8 concise reference notes for mathematics. These are public reference material, not private user records or a transfer of language-model weights.
Exact rational arithmetic
A rational number is represented as an integer numerator divided by a nonzero integer denominator. Reducing by their greatest common divisor preserves the value. Exact fraction arithmetic avoids binary floating-point rounding in supported arithmetic operations.
Authored reference note; not externally verified product advice · ref-089Floating-point arithmetic
Finite computer number formats cannot represent every real number exactly. Rounding can affect comparisons and decimal display. Use the exact tool for supported fraction and decimal arithmetic when exact rational results are required.
Authored reference note; not externally verified product advice · ref-090Bayesian updating
A posterior distribution combines prior probabilities with evidence likelihoods and then normalizes. Likelihood means the probability of the evidence under a hypothesis, not the probability of the hypothesis given that evidence. Results depend on the supplied model.
Authored reference note; not externally verified product advice · ref-091Conditional independence
Multiplying separate evidence likelihoods assumes the observations are conditionally independent under each hypothesis unless joint likelihoods are supplied. Correlated evidence can otherwise be counted more than once.
Authored reference note; not externally verified product advice · ref-092Impossible evidence
If every hypothesis assigns zero probability to an observation, normalizing the posterior is undefined. Reconsider the hypotheses or likelihood model instead of presenting an arbitrary confidence distribution.
Authored reference note; not externally verified product advice · ref-093Entropy
For a discrete probability distribution, Shannon entropy in bits is minus the sum of p times log base two of p, with zero-probability terms contributing zero. It measures uncertainty under that distribution, not truth or usefulness by itself.
Authored reference note; not externally verified product advice · ref-094Optimization bounds
A bound on the best possible remaining objective can prune a search branch that cannot improve the current result. If search stops before all necessary branches are resolved, report the best found solution without claiming proven optimality.
Authored reference note; not externally verified product advice · ref-095Dimensional consistency
Addition and comparison generally require compatible quantities. A number without its unit can be ambiguous. Unit conversion requires a defined relationship between compatible dimensions; fixed physical conversions are different from changing market exchange rates.
Authored reference note; not externally verified product advice · ref-096In chat, use knowledge: topic. Browse all reference topics.