Integral
Named by 21 essays across 6 fields — each of them below, with the objects they name alongside it.
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.
A rectangle grown on two sides
A product of two changing quantities is the area of a rectangle whose sides both move. The extra area is two strips and a corner, and the whole of the product rule is the observation that the corner is negligible and the strips are not.
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.
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.
The error that does not care how many dimensions
A grid gets rapidly better in one dimension and hopelessly worse in twenty. Random points get better at the same slow rate whatever the dimension, which is why a method that is bad everywhere ends up being the only one that works.
Sampling where the answer lives
Monte Carlo error cannot be made to fall faster than the square root, so the only thing left to attack is the constant in front of it. Drawing points where the integrand is large, and dividing by how often they were drawn, leaves the answer alone and can shrink the noise many times over.
Points too even to be random
Independent random points clump, and the clumping is what makes the error fall only as the square root. Points chosen to be evenly spread rather than independently beat that rate, and the price is that nothing about them is random at all.
A staircase with no steps
A function that rises from nought to one, is continuous everywhere, and has derivative zero at almost every point. All of its climbing happens on a set of no length at all, which is possible because that set has uncountably many points.
Sums of powers, read off a staircase
Add the first n squares, or cubes, or seventh powers, and the answer is always a polynomial in n. Its first term is the area under a curve, its second is half of the last step, and every term after that is a correction for the corners of a staircase — which is where the Bernoulli numbers come from, and why they eventually grow without bound.
One number under every bell
The area under e^(−x²) has no formula in terms of the usual functions, and yet the area under e raised to any downward quadratic is known exactly. Completing the square in the exponent moves and squeezes every such curve into the same one, so a single number — √π — pays for all of them.
An endless region with a finite area
A region that runs off to infinity can still have a finite area, and for the curves 1/xᵖ the exponent that makes the far end finite is exactly the one that makes the end at zero infinite. 1/x fails at both, no power succeeds at both, and a horn can hold less than π while needing infinite paint.
A rectangle cut by a curve
Integration by parts is taught as the product rule run backwards. It is also a picture: an increasing curve cuts a rectangle into two pieces, one of them the area under the curve and the other the area beside it, and the formula says only that the pieces fill the rectangle. Run repeatedly, the same cut produces the factorials and Wallis's product for π.
The pattern in e's continued fraction
Written as a continued fraction, e is 2; 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8 — two ones, then the next even number, for ever. Euler found the pattern and proved it with a differential equation. A proof from 2006 needs only three integrals, each of which turns out to be exactly the error of one of e's own convergents.
The length the derivative never sees
The Cantor function climbs from 0 to 1 with a slope of zero almost everywhere, so the formula ∫√(1 + f′²) dx says its graph has length 1 — the length of a flat line. The inscribed polygons say 2. Mix it half and half with the diagonal and the length becomes exactly the golden ratio, while the formula still reports only the part the slope can see.
A whole number split into two that are not
Run a ribbon round a closed loop and its two edges link a whole number of times. That number is shared between two quantities that are nothing like whole numbers: how far the ribbon twists about its core, and how far the core coils about itself. Bend the loop and the twist and the coiling trade continuously, to three decimal places, while their sum stays fixed — the arithmetic behind a coiled telephone cord and a supercoiled loop of DNA.
Counting targets by their holes
The Euler characteristic adds up like an area: glue two shapes together and it is the sum of theirs minus that of their overlap. So it can be used to measure, and measuring with it does something no area can. A field of sensors that each report only how many targets they detect — not which, not where — can have its readings added up, weighted by the Euler characteristics of the regions where the count is high, and the answer is exactly the number of targets.
An area measured by walking round it
A surveyor's instrument from 1854 measures the area of any shape on a map by having its pointer steered once round the boundary; a small wheel rolls and slides, and its reading, times the length of one arm, is the area. Nothing touches the inside. The reason is that area can be written as an integral over the boundary — Green's theorem — and the instrument is that integral built in brass. Walk a curve that crosses itself and the same integral counts some regions twice.
No integrand sits on the border
The area under 1/x out to infinity is infinite and the area under 1/x² is finite, so it is natural to look for the dividing line. There is none. Divide any divergent integrand by its own running integral and it still diverges, more slowly; divide any convergent one by the square root of its tail and it still converges, more slowly. Paul du Bois-Reymond proved in 1873 that no single integrand can separate the two families — and the slow ones are slow beyond imagining.
Counting targets when the sensors make mistakes
Add up the Euler characteristics of the regions where sensor counts are high and the answer is exactly the number of targets — until one sensor misreads. Each wrong reading is a new piece or a new hole, so the error grows with the number of sensors instead of averaging away. Smooth the readings and count each piece and hole by how long it survives as the threshold is lowered, and the count comes back exactly, with half the readings wrong, at the price of having to know how small a real feature can be.
Only one curve through the factorials bends the right way
Infinitely many smooth curves pass through 1, 1, 2, 6, 24, 120, and some of them even obey the factorial's own rule, f(x + 1) = (x + 1)·f(x), at every x. Ask that the logarithm of the curve bend upwards everywhere and exactly one survives — Euler's integral. A wiggle of any size breaks the condition somewhere, the smaller the wiggle the further out, and the proof that nothing else survives is a squeeze that computes √π along the way.
The sine of a figure eight
Walk round a circle and read the height against the distance walked, and the reading is the sine. Do the same on Bernoulli's figure eight and the reading is a new function, with an arc integral that has t⁴ where the circle's has t², a length Gauss found inside an average, two periods instead of one — and exactly the same list of equal divisions that compass and straightedge can draw.
Named alongside it
The objects these essays reach for when they reach for this one.
DerivativeLogarithmAreaConvergence ratee, the numberLimitPiApproximationFundamental theoremHarmonic seriesMonte CarloSampling