Exact arithmetic
Named by 8 essays across 6 fields — each of them below, with the objects they name alongside 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.
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.
The remainders that count the roots
Run Euclid's algorithm on a polynomial and its derivative, flipping the sign of each remainder, and write down the signs of the whole chain at any point. The number of sign changes drops by exactly one each time the point passes a real root — so the roots in any interval can be counted, exactly, without finding a single one.
The sums of roots of unity that add to nothing
All n of the n-th roots of unity add to zero, and so does any regular polygon among them, turned. Those are not the only vanishing sums: six thirtieths of a turn close into a loop with no polygon in them. Which counts of roots can close at all is decided by the prime factors of n — no seven fifteenths ever add to zero — and the same question counts where the diagonals of a regular polygon cross.
What the signs allow
Descartes' rule reads the signs of a polynomial's coefficients and bounds its positive roots by how often they change — a bound set by the number of terms, not the degree, and proved by nothing more than Rolle's theorem. Each count the rule allows can be had. But not every pair of counts, positive and negative, that the rule allows together can be had together: the first combination that never occurs is at degree four, and the list of impossible ones is still being worked out.
The signs no completion can change
A quadratic in several variables can be completed square by square in many orders, and each order hands back different coefficients. What no order changes is how many come out positive and how many negative — and that count, read off a completion of A − tI, says how many eigenvalues lie below t without finding a single one.
What a missing string of digits leaves behind
Strike from the harmonic series every term whose denominator contains 42 and the rest adds up to 228.45. Strike out 99 instead and it adds up to 253.28. The difference is not about which numbers are lost but about waiting: 99 overlaps itself, so it takes 110 random digits on average to turn up where 42 takes 100, and the sum is almost exactly ln 10 times that wait.
What two points build in two rounds
Start with two points and draw every line and circle they allow; mark every crossing; do it again. The first round gives six points, the second gives 203, and the third gives over 1.7 billion. Computed exactly, the 203 points show the tower of square roots growing in real time — eleven rational points, seventy-six that need √3, and the rest needing one of six new square roots.
Named alongside it
The objects these essays reach for when they reach for this one.
Exhaustive searchConvergence rateDerivativeEigenvalueGeometric seriesPolynomialRootsSignBinary expansionBisectionClosureCompass and straightedge