# 7.S: The Properties of Real Numbers (Summary)

### Key Terms

Additive Identity | The additive identity is 0. When zero is added to any number, it does not change the value. |

Additive Inverse | The opposite of a number is its additive inverse. The additive inverse of a is −a . |

Irrational number | A number that cannot be written as the ratio of two integers. Its decimal form does not stop and does not repeat. |

Multiplicative Identity | The multiplicative identity is 1. When one multiplies any number, it does not change the value. |

Multiplicative Inverse | The reciprocal of a number is its multiplicative inverse. The multiplicative inverse of a is \(\frac{1}{a}\). |

Rational number | A number that can be written in the form \(\frac{p}{q}\), where p and q are integers and q ≠ 0. Its decimal form stops or repeats. |

Real number | A number that is either rational or irrational. |

### Key Concepts

#### 7.1 - Rational and Irrational Numbers

- Real numbers

#### 7.2 - Commutative and Associative Properties

- Commutative Properties
**Commutative Property of Addition**: If a, b are real numbers, then a + b = b + a**Commutative Property of Multiplication**: If a, b are real numbers, then a • b = b • a

- Associative Properties
**Associative Property of Addition**: If a, b, c are real numbers then (a + b) + c = a + (b + c)**Associative Property of Multiplication**: If a, b, c are real numbers then (a • b) • c = a • (b • c)

#### 7.3 - Distributive Property

**Distributive Property**:- If a, b, c are real numbers then
- a(b + c) = ab + ac
- (b + c)a = ba + ca
- a(b • c) = ab • ac

- If a, b, c are real numbers then

#### 7.4 - Properties of Identity, Inverses, and Zero

- Identity Properties
**Identity Property of Addition**: For any real number a: a + 0 = a, 0 + a = a- 0 is the additive identity

**Identity Property of Multiplication**: For any real number a: a • 1 = a, 1 • a = a- 1 is the multiplicative identity

- Inverse Properties
**Inverse Property of Addition**: For any real number a: a + (- a) = 0- - a is the additive inverse of a

**Inverse Property of Multiplication**: For any real number a: (a ≠ 0) a • \(\frac{1}{a}\) = 1- \(\frac{1}{a}\) is the multiplicative inverse of a

- Properties of Zero
**Multiplication by Zero**: For any real number a, a • 0 = 0, 0 • a = 0- The product of any number and 0 is 0.

**Division of Zero**: For any real number a, 0 a = 0, 0 + a = 0- Zero divided by any real number, except itself, is zero.

**Division by Zero**: For any real number a, \(\frac{0}{a}\) is undefined and a ÷ 0 is undefined.- Division by zero is undefined.

### Contributors

Lynn Marecek (Santa Ana College) and MaryAnne Anthony-Smith (Formerly of Santa Ana College). This content is licensed under Creative Commons Attribution License v4.0 "Download for free at http://cnx.org/contents/fd53eae1-fa2...49835c3c@5.191."