Numbers and Their Classifications

Types of Numbers

Understanding the different types of numbers is fundamental in algebra. Numbers are classified based on their properties and uses.

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

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

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

• Rational Numbers: Numbers that can be expressed as a fraction , where and are integers and .

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

• Real Numbers: All rational and irrational numbers.

Example: is an integer, rational, and real number.

Operations with Fractions and Decimals

Simplifying Fractions

To simplify a fraction, divide both the numerator and denominator by their greatest common divisor (GCD).

• Example:

Adding, Subtracting, Multiplying, and Dividing Fractions

• Add/Subtract: Find a common denominator, then add/subtract numerators.

• Multiply: Multiply numerators and denominators directly.

• Divide: Multiply by the reciprocal of the divisor.

Example:

Order of Operations

PEMDAS/BODMAS

Follow the order: Parentheses, Exponents, Multiplication/Division, Addition/Subtraction.

• Example:

Exponents and Powers

Exponent Rules

Exponents indicate repeated multiplication. Key rules include:

• Product Rule:

• Quotient Rule:

• Power Rule:

• Zero Exponent: (for )

Example:

Polynomials and Their Classification

Types of Polynomials

Polynomials are algebraic expressions with one or more terms, each consisting of a variable raised to a non-negative integer power.

• Monomial: One term (e.g., )

• Binomial: Two terms (e.g., )

• Trinomial: Three terms (e.g., )

Degree of a Polynomial: The highest exponent of the variable in the polynomial.

Example: has degree 3.

Solving Equations and Problem Solving

Linear Equations

Linear equations are equations of the first degree, meaning the variable is not raised to any power other than one.

• General Form:

• Solving: Isolate the variable using inverse operations.

Example:

Word Problems and Algebraic Expressions

Translating verbal statements into algebraic expressions is a key skill.

• Example: "Five times the sum of a number and 4" is

Properties of Operations

Distributive Property

The distributive property allows you to multiply a sum by multiplying each addend separately and then adding the products.

• Formula:

• Example:

Evaluating and Simplifying Expressions

Combining Like Terms

Like terms have the same variable raised to the same power. Combine them by adding or subtracting their coefficients.

• Example:

Numerical Coefficient

The numerical coefficient is the number multiplying the variable in a term.

• Example: In , the coefficient is -8.

Opposites and Reciprocals

Additive Inverse (Opposite)

The additive inverse of a number is the number that, when added to the original, yields zero.

• Example: The opposite of is $8$.

Reciprocal

The reciprocal of a nonzero number is .

• Example: The reciprocal of is .

Application and Problem Solving

Translating and Solving Word Problems

Many algebra problems require translating real-world situations into mathematical equations.

• Example: "The number of graduate students is 20,000 fewer than the number of undergraduate students." If is the number of undergraduates, then graduates = .

Tables: Number Classifications

Additional info:

• Some questions involve basic graphing and representation, which are foundational for later graphing topics.

• Problems include both computational and conceptual elements, suitable for exam preparation.