5 gives you 10, another natural number. This reliability supports the development of arithmetic algorithms, such as those for addition and multiplication, ensuring that calculations are consistent and accurate. Furthermore, this concept forms the foundation for learning more complex topics in mathematics, such as algebraic structures and proofs, where closure properties are crucial for understanding and establishing theorems. In practical applications, knowing that natural numbers are closed under these operations helps in creating reliable mathematical models, like budgeting, where calculations must stay within the set of natural numbers. Additionally, this knowledge underpins mathematical proofs, helping you to construct and understand arguments about number properties. Thus, the closure property not only makes arithmetic straightforward but also prepares you for deeper mathematical exploration and practical problem-solving.

Understanding the concept of closure in mathematics is essential for elementary school teachers for several reasons.

1. Building a Strong Mathematical Foundation

Understanding closure helps elementary school teachers lay a solid foundation for their students' future math education. When teachers grasp how basic operations like addition and multiplication work within specific sets of numbers, they can more effectively teach these concepts to young learners. For example, knowing that addition is closed for whole numbers allows teachers to confidently explain to students that adding two whole numbers always results in another whole number. This foundational understanding is crucial for teaching arithmetic, which is the basis for more advanced math concepts.

Example: If a teacher understands that whole numbers are closed under addition, they can confidently explain to students that adding two whole numbers, such as 3 and 4, will always give another whole number, such as 7. This concept helps students grasp that the results of their arithmetic operations will remain within the set of whole numbers.

2. Developing Effective Teaching Strategies

Teachers who understand closure can develop more effective teaching strategies and lesson plans. By recognizing which operations are closed for different sets of numbers, teachers can create appropriate exercises and problems for their students. For instance, if a teacher knows that subtraction is not closed for whole numbers, they can design activities to help students explore what happens when subtraction results in a negative number and explain that the set of whole numbers does not include negative results.

Example: A teacher might use a visual aid to show that subtracting 3 from 2 results in -1, which is not a whole number. This helps students understand that not all operations will always stay within the same set of numbers.

3. Fostering Critical Thinking Skills

Teaching the concept of closure encourages students to think critically about numbers and operations. When students understand that operations can be performed within a set but may not always stay within that set, they learn to question and explore mathematical properties. This critical thinking is a key component of problem-solving skills and helps students develop a deeper understanding of mathematics.

Example: By explaining that multiplication is closed for natural numbers, teachers can encourage students to explore and predict outcomes for various multiplication problems, which fosters critical thinking and problem-solving abilities.

4. Supporting Effective Problem-Solving Techniques

An understanding of closure equips teachers with the knowledge to help students solve problems correctly. For example, when students learn that the set of integers is closed under addition and multiplication, teachers can guide students in solving problems involving these operations, knowing that their answers will always be integers.

Example: A teacher can show students that adding or multiplying any two integers will always result in another integer, which helps students understand and solve problems involving these operations.

5. Bridging to More Advanced Concepts

Early exposure to the idea of closure prepares students for more advanced mathematical concepts in the future. When teachers introduce these concepts early on, they help students develop a mindset for exploring and understanding more complex mathematical ideas as they advance in their education.

Example: By demonstrating that addition is closed for whole numbers and introducing similar concepts, teachers help students build a framework for learning about algebraic structures and operations in higher grades.

Imagine teaching a child to play a game with rules. If you know the rules work the same way every time (like always getting a red card in the game means you stay in the game), you can teach the child the rules confidently and help them understand how the game works. Similarly, understanding closure helps teachers explain math rules clearly and consistently, which is crucial for effective teaching and learning.

For elementary school teachers, understanding the concept of closure is crucial because it underpins the way they teach basic arithmetic operations and mathematical properties. It helps them create effective lesson plans, support students in developing critical thinking skills, and lay the groundwork for future math concepts. By grasping how and why operations work within certain sets, teachers can provide clearer explanations and better prepare students for more advanced mathematics.

Citations on Attributions

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