Algebra Topics Overview

Types of Numbers

• Natural Numbers: Whole numbers greater than zero (e.g., 1, 2, 3).
• Whole Numbers: Natural numbers including zero (e.g., 0, 1, 2).
• Integers: Whole numbers that can be negative.
• Rational Numbers: Numbers that can be expressed as a fraction (e.g., (\frac{2}{3})).
• Irrational Numbers: Numbers that cannot be expressed as a simple fraction (e.g., (\sqrt{5})).
• Real Numbers: All numbers on the number line, including rational and irrational numbers.
• Imaginary Numbers: Numbers involving (i), where (i = \sqrt{-1}).

Basic Operations

• Addition/Subtraction: Use number line for visualization.
  • Adding moves to the right.
  • Subtracting moves to the left.
• Multiplication/Division:
  • Multiplying large numbers by breaking down into smaller parts.
  • Division often involves long division.

Fractions

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

Converting Numbers

• Improper Fractions to Mixed Numbers: Divide numerator by denominator.
• Mixed Numbers to Improper Fractions: Multiply whole number by denominator, add numerator.
• Fractions to Decimals: Use long division.
• Decimals to Fractions: Multiply top and bottom by power of 10.
• Percentages to Decimals/Fractions: Divide by 100.

Variables and Exponents

• Multiplying Variables: Add exponents.
• Dividing Variables: Subtract exponents.
• Raising Powers: Multiply exponents.
• Negative Exponents: Inverse the base.

Radical Expressions

• Converting Exponential to Radical:
  • (x^{\frac{m}{n}} = \sqrt[n]{x^m})
• Simplifying Radicals: Divide exponent by index.

Solving Equations

• Linear Equations: Isolate the variable.
• Quadratic Equations: Factor or use quadratic formula.
• Radical Equations: Square both sides to eliminate radicals.
• Exponential Equations: Convert bases to the same and equate exponents.

Functions

• Evaluating Functions: Substitute given value into the equation.
• Composite Functions: Evaluate inner function first, then substitute into outer function.

Logs and Logarithmic Functions

• Basic Properties:
  • (\log_a(b) = c \rightarrow a^c = b)
  • (\ln(a) + \ln(b) = \ln(ab))
  • (\ln(a) - \ln(b) = \ln\left(\frac{a}{b}\right))
• Solving Logarithmic Equations: Convert log form to exponential form.
• Expanding/Condensing Logs: Use properties to expand or condense expressions.

Problem Solving

• Rational Equations: Use common denominators or multiply through to eliminate fractions.
• Quadratic Substitution: Replace expressions to simplify quadratic equations.
• Natural Logarithms: Use natural logs for equations involving (e).

This overview summarizes key algebraic concepts, including types of numbers, basic operations, fractions, conversions, exponents, radicals, solving a variety of equations, and working with functions and logarithms. This should serve as a comprehensive study aid for reviewing foundational and advanced algebra topics.