Algebra II: Real Numbers and Their Properties

Classification and Subsets of Real Numbers

Real numbers encompass several distinct subcategories, each defined by specific mathematical properties:

• Natural Numbers: Positive whole numbers.
  • Examples: $1$, $2$, $3$, $4$
• Whole Numbers: All natural numbers together with the number $0$.
  • Examples: $0$, $1$, $2$, $3$, $4$
• Integers: All whole numbers and their negative whole number counterparts.
  • Examples: $-2$, $-1$, $0$, $1$, $2$
• Rational Numbers: Any number that can be expressed as a fraction or ratio of integers.
  • Examples: $0.3$, $-\frac{5}{4}$, $-1.01254$
• Irrational Numbers: Numbers that cannot be written as a fraction of integers.

Properties of Real Numbers

Let $a$, $b$, and $c$ represent real numbers. The algebraic properties governing addition and multiplication are structured as follows:

• Closure Property:

  • Addition: $a + b$ is a real number.
  • Multiplication: $a b$ is a real number.

• Commutative Property:

  • Addition: $a + b = b + a$
  • Multiplication: $a b = b a$

• Associative Property:

  • Addition: $(a + b) + c = a + (b + c)$
  • Multiplication: $(a b) c = a (b c)$

• Identity Property:

  • Addition: $a + 0 = a$ and $0 + a = a$
  • Multiplication: $a \times 1 = a$ and $1 \times a = a$

• Inverse Property:

  • Addition: $a + (-a) = 0$
  • Multiplication: $a \times \frac{1}{a} = 1$ (where $a \neq 0$)

• Distributive Property:

  • $a (b + c) = a b + a c$

Additive and Multiplicative Inverses

• Opposite (Additive Inverse):

  • Definition: The sum of a number and its opposite is $0$
  • For $8$, the opposite is $-8$
  • For $400$, the opposite is $-400$
  • For $-\frac{4}{9}$, the opposite is $\frac{4}{9}$

• Reciprocal (Multiplicative Inverse):

  • Definition: The product of a number and its reciprocal is $1$
  • For $-4$, the reciprocal is $-\frac{1}{4}$
  • For $400$, the reciprocal is $\frac{1}{400}$
  • For $-\frac{4}{9}$, the reciprocal is $-\frac{9}{4}$

Absolute Value

• Definition: The distance that a number is from zero on a number line.
• Key Property: Distance can never be a negative number.
• Examples:
  • $|-4| = 4$
  • $|8| = 8$

Number Line Graphing and Assigned Practice

• Graphing on Number Line:

  • Specific numbers to graph: $4.3$, $3.7$, \root{}\frac{8}{} / $\text{sqrt}(8)$, specifically written as $\text{sqrt}(8)$ or \root\frac{8}{}.

• Reference Notes & Problem Sets:

  • Properties List: Copy the list of properties from Page 87.
  • Practice Problems: Page 90 #16 - 54 Left Hand Column.