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Concept

Logarithm

The exponent a base must be raised to in order to give a number, which turns multiplication into addition. Turning products into sums is exactly why it appears in the prime counting theorem and in every entropy.

Named by 10 essays across 3 fields — each of them below, with the objects they name alongside it.

eˣ and its tangent lines. The exponential curve with tangent lines at several points; at each point the slope equals the height.

The curve that is its own slope

There is exactly one shape of exponential curve whose steepness at every point equals its height at that point. The number that produces it is 2.71828…, and it was not chosen for elegance.

analysis · The exponential Terms that vanish, a total that does not. The first 24 terms of the harmonic series as bars, with the running total above them. The last bar is 0.042 tall and the total has reached 3.776.

A sum whose terms vanish and whose total does not

Add a half, a third, a quarter, and keep going. The terms shrink to nothing and the total passes every number there is — but so slowly that no computation will ever watch it happen.

analysis · Harmonic series 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 The Lyapunov exponent, 2.8 to 4. The average rate at which nearby orbits separate, plotted against the parameter.

How fast two orbits part

The word "sensitive" is an adjective. Averaging the logarithm of one derivative along an orbit turns it into a number — one that says how many steps of prediction the map allows, and whose sign says whether it allows any.

dynamics · Sensitive dependence The Collatz orbit of 27. Halve an even number, triple an odd one and add one; the sequence plotted on a logarithmic scale.

The question nobody can answer

Halve it if it is even, triple it and add one if it is odd. Every number anyone has tried comes down to one. Nobody can prove they all do, and the reason is not that the problem is hard to state.

dynamics · Collatz π(x) against its two estimates, up to 20,000. The ratio of the prime counting function to x over the logarithm of x, and to the logarithmic integral, plotted against x. The first is above one and coming down slowly; the second is close to one throughout.

Counting what has no formula

There is no expression that gives the nth prime, and yet the number of primes below a bound is predictable to within a fraction of a per cent — by a function that is not a formula for the primes but an integral of the wrong-looking quantity.

number · Prime distribution eˣ and its inverse, reflected in the diagonal. A curve, the line y = x, and the curve reflected in it — which is the graph of the inverse function. Tangents are drawn at matched pairs of points, and the two slopes at each pair multiply to one.

The slope of the mirror image

Undoing a function is reflecting its graph in the diagonal, and a reflection turns a slope into its reciprocal. That single observation supplies the derivative of every inverse — the logarithm, the roots, the inverse trigonometric functions — without differentiating any of them.

analysis · The derivative The area that names the number. The curve 1/x with the area under it from 1 to 2.7183 shaded, measuring 1.0000.

The area that names the number

The number e can be defined without mentioning slopes at all. Slide right along the curve 1/x until the area underneath reaches exactly one, and stop. That is where e is, and the reason logarithms turn multiplication into addition is visible in the same picture.

analysis · The exponential A closer start buys time and nothing else. The logarithm of the separation between two Lorenz trajectories plotted against time, for three different initial separations. The three curves are straight and parallel over most of their length, with the same fitted slope.

A closer start buys only time

Two trajectories from almost the same place separate exponentially, and the rate does not depend on how close they began. Halving the initial error buys one fixed interval of extra agreement, and no amount of precision buys more than a fixed number of those.

dynamics · Strange attractor Downhill on average, and never on purpose. The logarithms of 4 Collatz orbits plotted against step number, each wandering upward and downward and each ending at one. A separate sample of four thousand starts gives an average fall of -0.15 per step.

The heuristic that cannot be a proof

There is a two-line argument that the Collatz conjecture is true, it is convincing, and everybody who works on the problem believes it. It also cannot be turned into a proof, and understanding exactly where it fails is more instructive than the argument itself.

dynamics · Collatz

Named alongside it

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

DerivativeLimitHarmonic seriesIterationNatural logarithmApproximationChaosCollatzContinuityConvergenceDivergencee, the number

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