Algebra

Unit I — Expressions
A variable as a name for a quantity · Translating words into expressions · Evaluating and simplifying expressions · The distributive law and combining like terms

Unit II — Equations in One Variable
The equals sign as a balance · One-step and two-step equations · Equations with variables on both sides · Checking a solution and spotting no-solution cases

Unit III — Inequalities
Reading and graphing inequalities on a number line · Solving linear inequalities · Why multiplying by a negative flips the sign

Unit IV — Lines on the Plane
The coordinate plane and plotting solutions · Slope as a rate of change · Slope-intercept form and graphing lines · Reading a real situation off a linear graph

Unit I — Functions as Objects
Function notation, domain, and range · Transformations: shifting, stretching, reflecting · Inverse functions and when they exist

Unit II — Quadratics
Factoring, completing the square, and the quadratic formula · The vertex and what the discriminant reports · Projectiles and other parabolic situations · Complex numbers, met at the point of need

Unit III — Polynomials & Rational Functions
End behavior and the shape of polynomial graphs · Roots, factors, and the remainder theorem · Rational functions and asymptotes

Unit IV — Exponentials & Logarithms
Exponential growth and decay · The logarithm as the exponent you are missing · Log laws and solving exponential equations · Compound interest and half-life as applications

Unit V — Systems
Systems of linear equations: substitution and elimination · Systems mixing lines and curves · Modeling with systems: mixture and rate problems

Unit I — Vectors
Vectors as arrows and as lists of numbers · Linear combinations and span · The dot product, length, and angle · Linear independence

Unit II — Systems & Elimination
Systems of linear equations as geometry · Gaussian elimination and row echelon form · Pivot positions, free variables, and solution sets

Unit III — Matrices as Maps
A matrix as a function on vectors · Matrix multiplication as composition · Rotations, projections, and shears · Inverse matrices and when they fail to exist

Unit IV — Determinants & Subspaces
The determinant as a volume factor · Column space, null space, and rank · The rank-nullity theorem

Unit V — Eigenvalues
Eigenvectors: directions a map merely stretches · Computing eigenvalues from the characteristic polynomial · Diagonalization and matrix powers · A preview: Markov chains and principal components

Unit I — Symmetry & the Group Axioms
Symmetries of a square as a worked object · The group axioms and first examples · Cyclic groups and orders of elements · Permutations and the symmetric group

Unit II — Subgroups & Cosets
Subgroups and how to recognize them · Cosets and Lagrange's theorem · Normal subgroups and quotient groups

Unit III — Homomorphisms
Structure-preserving maps · Kernels and images · The first isomorphism theorem

Unit IV — Rings & Ideals
Rings: two operations, one story · Integral domains and zero divisors · Ideals and quotient rings · Polynomial rings

Unit V — Fields
Fields and field extensions · Constructibility: why angle trisection fails · A glimpse of Galois theory and the unsolvable quintic

Unit I — Divisibility
Divisibility and the division algorithm · Greatest common divisors and the Euclidean algorithm · Primes and the fundamental theorem of arithmetic · The infinitude of primes, proved two ways

Unit II — Congruences
Modular arithmetic as clock arithmetic made rigorous · Solving linear congruences · The Chinese remainder theorem

Unit III — Fermat & Euler
Fermat's little theorem · Euler's totient function and Euler's theorem · Primality testing in outline

Unit IV — Applications & Open Doors
RSA: public-key cryptography from first principles · Pythagorean triples and Fermat's last theorem, told honestly · Perfect numbers and other unfinished business