5 available now · 11 on the way
Complex numbers
Numbers as points in a plane, where multiplying means rotating. The arithmetic behind phase, spectra, and the FFT.
Euler’s formula
Why e appears in a formula about circles, and why eⁱᶿ means “rotate by θ”. The spiral construction makes both visible.
Parameterization and substitution
Two different moves hide inside “substitute and simplify”: composing with a new input, as in θ = 2πft, and extending a definition, as in eⁱᶿ. Which move is in play determines what stays true.
Logarithms and exponents
The inverse of raising-to-a-power: why logs turn multiplication into addition, what log₂ counts, and how to read a log the way you read an exponent.
Probability: events and independence
Events, the probability of an event, and the product rule: why independent chances multiply, the fact behind every “bits add” argument.
Probability distributions
The whole table of outcomes and their probabilities as one object: the thing entropy is a property of, and the thing a model predicts.
Expected value
The probability-weighted average: what “on average” means precisely, and how Σ p(x)·f(x) computes it.
Summation notation (Σ)
Reading Σ like a for-loop: index, range, body, including what an infinite sum means.
Sine waves
Amplitude, frequency, phase, and period: the four knobs of a pure tone, and why T = 1/f.
Vectors and the dot product
Signals as arrows in N dimensions: the dot product as similarity, and orthogonality as “zero overlap”, the geometry under correlation and the DFT.
Trig identities from the rotating point
Sum-to-product, product-to-sum, and sin·cos, derived geometrically from circular motion instead of memorized.
Decibels (dB)
How dB turns enormous loudness ratios into small, readable numbers, and why +6 dB means double.
Signal-to-noise ratio
What SNR measures, how it is quoted in dB, and where the 6-dB-per-bit rule comes from.
Logarithms and log scales
Reading logarithmic axes, octaves, and orders of magnitude, the scale most charts on this site use.
Sinc and band-limited reconstruction
What “band-limited” means, why exactly one band-limited curve passes through a valid set of samples, and how sinc pulses rebuild it.
Bits and binary counting
Why N bits give 2ᴺ labels, and how binary counting underlies bit depth, bitrate, and “bits” as a unit of information.
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