Concept

Exact arithmetic

Computation carried out in whole numbers or fractions, with no rounding anywhere. It is what a claim of strict inequality requires, because an arithmetic that has already rounded a quantity to the edge of an interval cannot say whether it was strictly inside.

Named by 2 essays across 2 fields — each of them below, with the objects they name alongside it.

The tail that would have to be a whole number. For each denominator, the value of q! times the tail of the series for e, plotted against the band between zero and one where no whole number lies, with the bound 1/q above it.

A tail too small to be a whole number

If e were a fraction with denominator q, then q! times e would be a whole number. It splits into a whole part and a tail, the tail is squeezed strictly between nothing and one, and there is no whole number there.

number · irrationality $The orbit a computer draws, and the orbit. Two orbits of the tent map from the same starting fraction plotted against the step number — one computed exactly in whole-number arithmetic and periodic, one computed in double precision and reaching zero.$

The orbit a computer draws

A chaotic orbit computed in floating point is not the orbit of the point it started from. Sometimes it is the true orbit of a nearby point, which is enough; sometimes the arithmetic simply runs out, and the picture is of the rounding.

dynamics · iteration

Named alongside it

The objects these essays reach for when they reach for this one.

Binary expansion Convergence rate Exponential constant Factorial Floating point Geometric series Integrality Irrationality Iteration Numerical artefact Periodic orbit Proof by contradiction

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