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Concept

e, the number

The number whose exponential curve has slope equal to its own height everywhere, near enough 2.71828. It is the base at which the exponential is its own derivative, and it is the limit of compounding a hundred per cent interest ever more often.

Named by 8 essays across 2 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 All 24 arrangements of 4 objects, and the 9 that move every one. Every permutation of 4 objects drawn as a grid of cells, with the diagonal — where an object stays where it began — shaded, and the arrangements that avoid it entirely marked.

Nobody gets their own hat

Hand back a pile of hats at random and ask for the chance that not one person gets their own. The answer barely moves as the crowd grows — it is a third and a bit at four people, and a third and a bit at four thousand.

probability · Inclusion exclusion 5,040 orders, 7 thresholds, one best rule. For each number of candidates passed over, the share of the 5,040 possible arrival orders in which the rule ends up with the best of the 7. The count is exhaustive.

When to stop looking

Candidates arrive one at a time in a random order. Each must be accepted or rejected on the spot, with no going back and no way to know what is still to come. The best possible rule is to look at about a third of them and then take the first one that beats everything seen — and it works about a third of the time, however many there are.

probability · Optimal stopping 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 The slopes the equation demands, and the curves that obey them. A field of short segments whose slope at each point is 0.9 times the height there, with 3 solution curves integrated through it; each doubles over an interval of 0.770 wherever that interval is taken.

The equation with only one answer

A rate of change proportional to the current amount is the most common description in nature, and it pins down the function completely. There is exactly one curve through each starting point, and a half-life and a doubling time are the same measurement.

analysis · The exponential The flow of a linear equation, and the matrix that runs it for one unit of time. Paths of points moving so that their velocity is [0.25, −1.2, 1.2, 0.25] applied to their position, with the position after time 1 marked on each; the matrix taking start to finish is e^A.

The exponential of a square

The series for e makes perfect sense with a matrix in it. What comes out solves a system of equations the way the ordinary exponential solves one, and a skew matrix exponentiates into a rotation with no trigonometry anywhere.

analysis · The exponential The share of arrangements that fix nothing, up to 8 objects. A bar per number of objects, giving the proportion of its arrangements that leave nothing in place, against the horizontal line at 1/e.

The constant that counts what does not happen

Nothing grows in a shuffled pack of cards, and nothing grows in a factorial. Yet e sits in the middle of both — as the chance that a shuffle leaves nothing in place, and as the base that makes n! nearly a power.

analysis · The exponential 58.6% at 46 candidates, against 37% without the values. The chance of ending with the best candidate when the values are shown, against the number of candidates, for 10 sizes. It falls towards 0.5802 rather than towards 1/e.

When the numbers are shown

The secretary rule wins a third of the time and cannot do better, because it is told only who is ahead. Show the actual values and say where they came from, and the same problem is won three times in five — by a standard that falls as the end approaches.

probability · Optimal stopping

Named alongside it

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

Counting argumentDerivativeLimitPower seriesApproximationConditional probabilityDecision procedureDerangementDifferential equationExpectationHarmonic seriesIntegral

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