7.9: The Properties of Real Numbers (Summary)

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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 $\dfrac{1}{a}$.
Rational number	A number that can be written in the form $\dfrac{p}{q}$, 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

$The image shows a large rectangle labeled “Real Numbers”. The rectangle is split in half vertically. The right half is labeled “Irrational Numbers”. The left half is labeled “Rational Numbers” and contains three concentric rectangles. The outer most rectangle is labeled “Integers”, the next rectangle is “Whole Numbers” and the inner most rectangle is “Natural Numbers”.$

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

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 • $\dfrac{1}{a}$ = 1
  - $\dfrac{1}{a}$ 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, $\frac{0}{a} = 0$, $0 \div a = 0$
  - Zero divided by any real number, except itself, is zero.
- Division by Zero: For any real number a, $\dfrac{a}{0}$ is undefined and a ÷ 0 is undefined.
  - Division by zero is undefined.

Contributors and Attributions

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."