Why exponentials are everywhere

There is a single principle behind all exponential phenomena: the variation proportional to the current value.

Observation — Why exponentials are everywhere

Every time a quantity $y$ grows (or decays) in a way proportional to its current value, the law that describes it is exponential. A few examples:

• Radioactive decay: $N(t) = N_0 \cdot \bigl(\tfrac{1}{2}\bigr)^{t/\tau}$, with $\tau$ the “half-life”.
• Compound interest: a capital $C_0$ that yields the rate $r$ per year becomes $C(t) = C_0\,(1+r)^t$ after $t$ years.
• Population growth or epidemics (initial phase): the number of individuals (or of infected people) doubles at constant intervals.

In the following pages we look more closely at these models, with the data and the numerical calculations that make them concrete.

Links

Topics: Exponential function
Concepts: Exponential growth · Exponential decay · Compound interest
Functions: Exponential function
Skills: Modelling