Algebra
Algebra is where mathematics stops being about specific numbers and starts being about relationships between numbers. Instead of asking "what is 3 + 4?", algebra asks "if I add the same amount to both sides of an equation, is it still true?" That shift — from computing answers to reasoning about structure — is the single most important step in a mathematical education, and everything from calculus to machine learning depends on it.
Learning Objectives
By the end of this subfield, you should be able to:
- Explain what a variable is and why letters are used to stand for numbers
- Simplify algebraic expressions using the order of operations and the distributive property
- Solve linear equations and inequalities, and interpret their graphs
- Solve quadratic equations by factoring, completing the square, and the quadratic formula
- Understand functions as input–output machines, and read information from their graphs
- Apply the laws of exponents and understand logarithms as their inverse
Quick Answer
Algebra uses symbols (usually letters) to represent numbers that are unknown or that vary. An equation states that two expressions are equal, and "solving" it means finding the value(s) of the variable that make the statement true. The golden rule is balance: whatever you do to one side of an equation, you must do to the other. The two most important equation types are linear (straight-line, highest power of x is 1) and quadratic (highest power is 2, producing a parabola). Above all, algebra introduces the function — a rule pairing each input with exactly one output — which becomes the central object of study in all higher mathematics.
Topics at a Glance
| Topic | What You'll Learn | Key Concepts |
|---|---|---|
| Variables and Expressions | Using symbols and simplifying expressions | Terms, coefficients, distributive property, like terms |
| Linear Equations | Solving straight-line equations and inequalities | Balance, slope-intercept form, graphing |
| Quadratic Equations | Solving second-degree equations | Factoring, quadratic formula, the discriminant, parabolas |
| Functions | Input–output rules and their graphs | Domain, range, notation, transformations |
| Exponents and Logarithms | Repeated multiplication and its inverse | Laws of exponents, logarithms, exponential growth |
Learning Path
Key Terms
| Term | Definition | Related Concept |
|---|---|---|
| Variable | A letter representing an unknown or changing number | Expression, equation |
| Coefficient | The number multiplying a variable (the 3 in 3x) | Term, like terms |
| Equation | A statement that two expressions are equal | Solving, balance |
| Linear | Highest power of the variable is 1; graphs as a straight line | Slope, intercept |
| Quadratic | Highest power is 2; graphs as a parabola | Factoring, discriminant |
| Function | A rule giving exactly one output per input | Domain, range |
Why Algebra Matters
Every formula you will ever use — the area of a circle, compound interest, the trajectory of a rocket — is algebra. When a spreadsheet computes a column from another column, that is a function. When an engineer sizes a beam or an economist forecasts demand, they are solving equations. Algebra is not a hoop to jump through; it is the reusable machinery of quantitative thinking.
Quick Revision
- Solving = isolating the variable using inverse operations, keeping both sides balanced.
- Linear:
y = mx + b, wheremis slope andbis the y-intercept. - Quadratic formula: for
ax² + bx + c = 0,x = (-b ± √(b² - 4ac)) / (2a). - Function: each input maps to exactly one output (the vertical-line test).
- Exponents:
xᵃ · xᵇ = xᵃ⁺ᵇ, and logarithms undo exponentials.
Related Topics
Prerequisites: arithmetic, fractions, negative numbers, order of operations.
Related: Geometry (coordinate graphing), Trigonometry.
Next: Calculus — which treats functions as things you can differentiate and integrate.