Algebra
The language the rest of mathematics is written in: expressions and the rules for rearranging them, equations and what it means to solve one, functions and their graphs, polynomials and factoring, quadratics and the complex numbers they force into existence, exponentials and logarithms, and systems solved several unknowns at a time, and the sums of ordered lists. Nine courses and 112 lessons are available.
- Courses
- 9
- Lessons
- 112
- Level
- Beginner → Advanced
- Assumes
- arithmetic is the only prerequisite
Overview
Foundations comes first because algebra is arithmetic with the numbers withheld, and every later move is one of a small set of properties applied to an expression. A reader who cannot say why -(x - 3) is -x + 3 will make that error in every course that follows, and will not know they made it.
Equations come second, and functions third, in that order rather than the reverse. Solving is a mechanical skill built on the properties; a function is an idea, and it is far easier to meet once you can already manipulate the formula that defines one.
Factoring sits in the middle of the path because it is the bottleneck. Rational expressions, quadratics, and every polynomial graph after Polynomials and Factoring are stated in terms of factors, and none of those techniques applies to a sum. It is placed where there is still time to practise it.
Exponentials and logarithms come seventh because a logarithm is an inverse function, and inverse functions are Lines, Functions and Graphs. Systems and matrices come last because row reduction is elimination performed on exact fractions, and it is unreadable to anyone not already fluent in both.
Courses
Each course assumes the ones before it and nothing else. Polynomials and Factoring factors the expressions Foundations of Algebra taught you to build; Rational and Radical Expressions cannot start until you can factor; Quadratics and Complex Numbers needs both, and Systems and Matrices needs all of them.
Arithmetic, made general. Why a letter can stand for a number, which rearrangements are always allowed and which only look allowed, and the exponent and radical rules that everything after this course leans on.
Find the value that makes a statement true — and know why each move you made kept it true. Linear equations, literal formulas, inequalities, compound and absolute value statements, and the notation for writing a solution set down.
Put algebra on a picture. Slope and every form of a line, then the definition that organises the rest of the path — a function — with its notation, its domain, the transformations that move its graph, and its inverse.
Multiply out, then learn to go back. The factored form is the one that answers questions — where a graph crosses, when a product is zero — and this course is about producing it reliably rather than by inspection.
What happens when a variable lands in a denominator or under a root. Both put values out of bounds, both create solutions that are not solutions, and both need the domain written down before the algebra starts.
One equation shape, four solution methods, and a number system that has to be extended so that every one of them always finishes. Completing the square is the centre of the course: it derives the formula and it puts a parabola in vertex form.
The variable moves into the exponent, and ordinary algebra stops working. A logarithm is the tool that brings it back down — defined as an inverse, with three laws that are the exponent laws read backwards.
Solve for several unknowns together, then find the notation that removes the bookkeeping. A matrix is elimination with the letters deleted, which is why this course does elimination by hand first and only then writes it down as rows.
A sequence is a function whose input is a position. Arithmetic and geometric families make that idea calculable: both have closed forms for a term and a finite sum, some geometric sums have a limit, and the same indexed notation leads to Pascal's triangle and binomial expansion.
Recommended background
Arithmetic, and specifically arithmetic with fractions and negative numbers. If -3 - (-5) and 2/3 ÷ 4/9 are not automatic, do those first — every course here runs on them, and Systems and Matrices runs on them for pages at a time.
No prior algebra. Foundations of Algebra starts from what a variable is. If you have met some of this before, the labs are still worth opening: they show the step list, and a wrong step list survives a lot of correct answers.
No calculus, and none is used. Quadratics and Complex Numbers finds a maximum with the vertex of a parabola rather than a derivative, which is the honest algebraic method and not a workaround.
No programming. Nothing here asks you to write code. The labs run so that you can change a coefficient and watch what moves, which is the one thing a printed page cannot do.