# The Properties of Real Numbers

In this lesson, we are going to go over the different properties of real numbers (ℜ). Understanding the properties of real numbers will help us simplify numerical and algebraic expressions, solve equations, and more as we progress in studying algebra.

For clarity, “properties” in this context refer to the characteristics or behaviors of real numbers under the operations of addition and/or multiplication that are accepted even without proof.

In fact, the terms axioms and properties can be used interchangeably here because axioms are properties that are self-evidently true. Therefore, the statements or propositions that will be presented here don’t require any proof. In other words, the properties or axioms of real numbers are just one of many basic foundations of mathematics.

To keep it organized, I decided to divide the properties of real numbers into three (3) parts. The first one involves the addition operation. The second involves the operation of multiplication. While the third combines the operations of addition and multiplication.

## Addition Properties of Real Numbers

Suppose [latex]a[/latex], [latex]b[/latex], and [latex]c[/latex] represent real numbers.

**1) Closure Property of Addition**

- Property: [latex]a + b[/latex] is a real number
- Verbal Description: If you add two real numbers, the sum is also a real number.
- Example: [latex]3 + 9 = 12[/latex] where [latex]12[/latex] (the sum of 3 and 9) is a real number.

**2) Commutative Property of Addition**

- Property: [latex]a + b = b + a[/latex]
- Verbal Description: If you add two real numbers in any order, the sum will always be the same or equal.
- Example: [latex]5 + 2 = 2 + 5 =10[/latex]

**3) Associative Property of Addition**

- Property: [latex](a + b) + c = a + (b + c)[/latex]
- Verbal Description: If you are adding three real numbers, the sum is always the same regardless of their grouping.
- Example: [latex](1 + 2) + 3 = 1 + (2 + 3) = 6[/latex]

**4) Additive Identity Property of Addition**

- Property: [latex]a + 0 = a[/latex]
- Verbal Description: If you add a real number to zero, the sum will be the original number itself.
- Example: [latex]3+0=3[/latex] or [latex]0+3=3[/latex]

**5) Additive Inverse Property**

- Property: [latex]a + (– a) = 0[/latex]
- Verbal Description: If you add a real number and its opposite, you will always get zero.
- Example: [latex]13+(-13)=0[/latex]

## Multiplication Properties of Real Numbers

Suppose [latex]a[/latex], [latex]b[/latex], and [latex]c[/latex] represent real numbers.

**6) Closure Property of Multiplication**

- Property: [latex]a × b[/latex] is a real number.
- Verbal Description: If you multiply two real numbers, the product is also a real number.
- Example: [latex]6 × 7 = 42[/latex] where [latex]42[/latex] (the product of 6 and 7) is a real number.

**7) Commutative Property of Multiplication**

- Property: [latex]a × b = b × a[/latex]
- Verbal Description: If you multiply two real numbers in any order, the product will always be the same or equal.
- Example: [latex]9 × 4 = 4 × 9 = 36[/latex]

**8) Associative Property of Multiplication**

- Property: [latex](a × b) × c = a × (b × c)[/latex]
- Verbal Description: If you are multiplying three real numbers, the product is always the same regardless of their grouping.
- Example: [latex](5 × 3) × 2 = 5 × (3 × 2) = 30[/latex]

**9) Multiplicative Identity Property of Multiplication**

- Property: [latex]a × 1 = a[/latex]
- Verbal Description: If you multiply a real number by one (1), you will get the original number itself.
- Example: [latex]25 × 1 = 25[/latex] or [latex]1 × 25 = 25[/latex]

**10) Multiplicative Inverse Property**

- Property: [latex]a × (1/a) = 1[/latex] but [latex]a \ne 0[/latex]
- Verbal Description: If you multiply a nonzero real number by its inverse or reciprocal, the product will always be one (1).
- Example: [latex]2×(1/2)=1[/latex]

## The Property of Multiplication together with Addition

**11) Distributive Property of Multiplication over Addition**

Suppose [latex]a[/latex], [latex]b[/latex], and [latex]c[/latex] represent real numbers.

- Property: [latex]a(b + c) = ab + ac[/latex]
or[latex](a+b)c = ac + bc[/latex] - Verbal Description: The operation of multiplication distributes over the addition operation.
- Example: [latex]4 (5 + 8) = 4 × 5 + 4 × 8[/latex] or [latex](5 + 8) 4 = 5 × 4 + 8 × 4[/latex]

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