Algebra Lecture Notes

Introduction

• Topics covered: solving equations, linear quadratic equations, factoring, exponents, graphing equations.
• Focus on numbers and their classifications.

Types of Numbers

• Natural Numbers: Whole numbers greater than 0 (e.g., 1, 2, 3).
• Whole Numbers: Natural numbers including 0.
• Integers: Whole numbers that can be negative.
• Rational Numbers: Numbers that can be expressed as a fraction of integers (e.g., 2/3, 4).
• Irrational Numbers: Cannot be expressed as a fraction (e.g., √5).
• Imaginary Numbers: Involve 'i' (e.g., i = √-1).

Basic Operations

• Addition & Subtraction: Use a number line for visualization.
• Multiplication & Division: Breaking into smaller parts helps manage larger numbers.

Fractions

• Addition/Subtraction: Find common denominators.
• Multiplication: Multiply across numerators and denominators.
• Division: Use "keep, change, flip".

Converting Numbers

• Improper Fractions to Mixed Numbers: Divide and express remainder.
• Fractions to Decimals: Use long division.
• Decimals to Fractions: Multiply to remove decimal places.

Percentages

• Convert percentages to decimals by dividing by 100.
• Calculate percentage of a number by multiplying.

Variables & Exponents

• Basic Rules:
  • Multiply variables: add exponents.
  • Divide variables: subtract exponents.
  • Negative exponents: reciprocal.
• Exponent Operations:
  • Raise to a power: multiply exponents.

Solving Equations

• Linear Equations: Isolate the variable.
• Quadratic Equations: Factor or use the quadratic formula.
• Exponential Equations: Use logarithms when no common base.

Functions

• Evaluate functions by substituting values.
• Composite functions: evaluate inner function first.

Logarithms

• Properties:
  • ln(a) + ln(b) = ln(ab).
  • ln(a) - ln(b) = ln(a/b).
  • ln(a^b) = b*ln(a).
• Solving logarithmic equations by converting to exponential form.*

Practice

• Solve for x in various scenarios such as equations involving fractions or exponents.
• Use factoring, quadratic formula, or substitution as needed.

Additional Topics (Brief Overview)

• Solving inequalities, absolute values, system of equations.
• Graphing equations and understanding the properties of graphs.
• More on functions: domain, range, and different types like linear, quadratic, and cubic.

Summary

This lecture covered fundamental algebraic concepts necessary for understanding and solving equations. By mastering these concepts, students can solve a wide range of problems related to algebra.