Exponential Decay Calculator

What this calculator does

Exponential decay is the mirror of exponential growth. Instead of a thing growing by a fraction of itself, it shrinks by a fraction of itself, so it falls quickly at first and then more and more gently, forever approaching zero without quite reaching it. Radioactive material, a drug clearing your bloodstream, a hot drink cooling, a car losing value.

Give it any three of the four quantities, the starting amount, the decay rate, the time, and the final amount, and it finds the fourth. It runs right here in the browser.

Using the calculator

1. Fill in any three of the four: initial amount, decay rate, time, and final amount.
2. Leave the one you want to find blank.
3. Set a precision if needed, then press Calculate.

Enter the decay rate as a positive decimal, so a continuous rate of 10 percent goes in as 0.1. The tool solves for the empty box. Reset clears them all.

What exponential decay is

The pattern is the same runaway curve as growth, but pointed downhill. Each step removes a fixed percentage of what is currently there, not a fixed amount. So a lot disappears early, when there is a lot to lose, and then less and less as the amount dwindles.

That is why decay never quite finishes. Halve something, then halve the half, then halve that, and there is always a little left. This slow, tapering tail is the signature of decay, and it is why a small trace of a substance can linger long after most of it has gone.

The formula, and the four things it links

This tool uses the continuous decay model:

A = A₀ × e^{(−r × t)}

where A₀ is the starting amount, r is the continuous decay rate, t is the time, A is the amount left, and e is about 2.718. It is exactly the growth formula with a minus sign in the exponent, which is all that turns growth into decay. The four quantities are locked together, so any three give the fourth.

Leave one blank and it finds it

Run in any direction you need:

• Missing the final amount? It multiplies the start by e^{(−r × t)}.
• Missing the decay rate or the time? It uses the natural logarithm to unwind the exponent.
• Missing the starting amount? It scales the final amount back up.

So the one tool answers "how much will be left", "how fast is it decaying", and "how long until it drops to that".

A worked example

Start with 100, decaying at a continuous rate of 0.1 (10 percent), over 5 units of time. Leave the final amount blank.

A = 100 × e^{(−0.1 × 5)} = 100 × e^−0.5 = 100 × 0.6065 = 60.65. So a little over 60 remains after 5 units of time, a bit under two-fifths of it gone.

The friendlier way to state a decay rate

A continuous decay rate like 0.1 is precise but hard to feel. There is a far more intuitive way to say the same thing: give the half-life, the time it takes for the amount to fall by half. Radioactivity and medicine almost always use half-life rather than a decay rate, because "this halves every 8 hours" is easier to reason about than a rate constant. The two are just different descriptions of the same decay, and the half-life calculator works in those terms. For the opposite direction, things that grow, see the exponential growth calculator.

On decimals and precision

Decayed amounts can become very small, so this tool runs the arithmetic through Decimal.js for exact decimal work, with the precision dropdown controlling how many places it keeps, and it shows very small results in exponential notation so they stay readable. The answer is only ever as precise as the rate and time you enter.

Questions people ask

What is exponential decay?

A process where a fixed percentage of the current amount is lost each step, so it falls fast at first and then ever more slowly. It follows A = A₀ × e^{(−r × t)}.

Does it ever reach zero?

Not exactly. Each step removes a fraction of what is left, so there is always a tiny amount remaining, however long you wait, even as it becomes vanishingly small.

How does this relate to half-life?

Half-life is the same decay described by the time to halve rather than by a rate. It is often the more natural way to state radioactive or drug decay, and the half-life calculator uses it.

Can it find the rate or the time?

Yes. Fill in any three of the four quantities and leave the one you want blank. It solves for the missing one.

Where does exponential decay show up?

Radioactive decay, a medicine clearing the body, a hot object cooling, sound and light fading, and asset depreciation.

References

A note on where this comes from. Decay in which the rate of loss is proportional to the amount remaining follows the exponential function A = A₀e^−rt. In physics it is the radioactive decay law, established by Ernest Rutherford and Frederick Soddy in the early 1900s, who showed that radioactive substances decay at a rate proportional to the amount present. For further reading, see Exponential decay.

1. The continuous exponential decay model, A = A₀e^−rt, in which the rate of loss is proportional to the amount remaining.
2. Rutherford and Soddy's radioactive decay law (early 1900s), the physical origin of the exponential decay model.
3. Decimal.js, an arbitrary-precision decimal library, used here so decayed amounts stay exact.