# Exponentials and continuous scaling

Authored explanation; editorial review pending.

The exponential function e^u, often written exp(u), is the unique mathematical function whose derivative with respect to its argument equals the function value itself. Whenever a physical growth or decay rate is proportional to the amount already present, the resulting trajectory is an exponential.

A fundamental rule of dimensional physics is that the argument of any transcendental function, including the exponential, must be a dimensionless number. In Wien's radiation law exp(-beta nu / T), the product beta nu has dimensions of temperature, so dividing by temperature T yields a pure dimensionless ratio. In the Brownian spreading Gaussian exp(-x^2 / (4Dt)), x^2 has units of squared metres and 4Dt has units of (m^2/s)*s = m^2, ensuring the exponent is dimensionless.

Exponentials also arise naturally from the multiplication of independent probabilities across repeated random steps or subdivided spatial cells.

## Worked example

$$
f(x,t) = \frac{1}{\sqrt{4\pi Dt}} \exp\left(-\frac{x^2}{4Dt}\right)
$$

Density f of x and t equals one over square root of four pi D t times exponential of minus x squared over four D t.

At the center x = 0, the exponential factor exp(0) = 1. At one standard deviation x = sqrt(2Dt), the exponent is -1/2 and exp(-0.5) approx 0.6065. At x = sqrt(4Dt), the exponent is -1 and exp(-1) approx 0.367879, showing symmetric bell-shaped decay.

The argument inside an exponential function must always be a dimensionless pure number.