Mathematics / foundation
The language of change
Build a working language for quantities, functions, change, accumulation, and rigorous reasoning.
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Inside this module
Module 01 foundation
The language of change
Build a working language for quantities, functions, change, accumulation, and rigorous reasoning.
- 01
Mathematical language: making a claim precise
Read variables, sets, sums, units, and assumptions before manipulating a formula.
16 min - 02
Functions and graphs: describing dependence
Understand inputs, outputs, domains, and composition before treating a curve as a picture.
17 min - 03
Algebra and scaling: preserving meaning while rearranging
Solve equations reversibly, compare growth laws, and expose scale with dimensionless ratios.
18 min - 04
Trigonometry and phase: turning rotation into a wave
Connect the unit circle to sine, cosine, frequency, and relative timing.
18 min - 05
Vectors and components: keeping direction in the calculation
Add directed quantities, resolve components, and use a dot product to measure alignment.
19 min - 06
Complex numbers: arithmetic with rotation
Use two real coordinates to describe magnitude, phase, conjugation, and interference.
19 min - 07
Limits and continuity: reasoning about approaching
Separate nearby behavior from a value at a point, and read continuity as an explicit condition.
19 min - 08
Derivatives and local change: the best nearby straight line
Build the derivative from a difference quotient and interpret its sign, units, and local approximation.
20 min - 09
Integrals and accumulation: adding a continuously varying rate
Build a definite integral from local contributions and connect accumulation to differentiation.
20 min - 10
Proof and counterexample: knowing what evidence establishes
Distinguish examples from proof, build a direct argument, and refute universal claims precisely.
19 min