Generator

eˣ and its tangent lines

A generator in the analysis library, called 26 times across 10 essays. Below: what it draws at its defaults and at each mode an essay asks for, what it checks while drawing, and everywhere it is used.

exponential is one function. Everything below came out of it during this build, at parameters taken from the essays rather than invented for this page — so a figure here is the same figure a reader meets in an essay, and if the generator changes, this page changes with it.

At its defaults

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

`show: "series"`

The series for e, 9 terms. Each term of 1 + 1/1! + 1/2! + … added in turn. The factorials in the denominators make the tail vanish very fast.

`show: "stirling"`

The factorial, and the formula with e inside it. A table of n, n factorial, the value of (n/e)ⁿ√(2πn), their ratio and n times the excess, for n up to 11; the ratio falls from 1.0844 to 1.007602.

`show: "logarea"`

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

`show: "compound"`

Compounding toward e. Interest compounded 1, 2, 4, 12 and 365 times a year, each staircase ending a little higher than the last, and the last of them within 0.0037 of e.

`show: "field"`

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.

What it checks while it draws

Collected by running the family and recording what it asserted, not written here. The count is how many separate times the claim was put to the test while these drawings were made.

so over 0.7702 exactly the factor is 2 ×4
a base that is not e does not have slope equal to height ×1
and at base e the slope is the height ×1
and is what the closed form gives over that interval ×1
and the gap left is smaller than the last term added ×1
and the limit of the staircase is e itself ×1
and the place where that area is exactly 1 is e ×1
and the ratio falls towards 1 every step ×1
both factors are above 1 and their product fits ×1
both intervals lie inside the integrated range ×1
each solution starts at a positive height of at most 4 ×1
every partial sum is below e ×1
more splits is closer to e ×1
no number of splits reaches e ×1
so by the end it is under one per cent ×1
so the two areas add to the area of the product ×1
the approximation is always an underestimate here ×1
the area is taken from 1 up to somewhere between 1 and 12 ×1
the area under 1/x from 1 is the logarithm ×1
the base is a positive number of at most 20 ×1
the curve integrated from the equation is the exponential ×1
the factor is the same wherever along the curve the interval starts ×1
the gap closes like one twelfth of 1/n, measured at the last row ×1
the growth rate is between a twentieth and three in size ×1
the interval over which the height changes by that factor is inside the window drawn ×1
the region from a to ab has the same area as the region from 1 to b ×1
the region runs from 1 to somewhere between 1 and 12 ×1
the table runs to between 4 and 20 ×1
the table shown gets e right to three decimals ×1
the tangent drawn is the curve's own slope ×1
the two factors are both above 1 and their product is at most 12 ×1
the view is one the family draws ×1
the year is split between 2 and 6 ways, each a whole number up to a million ×1

Where it is called

Changing this generator changes every figure on this list. That is what makes the list worth publishing rather than keeping in a check script.

Number

A tail too small to be a whole number

If e were a fraction with denominator q, then q! times e would be a whole number. It splits into a whole part and a tail, the tail is squeezed strictly between nothing and one, and there is no whole number there.

Computation

Nine thousand four hundred and eight

There are four Latin squares of order four once the first row and column are fixed, fifty-six of order five, and nine thousand four hundred and eight of order six. The exact answer is known for eleven orders and for no more — and yet a half-finished square can always be finished.

Analysis

One point's worth of information

A Taylor series claims that everything a function does, everywhere, is encoded in its behaviour at a single point. That claim is extraordinary, it is often true, and the cases where it fails are the interesting ones.

Analysis

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 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 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 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 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 slope of a single point

A slope needs two points. A derivative is the slope at one. The construction that bridges the gap is a sequence of secants, and the whole difficulty of calculus is in what "the limit of that sequence" is allowed to mean.

Analysis

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 whole library · What the figures prove