Mathematics

Exponentials & Euler’s number:

• exponential-function — Definition $a^x$, the identity $a^{x+y}=a^x{a}^y$, and why every exponential is proportional to its own derivative
• eulers-number — $e\approx 2.71828$ as the unique base where $\frac{d}{dt}{e}^t=e^t$; rewriting any $a^t$ as $e^{ct}$
• natural-logarithm — Inverse of $e^x$; resolves the mystery proportionality constants as $\ln (a)$
• complex-exponential-rotation — $e^{it}$ traces the unit circle; $e^{i\pi }=-1$ via the dynamics view (velocity = $i\times$ position) and the group-theory view ($\exp$ as homomorphism)
• group-theory-intro — Groups as sets of symmetry actions; numbers as additive (sliding) and multiplicative (stretching/rotating) groups; $\exp$ as the homomorphism between them

Signal processing chain (read in order for the Shazam implementation):

• fourier-transform — Decomposes a signal into complex sinusoids; convolution theorem is the key engineering result
• discrete-fourier-transform — N-point DFT: formula, twiddle factors, matrix form, O(N²) complexity, frequency bin interpretation
• fast-fourier-transform — Cooley-Tukey FFT: butterfly diagram, bit-reversal, O(N log N); the algorithm behind scipy.fft
• spectrogram — STFT: sliding-window DFT, Hann window, time-frequency trade-off, log-magnitude display
• audio-fingerprinting — Shazam’s constellation-map algorithm: peak detection, hash pairs, voting-based lookup

Image compression chain (read in order for the JPEG implementation):

• discrete-cosine-transform — DCT-II: real-valued, energy compaction, relationship to DFT, 2D separable form
• jpeg-compression — Full JPEG pipeline: YCbCr, 8×8 DCT, quantization tables, zig-zag, RLE, Huffman coding

• probability-theory-intro — Probability fundamentals: frequentist vs Bayesian, permutations, expected value

Read in order:

• linalg-set-notation — Set notation and intro to linear algebra
• vector-operations — Vectors: addition, scalar multiplication, dot/outer/Hadamard/cross products
• matrix-operations — Matrices: operations and 4 multiplication approaches
• matrix-composition-transformations — Matrix multiplication as composition of linear transformations
• square-matrices-determinant-trace — Square matrices: determinant (geometric intuition + formulae), trace, identity
• matrix-inverse-linear-systems — Systems of linear equations and matrix inverse
• vector-spaces-subspaces-rank — Spans, spaces, subspaces, rank, and non-square matrices
• matrix-decompositions — Gram-Schmidt, LU, QR, Cholesky decompositions
• eigendecomposition-diagonalisation — Eigendecomposition, eigenbasis, and diagonalisation

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• hub-ai-ml — linear algebra and probability are core ML prerequisites
• hub-cryptography — number theory is the foundation of public key cryptography
• hub-physics — analysis and differential equations underpin classical and quantum physics