Every essay — page 4
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Analysis
Limits, curves, and what happens when the going does not stop.
The sum that fits in one square
Half, then a quarter, then an eighth, forever. Adding infinitely many things sounds like it should give infinity, and the picture that says otherwise is a square with a corner left uncut.
The staircase that is not the diagonal
A staircase can be made to follow a quarter circle as closely as anyone likes. Its length is 2 at every stage and the arc's length is 1.5708, and no amount of refinement closes the gap — which is a fact about length rather than about staircases.
A curve with a corner at every point
Continuity means a curve can be drawn without lifting the pen. Differentiability means it has a tangent. The first was assumed to nearly imply the second until 1872, when Weierstrass exhibited a curve that is continuous everywhere and has a tangent nowhere — and it is a sum of cosines.
Almost none of it left, and still uncountably many
Remove the middle third of an interval, then the middle third of each piece left, and keep going. The lengths removed add to exactly the whole interval, so nothing measurable survives — and what survives can be paired off one for one with every point of the interval that was started with.
Where the coefficients come from
The recipe for a square wave has a four over pi in front and a one over three on the second term, and the essay that built a square wave from sines used them without saying where they came from. They come from multiplying by one harmonic and taking the area.
When the period grows without bound
A repeating signal has a spectrum of separate lines. Stretch the gap between repeats and the lines crowd together while the curve they sit on stays exactly where it is — and at infinite period the lines are gone and the curve is the whole answer.
The corners go first
Fourier was not decomposing waves for the pleasure of it. He was solving the flow of heat, and the whole apparatus exists because each harmonic fades at a rate set by the square of its frequency — which is why a sharp profile smooths instantly and why the flow cannot be run backwards.
A map that shrinks everything
One extra hypothesis — that every distance is shortened by at least a fixed factor — turns the existence of a fixed point into its uniqueness, an algorithm for finding it, and a bound on the error after any number of steps.
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 flat map that fits closest
A derivative is usually met as a number, which works because a line through a point is described by one. In more than one dimension the object that plays the same role is a linear map, and the number was always a one-by-one instance of it.
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.
The curve of the average, and the average of the curve
A curve that bends upwards keeps every one of its chords above it. That single fact, applied to a weighted average instead of a midpoint, turns into an inequality that produces the arithmetic–geometric mean inequality, Cauchy–Schwarz and the entropy bound as special cases.
A limit that forgets to be continuous
Every one of the functions x, x², x³, … is as smooth as anything could be, and every column of the picture settles down. What they settle on has a jump in it — and the quantity that sees the difference is the largest gap anywhere, which is a number about the whole graph rather than about any point of it.
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 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.](/f/the-flow-of-a-linear-equation-and-the-matrix-that-runs-it-15ef5202.svg)
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.
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.
No interval in it, and length to spare
The middle-thirds set has no length because the removed pieces add to one. Remove shrinking middles instead and they add to a half — leaving a set that still contains no interval anywhere, and still has half the length it started with.
Covering a set from outside
To say how long a set is, cover it with intervals and add their lengths, then take the smallest total any covering achieves. That definition is short, obviously right for an interval, and gives the rationals a length of nothing.
Which functions can be added up
Riemann's integral works when the upper and lower sums close on each other. The exact condition for that, found once measure existed to state it in, is that the points where the function jumps have measure zero — which some nowhere dense sets fail.
A set that has no size at all
Slide the unit interval along itself by every rational and the points fall into classes. Choose one point from each and the resulting set has no length — not zero, not positive, none: countably many disjoint copies of it would have total length nought or infinity, and the union needs something in between.
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.
A line under every point
The chord above the curve is one definition of convexity. There is a second — a line under the curve at every point, staying under everywhere — and it is the one that turns a statement about a derivative at a point into a statement about the whole function.
The function seen from its tangents
A convex function is the upper envelope of its own tangent lines, so it can be described by giving, for each slope, how far the line of that slope has to be pushed down. That description is a second function, and applying the construction twice returns the original.