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Learning path · 12 parts

Calculus & Transforms

From rotating complex numbers to approximating any function and decomposing any signal — the analytic backbone of modern math.

About 2 hours of reading · 24 widgets

Start with The Derivative
  1. Mathematics10 min

    Part 1: The Derivative

    Speed is distance over time, yet a speedometer reads a speed right now. Resolving that paradox is the derivative — the foundation under optimization, physics, and machine learning.

  2. Mathematics10 min

    Part 2: The Integral: Adding Up the Infinitely Small

    The integral is not just a reversed derivative. It is the limit of a sum that measures accumulated area, and the Fundamental Theorem is what ties those two ideas together.

  3. Mathematics10 min

    Part 3: Logarithms: Taming Huge Numbers

    Logarithms are the inverse of exponents: they answer what power produces a number, turn multiplication into addition, and — despite the name — grow astonishingly slowly.

  4. Mathematics10 min

    Part 4: Infinity: Not All Infinities Are Equal

    Infinity is not a number. Learn how bijections measure the size of infinite sets, why the naturals and the reals differ, and Cantor's diagonal proof.

  5. Mathematics10 min

    Part 5: Complex Numbers

    Nobody invented i to solve x² = −1 — the cubic formula forced it on them. How a bookkeeping trick became a geometric plane where multiplication is rotation.

  6. Mathematics9 min

    Part 6: Euler's Formula

    Why eˣ has anything to do with circles — and what rotating arrows have to do with waves.

  7. Mathematics9 min

    Part 7: Taylor Series

    Turn sine, cosine, and eˣ into polynomials — and discover why this works everywhere.

  8. Mathematics8 min

    Part 8: Newton's Method

    The algorithm that squares the number of correct decimal places with every iteration.

  9. Mathematics10 min

    Part 9: Gradient Descent

    How repeatedly stepping in the direction of steepest descent minimizes functions in millions of dimensions — and trains essentially every modern neural network.

  10. Mathematics10 min

    Part 10: Matrices: Functions That Move Space

    A matrix is a linear transformation of space in disguise. Its columns tell you where the basis vectors land, and multiplication is just composition.

  11. Mathematics10 min

    Part 11: Eigenvectors and Eigenvalues

    Almost every vector gets knocked off its line by a matrix. The rare few that don't reveal what the transformation is really doing.

  12. Mathematics10 min

    Part 12: The Fourier Transform

    Decompose any wave into its frequency components — and see why this changes everything.