All living things contain carbon, and there are different types of carbon.
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One is called carbon-12, which is the “normal” type of carbon.
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Another is carbon-14, which is a radioactive type of carbon (called an isotope), which decays over time into a different element.
When a plant or animal is alive, it always has roughly the same amount of carbon-14. However, when it dies, it stops taking in new carbon-14 (or any carbon at all). Over time, its amount of carbon-14 decreases. It turns out that it decreases exponentially.
To measure the rate of exponential decay, instead of using a percentage, we use its half-life, which is the time it takes for the amount to decay to half its initial amount.
Carbon-14 has a half-life of 5730 years. This means that:
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After 5730 years, half of the carbon-14 is gone.
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After another 5730 years, half of what’s left is gone (so 25% of the initial amount is left).
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Then half again, and again, and so on.
This means if we start with 100% of the carbon-14, then the percent of carbon-14 remaining after \(t\) years is given by,
\begin{gather*}
P(t) = 100 \brac{\frac{1}{2}}^{\frac{t}{5730}}
\end{gather*}
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\(\frac{t}{5730}\) represents the number of half-lives in \(t\) years. For example, if \(t=5730\text{,}\) that is \(\frac{5730}{5730}=1\) half-lives.
The idea behind carbon dating is that if we measure how much carbon-14 the dead organism has, and also know how much it had when alive, we can estimate how long the carbon-14 has been decaying for, and therefore how long ago it died.
