Calculus & Transforms

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

12 articles

1. 1
   Mathematics10 min

   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. 2
   Mathematics10 min

   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. 3
   Mathematics10 min

   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. 4
   Mathematics10 min

   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. 5
   Mathematics10 min

   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. 6
   Mathematics9 min

   Euler's Formula

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

7. 7
   Mathematics9 min

   Taylor Series

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

8. 8
   Mathematics8 min

   Newton's Method

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

9. 9
   Mathematics10 min

   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. 10
    Mathematics10 min

    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. 11
    Mathematics10 min

    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. 12
    Mathematics10 min

    The Fourier Transform

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