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Concept

Natural logarithm

The power to which e must be raised to reach a given number. Its derivative is one over the input, which is what makes it appear whenever a sum over the whole numbers is estimated.

Named by 4 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 Waiting for all 6 kinds. One bar per new kind: the expected number of draws needed to see a kind not yet seen, rising as fewer of them are left, and adding to 14.70 draws in total.

How long until every one turns up

Draw at random from six equally likely kinds until all six have appeared. The wait is not six draws, and it is not sixty; it is fourteen point seven, and the number is a harmonic sum wearing a hat.

probability · Expectation 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 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

Named alongside it

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

LogarithmApproximationConvergence rateDerivativee, the numberHarmonic seriesIntegralAreaCollisionCompound-interestContinuityCounting argument

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