Reference and Synthesis
Foundation Mathematics: the Study Pathway
How the collection fits together, which order to work through it in, and where each stream leads.
What this page covers
- See the structure of the whole collection
- Choose a route suited to a particular goal
- Understand which topics depend on which
- Identify the load-bearing ideas
What the collection covers
Twelve weeks of foundation mathematics, from the number system through to matrix inverses. The material is what calculus and engineering mathematics assume but rarely teach.
| Stream | Covers | Depends on |
|---|---|---|
| Number and Algebraic Foundations | Number sets, field laws, fractions | Nothing |
| Polynomials and Factoring | Degree, products, factoring, division, the factor theorem | Field laws |
| Indices, Radicals and Rational Expressions | Index laws, roots, rationalising, complex numbers | Field laws, fractions |
| Equations | Linear, quadratic, cubic, literal, radical | Factoring, indices |
| Inequalities and Absolute Value | Intervals, sign diagrams, absolute value | Equations |
| Analytic Geometry | Distance, midpoint, symmetry, the circle | Pythagoras, equations |
| Functions | Domain, range, notation, parity | Equations, inequalities |
| Linear Functions | Slope, four forms, parallel and perpendicular | Functions, analytic geometry |
| Quadratic and Polynomial Functions | Parabolas, vertices, polynomial graphs | Completing the square, functions |
| Trigonometric Functions | Unit circle, radians, exact values, graphs | The circle, functions |
| Trigonometric Identities and Triangles | Identities, addition formulae, sine and cosine rules | Trigonometric functions |
| Vectors | Components, magnitude, the dot product | Analytic geometry, trigonometry |
| Systems of Equations | Substitution, elimination, three variables | Linear equations |
| Matrices and Determinants | Row reduction, multiplication, inverses, determinants | Systems |
| Reference and Synthesis | Notation, common errors, source corrections, this page | Everything |
The four load-bearing ideas
Most of the collection is built on four ideas. Understanding these properly makes the rest far easier.
The distributive law
Expanding and factoring are one law used in two directions. It underlies every special product, every factoring method, and the whole of polynomial arithmetic.
The index laws
Three laws, derived by counting factors, which then define the zero, negative and fractional cases by requiring consistency. The method of extension matters as much as the results.
The unit circle
Sine and cosine are coordinates, not ratios. Every trigonometric identity, graph and equation follows from that single picture.
Elimination
Adding a multiple of one equation to another. Two-variable elimination, three-variable elimination, Gaussian elimination and Gauss-Jordan are one technique at four levels of formalisation.
A fifth idea recurs without being a topic: Pythagoras. The distance formula, the equation of a circle, the magnitude of a vector and sin2 + cos2 = 1 are the same theorem four times over.
Suggested routes
| Goal | Route | Streams |
|---|---|---|
| Complete course | Straight through in order | All fifteen, roughly one stream per sitting |
| Preparing for calculus | Foundations, then functions and graphs | Foundations, Polynomials, Indices, Equations, Functions, Linear, Quadratic, Trigonometric |
| Preparing for linear algebra | Foundations, then straight to systems | Foundations, Equations, Linear Functions, Vectors, Systems, Matrices |
| Filling specific gaps | Start at the topic; follow its prerequisites backwards | Use the dependency column above |
The reference stream can be read at any point. The notation page in particular is worth reading early, and the common errors page is worth revisiting after each stream rather than once at the end.
How the pages are built
Every page follows the same structure, so it is possible to navigate one without reading it in full.
Worked examples are drawn from the teaching notes wherever the source provides one, and are checked independently. Where the source is wrong, the correction is shown on the page in a labelled panel rather than applied silently — every instance is catalogued in Corrections and Judgement Calls.
Where this leads
| This collection | Leads to |
|---|---|
| Functions, limits of algebraic manipulation | Differential calculus |
| Trigonometric identities and graphs | Integral calculus; Fourier analysis; wave mechanics |
| Vectors and the dot product | Statics, dynamics, three-dimensional geometry |
| Systems and matrices | Linear algebra; structural analysis; numerical methods |
| Complex numbers | Alternating-current analysis; control theory; signal processing |
| Polynomial division and the factor theorem | Partial fractions; transfer functions; Laplace transforms |
The right-hand column is why this material is worth doing properly rather than quickly. Every one of those subjects assumes the whole of the left-hand column as background and moves at a pace that does not allow for gaps.
Check by substituting a number. It works in every one of the subjects above, it costs seconds, and it catches the great majority of errors. Of everything in this collection, that habit is probably the most transferable.
Frequently asked questions
Do I have to work through it in order?
No, but the dependencies are real. Factoring needs the distributive law; the quadratic formula needs completing the square; the trigonometric identities need the unit circle.
Which topics are load-bearing?
Four: the distributive law, the index laws, the unit circle, and elimination. Almost everything else is built from one of them.
How long is each page?
Between five and ten minutes of reading, with worked examples. A stream of three to seven pages is a single sitting.
Where does this lead?
To calculus, linear algebra and engineering mathematics. The material here is what those subjects assume without stating.
Source. Compiled from the whole source: twelve weeks of teaching notes and nineteen supplementary sheets.
This page is an original exposition prepared for the KEVOS® knowledge library. It restates, reorganises and verifies the mathematics of the supplied teaching notes; it is not a reproduction of them. Numerical values taken from the notes are identified as source examples and are not presented as engineering standards.