Real Numbers and Properties

You have been using numbers your entire life - counting change, measuring ingredients, checking temperatures, or figuring out how far you have driven. What you might not realize is that all these different kinds of numbers belong to a carefully organized system, and that system follows a set of reliable rules that make algebra possible. Understanding these rules is like learning the grammar of mathematics: once you know how the pieces fit together, everything else becomes easier.

In this lesson, we are going to explore the real number system from the ground up. You will learn how different types of numbers relate to each other, and you will discover the properties that govern how numbers behave when you add, subtract, multiply, and divide them. These properties are not arbitrary rules invented to make your life difficult - they are patterns that numbers naturally follow, and they will become powerful tools as you move deeper into algebra.

Core Concepts

The Real Number System

Think of the real number system as a family tree, where each type of number is contained within a larger category. Let us start with the simplest numbers and work our way outward.

Natural Numbers are the counting numbers you learned as a child: $1, 2, 3, 4, 5, …$ These are the numbers you use when you count objects. Natural numbers do not include zero, fractions, or negatives.

Whole Numbers are natural numbers plus zero: $0, 1, 2, 3, 4, …$ Adding zero to our collection might seem like a small step, but zero is essential - it represents “nothing” or an empty quantity.

Integers expand our system further by including negative whole numbers: $…, -3, -2, -1, 0, 1, 2, 3, …$ Integers allow us to represent opposites, deficits, and values below a reference point.

Rational Numbers are any numbers that can be expressed as a fraction $\frac{a}{b}$ where $a$ and $b$ are integers and $b \neq 0$. This includes:

• All integers (since $5 = \frac{5}{1}$)
• Terminating decimals like $0.75 = \frac{3}{4}$
• Repeating decimals like $0.333… = \frac{1}{3}$

Irrational Numbers are numbers that cannot be written as a fraction of integers. Their decimal expansions go on forever without repeating. Famous examples include:

• $\sqrt{2} \approx 1.41421356…$
• $\pi \approx 3.14159265…$
• $e \approx 2.71828182…$

Real Numbers include all rational and irrational numbers combined. Every point on the number line corresponds to a real number, and every real number corresponds to a point on the number line.

Classifying Numbers

When you encounter a number, you can classify it by determining which categories it belongs to. Here is the key insight: smaller sets are always contained within larger sets.

$$\text{Natural} \subset \text{Whole} \subset \text{Integers} \subset \text{Rational} \subset \text{Real}$$

This means every natural number is also a whole number, every whole number is also an integer, and so on. However, irrational numbers sit separately within the real numbers - they are real but not rational.

For example, the number $7$ is:

• A natural number (yes, it is a counting number)
• A whole number (yes, it is a natural number or zero)
• An integer (yes, it is a whole number or its negative)
• A rational number (yes, $7 = \frac{7}{1}$)
• A real number (yes, it can be placed on the number line)

But $7$ is NOT an irrational number.

The Number Line and Absolute Value

The number line provides a visual representation of all real numbers. Zero sits at the center, positive numbers extend to the right, and negative numbers extend to the left.

$$\leftarrow ;-4 \quad -3 \quad -2 \quad -1 \quad 0 \quad 1 \quad 2 \quad 3 \quad 4; \rightarrow$$

Every real number has a specific location on this line. Rational numbers like $\frac{1}{2}$ sit exactly halfway between $0$ and $1$. Irrational numbers like $\sqrt{2}$ also have precise locations, even though we cannot write their exact decimal values.

Absolute value measures the distance from a number to zero on the number line. Since distance is always positive (or zero), absolute value is always non-negative.

We write absolute value using vertical bars: $|x|$

• $|7| = 7$ (seven is 7 units from zero)
• $|-7| = 7$ (negative seven is also 7 units from zero)
• $|0| = 0$ (zero is 0 units from itself)

Think of absolute value as answering the question: “How far?” without caring about direction.

Properties of Real Numbers

The real numbers follow predictable patterns that we call properties. These are not rules you need to memorize blindly - they describe how numbers naturally behave. Understanding these properties will help you simplify expressions and solve equations.

Commutative Properties

The commutative property says that order does not matter for addition and multiplication.

Commutative Property of Addition: $a + b = b + a$

Example: $3 + 5 = 5 + 3 = 8$

Commutative Property of Multiplication: $a \times b = b \times a$

Example: $4 \times 7 = 7 \times 4 = 28$

Important: Subtraction and division are NOT commutative. $5 - 3 \neq 3 - 5$ and $12 \div 4 \neq 4 \div 12$.

Associative Properties

The associative property says that grouping does not matter for addition and multiplication.

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

Example: $(2 + 3) + 4 = 2 + (3 + 4) = 9$

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

Example: $(2 \times 3) \times 4 = 2 \times (3 \times 4) = 24$

This property lets you regroup terms to make calculations easier.

Distributive Property

The distributive property connects multiplication and addition. It says that multiplying a sum is the same as multiplying each addend separately and then adding.

$$a(b + c) = ab + ac$$

This property also works with subtraction:

$$a(b - c) = ab - ac$$

Example: $5(2 + 3) = 5 \times 2 + 5 \times 3 = 10 + 15 = 25$

The distributive property is one of the most frequently used tools in algebra. It allows you to expand expressions, factor, and simplify.

Identity Properties

An identity is a number that leaves other numbers unchanged under a specific operation.

Additive Identity: The number $0$ is the additive identity because adding zero to any number gives you back that number.

$$a + 0 = a$$

Example: $15 + 0 = 15$

Multiplicative Identity: The number $1$ is the multiplicative identity because multiplying any number by one gives you back that number.

$$a \times 1 = a$$

Example: $15 \times 1 = 15$

Inverse Properties

An inverse is a number that, when combined with another number under a specific operation, produces the identity.

Additive Inverse: For any number $a$, its additive inverse is $-a$. Adding a number to its additive inverse gives zero.

$$a + (-a) = 0$$

Example: $8 + (-8) = 0$

Multiplicative Inverse: For any nonzero number $a$, its multiplicative inverse (or reciprocal) is $\frac{1}{a}$. Multiplying a number by its multiplicative inverse gives one.

$$a \times \frac{1}{a} = 1 \quad (a \neq 0)$$

Example: $5 \times \frac{1}{5} = 1$

Note: Zero has no multiplicative inverse because division by zero is undefined.

Notation and Terminology

Term	Meaning	Example
Natural numbers	Counting numbers	1, 2, 3, 4, …
Whole numbers	Natural numbers plus zero	0, 1, 2, 3, …
Integers	Whole numbers and negatives	…, -2, -1, 0, 1, 2, …
Rational numbers	Can be written as a fraction	$\frac{3}{4}$, $-2$, $0.5$
Irrational numbers	Cannot be written as a fraction	$\sqrt{2}$, $\pi$
Absolute value	Distance from zero	$|-5| = 5$
Commutative	Order does not matter	$a + b = b + a$
Associative	Grouping does not matter	$(a + b) + c = a + (b + c)$
Distributive	Multiply across addition	$a(b + c) = ab + ac$
Identity	Number that leaves values unchanged	$0$ for addition, $1$ for multiplication
Inverse	Number that produces the identity	$-a$ for addition, $\frac{1}{a}$ for multiplication

Examples

Key Properties and Rules

Properties of Addition and Multiplication

Property	Addition	Multiplication
Commutative	$a + b = b + a$	$a \times b = b \times a$
Associative	$(a + b) + c = a + (b + c)$	$(a \times b) \times c = a \times (b \times c)$
Identity	$a + 0 = a$	$a \times 1 = a$
Inverse	$a + (-a) = 0$	$a \times \frac{1}{a} = 1$ (when $a \neq 0$)

The Distributive Property

$$a(b + c) = ab + ac$$ $$a(b - c) = ab - ac$$ $$(b + c)a = ba + ca$$

The Real Number Hierarchy

$$\text{Natural Numbers} \subset \text{Whole Numbers} \subset \text{Integers} \subset \text{Rational Numbers} \subset \text{Real Numbers}$$

$$\text{Irrational Numbers} \subset \text{Real Numbers}$$

$$\text{Rational Numbers} \cap \text{Irrational Numbers} = \emptyset$$

(Rational and irrational numbers do not overlap - a number is one or the other, never both.)

Absolute Value Rules

• $|a| \geq 0$ for all real numbers $a$
• $|a| = |-a|$
• $|a| = 0$ if and only if $a = 0$

Real-World Applications

Temperature Changes

Temperature is a natural context for understanding positive and negative numbers and their properties.

Situation: The temperature in the morning was $-8°F$. By afternoon, it rose by $15$ degrees. What was the afternoon temperature?

Using the additive identity and our knowledge of integers: $$-8 + 15 = 7°F$$

The afternoon temperature was $7°F$.

Financial Transactions

Banking provides clear examples of how real numbers and their properties work in everyday life.

Situation: You have $250 in your account. You write checks for $75, $40, and $35. What is your balance?

We can use the associative property to group these transactions: $$250 - (75 + 40 + 35) = 250 - 150 = 100$$

Or calculate step by step: $$250 - 75 - 40 - 35 = 175 - 40 - 35 = 135 - 35 = 100$$

Either way, your balance is $100.

Elevation Above and Below Sea Level

Elevation naturally uses positive numbers (above sea level) and negative numbers (below sea level).

Situation: A hiker starts at an elevation of $-50$ feet (in Death Valley, which is below sea level). She climbs $1200$ feet. What is her final elevation?

$$-50 + 1200 = 1150 \text{ feet above sea level}$$

The absolute value interpretation: How far did she travel vertically? The absolute value of her change in elevation is $|1200| = 1200$ feet.

Using Properties to Solve Real Problems

Situation: A store is having a 20% off sale. You buy 3 items priced at $15, $25, and $10. Calculate your total after the discount.

Using the distributive property: $$0.80(15 + 25 + 10) = 0.80 \times 50 = 40$$

Or distribute first: $$0.80(15) + 0.80(25) + 0.80(10) = 12 + 20 + 8 = 40$$

Both approaches give the same answer: $40.

Self-Test Problems

Summary

Here is what you need to remember about real numbers and their properties:

• The real number system is organized into categories: natural numbers, whole numbers, integers, rational numbers, irrational numbers, and real numbers. Each smaller category is contained within the larger ones.

• Rational numbers can be written as fractions (including integers and terminating or repeating decimals). Irrational numbers cannot be written as fractions (their decimals never terminate or repeat).

• Absolute value $|a|$ measures distance from zero and is always non-negative.

• Commutative properties tell us that order does not matter for addition and multiplication: $a + b = b + a$ and $ab = ba$.

• Associative properties tell us that grouping does not matter for addition and multiplication: $(a + b) + c = a + (b + c)$ and $(ab)c = a(bc)$.

• The distributive property connects multiplication with addition: $a(b + c) = ab + ac$. This is essential for expanding and simplifying expressions.

• Identity properties: Adding zero or multiplying by one leaves a number unchanged.

• Inverse properties: Adding a number to its opposite gives zero; multiplying a number by its reciprocal gives one.

• These properties are not arbitrary rules - they describe how numbers naturally behave. Understanding them gives you powerful tools for simplifying expressions, solving equations, and working efficiently with algebra.

The real number system and its properties form the foundation of algebra. Every technique you learn going forward will rely on these fundamental ideas. When you get stuck on a problem, come back to these properties - they are often the key to finding a solution.