Math

Where does the math show up?
Make the numbers useful.

Use math to make software behavior, measurements, and engineering trade-offs easier to reason about: units, probability, statistics, and the models behind AI.

Work in progress. 50 of 50 lessons are drafted and open to read; they're still being reviewed.

No calculus required. Each lesson starts from a concrete engineering question and works through it with small TypeScript and Go calculations.

50 topics

Still being written 50 of 50 lessons are drafted and open to read while they're reviewed. Planned topics show where the catalog is going.

Quantities and representation

Start here

Choose useful quantities, keep units straight, and understand how values are represented.

Concept authoring

Units, dimensions, and conversions

Keep bytes and bits, seconds and milliseconds, and decimal and binary prefixes distinct.

Concept authoring

Ratios, rates, and proportions

Calculate throughput, requests per second, utilization, and scaling with matching units.

Concept authoring

Linear interpolation and scaling

Map values between ranges for progress bars, chart axes, and easing, and clamp what falls outside.

Concept authoring

Percentages and percentage points

Read error rates, cache hit rates, rollout shares, and growth without losing the denominator.

Concept authoring

Powers of two and logarithms

Relate address space, binary sizes, search depth, and scale to repeated doubling.

Concept authoring

Binary, hexadecimal, and bit operations

Read subnet masks, flags, permissions, and protocol fields at the bit level.

Concept authoring

Boolean logic, sets, and predicates

Reason about firewall rules, filters, and eligibility conditions as combinations of true/false claims.

Concept authoring

Counting and combinatorics

Count permutations and combinations to see why test matrices and feature-flag combinations grow so fast.

Concept authoring

Signed numbers, overflow, and fixed-width values

Understand integer limits, counters, timestamps, and why language number models differ.

Concept authoring

Floating point and rounding

See why 0.1 + 0.2 isn't 0.3, compare floats safely, and store money as integer minor units.

Concept authoring

Modular arithmetic

Use remainders for hash buckets, ring buffers, and wraparound, and see how % treats negative numbers across languages.

Concept authoring

Text, Unicode, and encoded bytes

Distinguish characters, code points, and UTF-8 bytes when sizes and indexes matter.

Concept authoring

Entropy and bits of strength

Turn token length and alphabet size into bits to judge how guessable an ID or password is.

Change, uncertainty, and evidence

Build understanding

Use growth, probability, and statistics to interpret behavior and measurements.

Concept authoring

Linear, exponential, and logarithmic growth

Compare workload growth, repeated doubling, search depth, and algorithm cost.

Concept authoring

Sums and series

Add up geometric series to find total backoff wait time and the amortized cost of a growing array.

Concept authoring

Probability and expected value

Reason about failure likelihood, retries, random choices, and average outcomes across many trials.

Concept authoring

Collisions and the birthday problem

Estimate how long an ID or hash must be before collisions become likely.

Concept authoring

Conditional probability and Bayes’ rule

Update a belief from evidence and see why rare events produce surprising alert accuracy.

Concept authoring

Precision, recall, and the confusion matrix

Measure classifiers, alerts, and search results by what they catch and what they wrongly flag.

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Sampling and selection bias

Understand how collection choices can make logs, benchmarks, or surveys misrepresent the whole.

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Randomness and random number generators

Choose between seeded and cryptographic generators, and avoid modulo bias and broken shuffles.

Concept authoring

Mean, median, variance, and percentiles

Summarize latency and workload without allowing a long tail to disappear into an average.

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Distributions and long tails

See why latency isn't normally distributed and what heavy tails mean for capacity and alerting.

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Moving averages and smoothing (EWMA)

Smooth noisy rates and load signals with exponential weighting, and see how the decay factor trades responsiveness for stability.

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Confidence, variability, and repeated measurements

Compare noisy measurements and state what a small experiment does and does not establish.

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Error bars and uncertainty communication

Present estimates as ranges when the evidence does not support a precise point value.

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Weighted averages and aggregation

Choose the right denominator and see why averaging averages or percentiles can mislead.

Math in engineering decisions

Go underneath

Apply simple models to queues, reliability, retries, caches, capacity, and subnet plans.

Concept Pro authoring

Back-of-envelope estimation

Size a system in orders of magnitude from reference latency and throughput numbers before measuring anything.

Concept Pro authoring

Queue growth and Little’s Law intuition

Relate arrival rate, work in progress, and wait time while keeping stability assumptions visible.

Concept Pro authoring

Availability and failure probability

Calculate simple dependency and redundancy cases, then challenge the assumption that failures are independent.

Concept Pro authoring

Tail latency under fan-out

Calculate how a slow response in one call out of a hundred becomes the common case when requests fan out.

Concept Pro authoring

Retry amplification and backoff

Calculate how retries multiply work during partial failure and compare bounded backoff policies.

Concept Pro authoring

Rate limiting math

Model token buckets and sliding windows to set burst size and refill rate deliberately.

Concept Pro authoring

Cache hit rate and break-even cost

Estimate when cached work pays off and how update and invalidation costs affect the decision.

Concept Pro authoring

SLOs, error budgets, and burn rate

Turn an availability target into an error budget and use burn rate to decide when to page.

Concept Pro authoring

Load distribution and hashing

Compare modulo hashing, consistent hashing, and power-of-two-choices for spreading load across nodes.

Concept Pro authoring

Capacity, headroom, and saturation

Reason about demand, utilization, and why a heavily loaded queue may degrade quickly.

Concept Pro authoring

Network subnet planning

Allocate address ranges for current needs and growth, with each mask and boundary check shown.

Math behind AI

Build understanding

Understand the math inside embeddings, training, and model output.

Concept authoring

Vectors and dot products

Treat a list of numbers as a direction and a length, and see what a dot product measures.

Concept authoring

Similarity and distance

Compare cosine and Euclidean distance, and see why the metric and normalization must match the index.

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Matrices as transformations

Read matrix multiplication as applying one function to many vectors at once, the core of a neural network layer.

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Derivatives, gradients, and the chain rule

Treat a derivative as sensitivity and see how the chain rule passes it back through layered functions.

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Gradient descent and learning rate

Step downhill on a loss surface and watch the learning rate cause slow progress, overshooting, or divergence.

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Exponentials, softmax, and temperature

Turn raw scores into probabilities, reshape them with temperature, and keep the math stable with log-sum-exp.

Concept authoring

Log-probabilities and information theory

Use entropy, cross-entropy, perplexity, and KL divergence to measure how well a model predicts.

Concept authoring

Sampling strategies (greedy, top-k, top-p)

Compare decoding strategies as different ways of picking the next token from a distribution.

Concept authoring

Normalization

Rescale features with min-max and z-scores, and see how the fitted population shapes model inputs.

Concept authoring

High-dimensional intuition

See why distance behaves strangely in many dimensions and why approximate nearest-neighbor search exists.

Concept authoring

Quantization and numeric precision

Compare fp32, fp16, bf16, and int8 and what each trades in range, precision, and memory.

Concept authoring

Token and cost arithmetic

Estimate token counts from a declared text ratio, budget context windows, and calculate request cost.