Overview
This lecture introduces foundational algebra concepts including variables, expressions, equations, exponents, logarithms, inequalities, simultaneous equations, Sigma notation for summation, and an introduction to finding area under curves.
The Number Line & Variables
- The number line extends infinitely in both positive and negative directions.
- A variable (e.g., x) stores unknown values in algebraic expressions.
Algebraic Expressions & Simplification
- Algebraic expressions often use variables and coefficients (e.g., 2x means x multiplied by 2).
- When squaring a variable (x²), imagine the area of a square with side x.
- The exponent indicates how many times the base is multiplied by itself.
- Simplification makes expressions easier to solve by combining like terms.
Linear Equations
- The general form is y = mx + c, where m is the slope (steepness) and c is the y-intercept.
- For each x value, y is found by applying the slope and adding the intercept.
Order of Operations (PEMDAS/BEDMAS)
- Brackets first, then exponents, then multiplication/division, then addition/subtraction.
- Expand brackets by multiplying every term outside by every term inside (e.g., with FOIL: First, Outer, Inner, Last).
Exponents & Powers
- When multiplying with the same base, add exponents; when dividing, subtract exponents.
- If coefficients and bases on both sides are equal, exponents must also be equal.
Inequalities
- Inequality signs: < (less than), > (greater than), ≤ (less than or equal to), ≥ (greater than or equal to).
- When multiplying or dividing both sides by a negative, flip the inequality sign.
Simultaneous Equations
- Solve systems with more than one variable using elimination or substitution.
- Align coefficients to eliminate variables, then solve sequentially for each variable.
Logarithms
- Logarithms are the inverse of exponents; logâ‚‚8 = x means 2Ë£ = 8, so x = 3.
- log(x) = y implies 10ʸ = x; ln(x) = y implies eʸ = x, where e ≈ 2.718.
- Power rule: log(a^n) = n·log(a).
- Product property: log(xy) = log(x) + log(y); log(x/y) = log(x) - log(y).
Sigma Notation & Summation
- Sigma (Σ) notation represents summing a sequence of numbers from a lower to upper bound.
- Adjusting the expression inside Sigma changes the sequence being summed.
- Useful formula: sum of first n natural numbers is n(n+1)/2.
Area Under a Curve (Introduction)
- Divide the interval into n sub-intervals for better approximation.
- Use midpoints to estimate area under a curve with summation.
- Approximating the area under curves leads to definite integrals.
Key Terms & Definitions
- Variable — a symbol for an unknown value.
- Coefficient — number multiplied by a variable.
- Exponent — shows how many times a base is multiplied by itself.
- Slope (m) — steepness of a line in y = mx + c.
- Y-intercept (c) — where the line crosses the y-axis.
- FOIL — method for expanding (First, Outer, Inner, Last) products of binomials.
- Logarithm — inverse function to exponentiation.
- Sigma Notation (Σ) — shorthand for summing sequences.
- Interval — section of the number line between two points, possibly inclusive.
Action Items / Next Steps
- Practice simplifying algebraic expressions and expanding brackets.
- Solve sample simultaneous equations using elimination and substitution.
- Try problems involving exponents and logarithms.
- Use Sigma notation to sum sequences.
- Read about definite integrals for further study on area under curves.