Skip to content
Axioma

Field book

From counting to categories.

Mathematics in the order it is actually learned, from arithmetic through the frontier, then the formulas physics, chemistry, and biology compute with. Each chapter explains the idea, states the formulas with their symbols, and works an example.

28 chapters · 137 formulas · progress stays in this browser

Start with arithmeticOpen the formula atlas

Four ways through

The foundationsNumber, algebra, shape, and the functions calculus needs.5 chaptersCalculus and linear algebraChange, accumulation, and the geometry of linear maps.5 chaptersPure mathematicsProof, number, limit, structure, shape, and the frontier.8 chaptersThe sciencesThe formulas physics, chemistry, and biology compute with.9 chapters

Mathematics

From counting and shape to proof, structure, and the frontier.

  1. 1Arithmetic and number senseNumbers, the four operations, fractions, percent, factors, and the idea that every integer has a unique prime recipe.
  2. 2AlgebraLetters stand for numbers. Equations are balances. The quadratic formula closes the story of every second-degree polynomial.
  3. 3GeometryShape, measurement, and the first real proofs: congruence, similarity, and the Pythagorean theorem.
  4. 4TrigonometrySine and cosine as coordinates on the unit circle, the identities that follow, and the laws that solve any triangle.
  5. 5Functions, exponentials, and logarithmsA function is a rule with one output per input. Composition, inverses, and the exponential–logarithm pair are the machinery calculus will differentiate.
  6. 6Differential calculusThe derivative is the instantaneous rate of change: a limit of slopes. The rules let you compute it without returning to the limit each time.
  7. 7Integral calculusThe integral accumulates. The fundamental theorem says differentiation and integration are inverse operations.
  8. 8Linear algebraVectors, matrices, and the one question they all answer: when does a linear equation have a solution, and what does the solution space look like?
  9. 9Multivariable calculusFunctions of several variables, partial derivatives, the gradient, and integrals over regions of the plane and space.
  10. 10Probability and statisticsProbability is the mathematics of uncertainty. Statistics is the reverse: data in hand, what can you say about the process that made it?
  11. 11Discrete mathematicsThe mathematics of separate things: sets, counting, graphs, and the first careful proofs by induction.
  12. 12The language of proofDefinitions, quantifiers, and the handful of proof shapes — direct, contrapositive, contradiction, counterexample — that every later course assumes.
  13. 13Differential equationsAn equation relating a function to its derivatives. Separable equations, first-order linear equations, and the harmonic oscillator.
  14. 14Number theoryThe integers, studied for their own structure: divisibility, congruences, and the first theorems about primes.
  15. 15Real analysisThe calculus, rebuilt on definitions. Limits, completeness of the reals, continuity, and what a derivative really is.
  16. 16Complex analysisCalculus over the complex numbers. Differentiability once, in the complex sense, is so strong that it creates infinite smoothness and the residue calculus.
  17. 17Abstract algebraGroups, rings, and fields: the study of operations by the axioms they satisfy, not by what the elements happen to be.
  18. 18TopologyWhat remains of geometry when you may bend and stretch but not tear: open sets, continuity, compactness, and connectedness.
  19. 19The frontier: analysis, geometry, and categoriesThree doors past the undergraduate core — functional analysis, algebraic geometry, and category theory — stated honestly as definitions and one theorem each.

Physics

Motion, energy, fields, light, and the constants that set their scale.

  1. 1Classical mechanicsMotion under forces: kinematics in one dimension, Newton’s laws, momentum, and circular motion.
  2. 2Energy, heat, and wavesWork and mechanical energy, the ideal gas, heat capacity, and the description of a traveling wave.
  3. 3Electromagnetism and opticsCoulomb’s law, circuits, the magnetic force on a charge, and the wave picture of light.
  4. 4Relativity and quantum ideasThe two twentieth-century corrections: c is a speed limit built into spacetime, and energy and matter are quantized.

Chemistry

Amounts, gases, equilibrium, energy, acids, and charge.

  1. 1Amounts, stoichiometry, and gasesThe mole, balancing reactions, concentration, and the gas laws chemists actually compute with.
  2. 2Equilibrium, energy, and ratesThe equilibrium constant, the free-energy test for spontaneity, and the rate laws that say how fast you get there.
  3. 3Acids, bases, and electrochemistrypH, weak-acid equilibria, the Henderson–Hasselbalch equation, and the voltage of an electrochemical cell.

Biology

Growth, genes, enzymes, membranes, and the physics cells obey.

  1. 1Populations and geneticsExponential and logistic growth, and the Hardy–Weinberg accounting of alleles in a population that is not evolving.
  2. 2Enzymes, membranes, and scalingMichaelis–Menten kinetics, the Nernst potential, diffusion time, and the allometric scaling that compares animals of different sizes.