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mathematics, in pictures
Ladder

The exponential — the ladder

5 distinct arguments against one idea, from the one that introduces it to the one that assumes the rest.
  1. 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.

    rung 1 · analysis
  2. 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.

    rung 2 · analysis
  3. 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.

    rung 3 · analysis
  4. 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.

    rung 4 · analysis
  5. 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.

    rung 5 · analysis

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