Real Numbers and Their Classifications

Sets and Number Types

In mathematics, numbers are organized into sets based on their properties. Understanding these classifications is foundational for precalculus.

• Set: A collection of objects or numbers.

• Natural Numbers: Counting numbers: 1, 2, 3, ...

• Whole Numbers: Natural numbers plus zero: 0, 1, 2, 3, ...

• Integers: Whole numbers and their negatives: ..., -3, -2, -1, 0, 1, 2, 3, ...

• Rational Numbers: Any number that can be written as a fraction of two integers, where .

  • Includes terminating decimals (e.g., 0.6, 0.425978)

  • Includes repeating decimals (e.g., 0.555... = )

• Irrational Numbers: Numbers that cannot be written as fractions. Their decimal expansions are non-terminating and non-repeating.

  • Examples: , ,

  • Often called "messy" decimals

• Real Numbers: The set of all rational and irrational numbers.

Example: is irrational, is rational, -5 is an integer.

Classification Diagram

The following table describes the relationships among the main number sets:

Inequalities and Interval Notation

Understanding Inequalities

Inequalities compare the relative size of two numbers. They are fundamental in expressing ranges and domains in precalculus.

• : is less than

• : is greater than

• : is less than or equal to

• : is greater than or equal to

• : is positive or zero

• : is negative

• : is non-negative

Example: If , is any number greater than 2.

Interval Notation and Number Lines

Intervals describe sets of numbers between endpoints. They are often represented on number lines and with interval notation.

• Open Interval: excludes endpoints and .

• Closed Interval: includes endpoints and .

• Half-Open Interval: or includes one endpoint.

• Infinity: is used to indicate unbounded intervals, always with an open bracket: or .

Example: is written as .

Set Operations: Union and Intersection

Sets can be combined or compared using union and intersection.

• Union (): Combines all elements from both sets. Example:

• Intersection (): Includes only elements common to both sets. Example:

Example: and is .

Algebraic Expressions and Exponents

Structure of Algebraic Expressions

Algebraic expressions consist of terms formed by variables and constants combined using operations.

• Terms: Parts of an expression separated by addition or subtraction.

• Factors: Quantities multiplied together within a term.

• Variables: Symbols (e.g., , ) that stand in for numbers.

Example: has three terms.

Properties of Operations

• Distributive Property:

• Zero Product Property: If , then or

Exponent Rules

Exponents indicate repeated multiplication. Several rules govern their manipulation:

• Product of Powers:

• Power of a Power:

• Power of a Product:

• Quotient of Powers:

Example:

Simplifying Expressions with Exponents

To simplify expressions with exponents, apply the exponent rules and combine like terms.

• Example: Simplify

Step-by-step solution:

• Expand numerator:

• Expand denominator:

• Combine:

Interval Lengths

Finding the Length of an Interval

The length of an interval is the absolute value of the difference between its endpoints.

• Formula:

• Example: The length of is

Summary Table: Key Properties

Additional info: These notes provide foundational concepts for real numbers, inequalities, interval notation, set operations, algebraic expressions, and exponent rules, all of which are essential for success in precalculus.