Overview
This lecture introduces foundational algebra concepts, covering variables, expressions, equations, exponents, inequalities, logarithms, sigma notation, and basic area approximation under curves.
Variables and Expressions
- A variable (like x) stores an unknown value on the number line, which includes both positive and negative numbers.
- An algebraic expression combines variables and numbers, often with operations like addition or multiplication.
- The coefficient is the number multiplying the variable; for example, in 2x, 2 is the coefficient.
Exponents and Powers
- x² means x multiplied by itself (x*x), visually related to the area of a square with side x.
- The exponent indicates repeated multiplication of the base.
- When multiplying powers with the same base, add the exponents; when dividing, subtract them.
Order of Operations
- Operations follow this order: Brackets, Exponents, Multiplication/Division, Addition/Subtraction.
- In expressions with brackets, simplify inside before expanding, unless unknown variables prevent this.
Expanding and Simplifying Expressions
- To expand (x + a)(x + b), use FOIL: multiply First, Outer, Inner, and Last terms, then simplify.
- Simplification reduces expressions to their smallest, most efficient form.
Linear Equations and Graphs
- The equation y = mx + c is a straight line; m is the slope, c is the y-intercept.
- Slope (m) shows how steep the line is; y-intercept (c) is where the line crosses the y-axis.
Inequalities
- Inequalities use symbols: < (less than), > (greater than), ≤ (less than or equal), ≥ (greater than or equal).
- When multiplying or dividing both sides by a negative, flip the inequality sign.
Simultaneous Equations
- Simultaneous equations involve solving for variables in more than one equation at once.
- Use elimination (manipulate equations to cancel a variable) or substitution (replace a variable with its expression from another equation).
Logarithms and Natural Logs
- A logarithm (log) finds the exponent that a base must be raised to get a given number: log₂8 = 3 because 2³ = 8.
- Natural log (ln) uses the base e (≈2.718); ln(x) = y means e^y = x.
- Log rules: log(x*y) = log(x) + log(y); log(x/y) = log(x) - log(y); power rule allows moving exponents in/out of log.
Sigma Notation & Summation
- Sigma (Σ) notation represents summing a sequence: Σ from i = b to a adds each term in the sequence.
- The sum of the first n natural numbers: n(n+1)/2.
Area Under a Curve (Approximations)
- The area under a curve between two x-values can be estimated by dividing into many intervals and summing areas of rectangles.
- The more intervals, the more accurate the approximation.
Key Terms & Definitions
- Variable — Symbol representing an unknown value.
- Coefficient — Number multiplied by a variable.
- Exponent — Number indicating repeated multiplication of the base.
- Y-intercept — Where a graph crosses the y-axis.
- Slope (m) — The steepness of a line.
- Inequality — A statement comparing two values using <, >, ≤, or ≥.
- Logarithm — Operation that finds the exponent for a base to reach a number.
- Natural log (ln) — Logarithm with base e.
- Sigma (Σ) notation — Symbol for summing a sequence of numbers.
Action Items / Next Steps
- Practice expanding and simplifying expressions using FOIL.
- Solve example simultaneous equations using elimination and substitution.
- Apply sigma notation to sum sequences.
- Review order of operations and exponent rules.
- Try calculating a simple area under a curve using interval approximation.