Number Sets and Classification

Types of Numbers

Understanding the classification of numbers is foundational in precalculus. Numbers are grouped into several sets based on their properties.

• Natural Numbers: Counting numbers starting from 1 (e.g., 1, 2, 3, ...).

• Whole Numbers: Natural numbers plus zero (e.g., 0, 1, 2, 3, ...).

• Integers: Whole numbers and their negatives (e.g., ..., -3, -2, -1, 0, 1, 2, 3, ...).

• Rational Numbers: Numbers that can be written as a fraction of two integers, where the denominator is not zero (e.g., 1/2, -3, 0.75).

• Irrational Numbers: Numbers that cannot be written as a fraction of two integers (e.g., , ).

• Real Numbers: All rational and irrational numbers.

Example: is a rational number, is a rational number (since ), and is irrational.

Basic Arithmetic Operations

Addition, Subtraction, Multiplication, and Division

Mastery of arithmetic operations with integers, fractions, and decimals is essential for algebraic manipulation.

• Addition/Subtraction: Combine like terms or values. Pay attention to signs.

• Multiplication: Multiply numbers directly; for negatives, remember that a negative times a negative is positive.

• Division: Divide as usual; dividing by a negative changes the sign.

Example: ; ; .

Order of Operations

PEMDAS/BODMAS

When evaluating expressions, follow the order of operations:

• Parentheses first

• Exponents (powers and roots, etc.)

• Multiplication and Division (left to right)

• Addition and Subtraction (left to right)

Example:

Properties of Numbers

Opposites, Reciprocals, and Absolute Value

• Opposite (Additive Inverse): The number that, when added to the original, yields zero. For , the opposite is .

• Reciprocal (Multiplicative Inverse): The number that, when multiplied by the original, yields one. For , the reciprocal is .

• Absolute Value: The distance from zero on the number line, always non-negative. .

Example: The opposite of 12-12-\frac{1}{12}$.

Exponents and Powers

Exponent Rules

Exponents are used to represent repeated multiplication. Key rules include:

• Product Rule:

• Quotient Rule:

• Power Rule:

• Negative Exponent:

• Zero Exponent: (for )

Example: ; .

Radicals and Roots

Simplifying Radicals

Radicals represent roots, such as square roots or cube roots. Simplifying involves factoring out perfect squares or cubes.

• Square Root: is a number which, when squared, gives .

• Product Rule for Radicals:

• Quotient Rule for Radicals:

Example:

Polynomials

Operations with Polynomials

Polynomials are algebraic expressions with terms consisting of variables raised to whole number powers and their coefficients.

• Addition/Subtraction: Combine like terms.

• Multiplication: Use distributive property or FOIL for binomials.

• Special Products: ; ;

Example:

Factoring Polynomials

Factoring is expressing a polynomial as a product of its factors.

• Greatest Common Factor (GCF): Factor out the largest common factor from all terms.

• Factoring Trinomials: can often be factored into .

• Difference of Squares:

Example:

Solving Equations

Linear and Quadratic Equations

Solving equations involves finding the value(s) of the variable that make the equation true.

• Linear Equations: ; solve by isolating .

• Quadratic Equations: ; solve by factoring, completing the square, or using the quadratic formula:

Example: Solve by factoring:

Fractions and Rational Expressions

Operations with Fractions

• Add/Subtract: Find a common denominator.

• Multiply: Multiply numerators and denominators.

• Divide: Multiply by the reciprocal of the divisor.

Example:

Summary Table: Number Set Classification

Additional info: This guide covers foundational topics from Ch. 1 (Graphs, Number Sets), Ch. 2 (Functions and Their Graphs, as related to polynomials), Ch. 3 (Linear and Quadratic Functions), and Ch. 4 (Polynomial and Rational Functions) of a typical Precalculus curriculum.