Matpic
mathematics, in pictures
Concept

Commutativity

The condition that the order of two operands does not matter. It holds for ordinary addition and multiplication and fails for composition, for matrices and for the arithmetic of order types, where a step placed in front of infinitely many is absorbed and a step placed after them is not.

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

The same terms, with the signs alternating. The partial sums of 1 - 1/2 + 1/3 - 1/4 + …, out to 24 terms. They close on 0.69315 from both sides at once, and the gap between consecutive sums is the next term, so the answer is trapped.

The same terms, in a different order, adding to whatever is asked

Flip alternate signs in the harmonic series and it converges. Reorder the terms — add nothing, remove nothing — and it converges to any number chosen in advance. Addition stops being commutative, and the picture shows where it goes.

analysis · harmonic series Ordinal sums and products, in normal form. A table of ordinal expressions with their Cantor normal forms and whether the two sides of each pair are equal, above two tick lines drawing one such pair.

One step in front of infinitely many

Put one step before an infinite run of them and nothing has changed; put it after and something has. Ordinal addition records that difference, which is why it is not commutative — and why it keeps information that counting throws away.

logic · ordinals

Named alongside it

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

Absolute convergenceAlternating seriesAssociativityCardinalityConditional convergenceConvergenceCountingDivergenceHarmonic seriesHarmonicsLimitLimit ordinal

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