y = eˣ. Exponential.
It starts
with one.
At x = 0, the instrument reads 1. A quiet beginning. Increase x by one and the value becomes 2.718281828… Increase it again. The same multiplier returns. Something small is gathering scale.
to the value itself.
The value of the system at a given x.
Euler’s number. Approximately 2.71828. Its decimal expansion never ends or repeats.
The input. It might represent scaled time, distance, or simply a number.
Same beginning.
Different worlds.
Two systems begin at one. One adds the same amount at every step. The other multiplies. At first, their paths almost touch. Then the gap opens. The rule never changes. Only its consequences do.
Both start at 1.
Both have slope 1 at x = 0.
Only one keeps that slope.
Differentiate e to the x and the same expression looks back at you. Its rate of change is exactly its current value. The larger it becomes, the faster it grows. This is the defining property of the curve.
One.
Two.
Sixty-four.
Watch the field fill. Every doubling takes the same step in x: ln(2), approximately 0.693. One becomes two. Two becomes four. Six doublings produce sixty-four. Equal intervals. Ever larger additions.
A counting model: each light represents one unit of y.
x = 4.159
Even disappearance
has a pattern.
Change the exponent to −x. Now the rate follows what remains. Radioactive decay obeys this form on average: N(t) = N₀e⁻λᵗ. A single atom’s decay is unpredictable. A large population reveals the law.
λ is the decay constant. λt is dimensionless.
Energy.
Unfolding.
Particles. Atoms. Possibility.
The invisible leaves a trace.
In a cloud chamber, charged particles leave trails of condensed droplets. Matter writes a fleeting record of its passage. A line. A collision. A sudden branching. The event is gone. The evidence remains.
And then,
everything.
The equation has no upper limit. Physical systems do: fuel, space, time. Somewhere between a simple rule and the limits of reality, the experiment continues.
Back to the beginning