Exponential Decay vs Linear Decrease

What Linear Means

A linear change subtracts or adds the same amount every step. A tank draining at 10 litres per hour loses 10 in the first hour and 10 in the tenth. The rate is constant and the total change is simply rate multiplied by time.

What Exponential Means

An exponential change applies a constant ratio rather than a constant amount. Radioactive decay removes half of what remains each step, so the absolute amount lost gets smaller while the proportion lost stays constant.

Why the Distinction Matters

Linear decay always reaches zero in a predictable time, so 10 litres per hour from 100 litres empties in exactly 10 hours. Exponential decay never reaches zero. Planning that assumes a linear drain will run out of capacity far sooner than reality.

Reading a Decay Curve

On a graph of amount against time, exponential decay starts steep and flattens. The curve is the mirror image of exponential growth. The half life is the time taken to move from any point to the position half way toward zero.

The Half Life Is Scale Invariant

This is the property that makes half life so useful. From 100 to 50 takes one half life, and from 0.01 to 0.005 takes exactly the same time. The rule applies unchanged at every magnitude, which is why one number is enough to describe the whole process.

Everyday Exponential Examples

Radioactive decay, bacterial growth, the spread of a virus, the cooling of an object, the charge in a capacitor, and the way a drug leaves the bloodstream all follow exponential patterns. In each case a fixed proportion changes per unit time rather than a fixed amount.