Glossary

Terms used across the library, with the page that explains each in full.

Absolute error — the difference between a value and the truth, in the value's own units (±0.05 cm). What + and − propagate. See uncertainty propagation.

Almost everywhere — everywhere except on a set of measure zero. The Cantor function's slope is 0 almost everywhere, and the function still climbs from 0 to 1, so an almost-everywhere fact can miss the thing that matters. Probability's name for the same idea is almost surely. See the Cantor function.

Almost surely — with probability 1, which is not the same as certainly: the exceptions exist, and together they have measure zero. A number drawn at random from [0, 1] is almost surely irrational. See probability zero.

Cantor function — also the devil's staircase. Read x in base 3, cut after the first 1, turn 2s into 1s, and read the result in base 2. Continuous, climbing from 0 to 1, and flat on every gap of the Cantor set, so its whole rise happens on a set of length 0. See the Cantor function.

Cantor set — what survives when the open middle third of [0, 1] is deleted, then the middle third of every piece left, forever. Uncountably many points and total length 0; exactly the numbers that can be written in base 3 with only 0s and 2s. See the Cantor set.

Catastrophic cancellation — the loss of most significant figures when two nearly equal numbers are subtracted. It does not create error; it removes the leading digits that were hiding error already present. See catastrophic cancellation.

Conditioning — how much a problem's output changes for a small change in its input. A property of the problem, not of any algorithm; an ill-conditioned problem defeats every method. See catastrophic cancellation.

Countable — able to be written as a list — a first, a second, a third — with every member somewhere in it. The whole numbers and the rationals are countable; [0, 1] is not. Every countable set has measure zero. See countable sets.

Dense — found inside every interval, however short. The rationals are dense in the line and still have measure zero. See countable sets.

Exact number — one that was counted or defined rather than measured (ballots cast, inches per foot, π). Has infinitely many significant figures and never limits a calculation. See exact vs approximate.

Fat Cantor set — also the Smith–Volterra–Cantor set. Built like the Cantor set, but the gaps deleted at step n are 1/4ⁿ long. It contains no interval and still has length 1/2 — the proof that full of gaps does not mean measure zero. See the fat Cantor set.

Floating point — the machine's binary approximation of a real number. Its errors look like measurement errors and are unrelated to them: the value was known perfectly and the hardware could not hold it. What a float stores is covered in full by the sibling Rust library (What a float actually stores ↗); what happens when you subtract two of them is catastrophic cancellation.

Lebesgue's criterion — a bounded function on a closed interval has a Riemann integral exactly when the points where it is discontinuous form a set of measure zero. Also called the Lebesgue–Vitali theorem. See the fat Cantor set.

Measure zero — a set has measure zero if, for every ε > 0, it fits inside a list of intervals whose lengths add up to at most ε. With squares or cubes in place of intervals, the same definition gives zero area or zero volume. Also called a null set. See what measure zero means.

Nowhere dense — for a closed set on the line, containing no interval at all: there are gaps everywhere. Both Cantor sets are nowhere dense, but only the standard one has measure zero. See the fat Cantor set.

Quadrature — combining independent uncertainties as √(a² + b²) rather than a + b. Linear addition is the worst case and is correct only for perfectly correlated errors. See uncertainty propagation.

Relative error — error as a fraction of the value (0.81%). What × and ÷ propagate, and the reason their rule counts significant figures. See uncertainty propagation.

Rounding — the mechanical operation of cutting a number at some place. The action; significant figures are the argument for where the action must stop. See significant figures.

Scientific notation — writing a value as mantissa × 10ⁿ, so the mantissa carries the precision claim and the exponent carries the magnitude. The only unambiguous way to write trailing zeros. See significant figures.

Significant figures — the digits of a measurement that carry information about the instrument rather than about place value. A claim about knowledge, not a formatting choice. See significant figures.

Stability — whether a particular algorithm preserves the accuracy a well-conditioned problem allows. The textbook quadratic formula is unstable for one of its two roots; a conjugate rearrangement fixes it for free. See catastrophic cancellation.

Uncountable — too big to be written as a list. [0, 1] is uncountable, and so is the Cantor set, which still has measure zero. See the Cantor set.