Overview
This lecture covered key algebra concepts including number types, arithmetic operations, fractions, exponents, equations, factoring, functions, logarithms, and solving various algebraic equations.
Types of Numbers
- Natural numbers are positive whole numbers greater than zero (1, 2, 3, ...).
- Whole numbers include zero and natural numbers.
- Integers are whole numbers, including negatives.
- Rational numbers can be written as fractions using integers.
- Irrational numbers cannot be written as simple fractions; their decimals do not terminate or repeat.
- Imaginary numbers involve the square root of negative numbers; i = √(-1).
- The root of a negative under an even index is imaginary, under odd index is real.
Basic Operations: Addition, Subtraction, Multiplication, Division
- Use the number line: move right to add, left to subtract.
- Carry and borrow when adding or subtracting larger numbers.
- Multiply using standard algorithm, carry over as needed.
- Long division: divide, multiply, subtract, bring down next digit, repeat.
Working with Fractions
- To add/subtract, find common denominators.
- Multiply fractions by multiplying numerators and denominators.
- Divide fractions by keeping the first, changing division to multiplication, and flipping the second.
- Simplify complex fractions by multiplying numerator and denominator by the least common denominator.
- Convert improper fractions to mixed numbers and vice versa.
- Use long division to convert fractions to decimals.
Percentages
- Convert percent to decimal by dividing by 100.
- To find a percentage of a value, multiply its decimal form by the value.
Exponents and Radicals
- When multiplying like bases, add exponents; when dividing, subtract exponents.
- Power to a power, multiply exponents.
- Negative exponents mean reciprocal.
- Fractional exponents: numerator is power, denominator is root.
- Simplify radicals by dividing exponents by the index.
Solving Equations
- Isolate the variable by using inverse operations.
- With variables on both sides, combine like terms before isolating.
- For fractions in equations, clear denominators by multiplying through.
- For exponents, use roots or reciprocal exponents as needed.
- Quadratic equations can be solved by factoring or quadratic formula.
Factoring Techniques
- Factor out the greatest common factor (GCF) first.
- Difference of squares: a² - b² = (a-b)(a+b).
- For trinomials ax² + bx + c: find two numbers multiplying to ac and adding to b.
Functions
- f(x) replaces y; substitute input values to evaluate.
- Composite functions: evaluate inner function first, then outer.
- Functions can be written in terms of variables and evaluated at numbers or expressions.
Logarithms
- logₐb = c ↔ aᶜ = b.
- log rules: logₐ(bc) = logₐb + logₐc; logₐ(b/c) = logₐb - logₐc; logₐ(bⁿ) = n logₐb.
- To solve log equations, rewrite in exponential form.
- Natural log (ln) is log base e.
- Use logs to solve exponential equations when bases can't be matched.
Key Terms & Definitions
- Natural number — positive integer > 0.
- Whole number — integer ≥ 0.
- Integer — positive, negative, or zero whole number.
- Rational number — can be expressed as a fraction of integers.
- Irrational number — cannot be written as a simple fraction.
- Imaginary number — involves √(-1), denoted as i.
- Exponent — power indicating repeated multiplication.
- Logarithm — inverse of exponentiation.
- Quadratic equation — polynomial equation of degree 2.
- GCF (Greatest Common Factor) — largest factor shared by terms.
Action Items / Next Steps
- Practice additional factoring and quadratic formula problems.
- Review function evaluations and composite functions.
- Study log and exponent properties for solving equations.
- Complete assigned algebra homework and prepare questions for next class.