Carbon dating explained
When an organism dies it ceases to replenish carbon in its tissues and the decay of carbon 14 to nitrogen 14 changes the ratio of carbon 12 to carbon 14.
Experts can compare the ratio of carbon 12 to carbon 14 in dead material to the ratio when the organism was alive to estimate the date of its death.
On the other hand, if tons of half-lives have passed, there is almost none of the sample carbon 14 left, and it is really hard to measure accurately how much is left.
Since physics can't predict exactly when a given atom will decay, we rely on statistical methods in dealing with radioactivity, and while this is an excellent method for a bazillion atoms, it fails when we don't have good sample sizes.
So in the real world, looking at a sample like say a bone dug up by an archaeologist, how do we know how much carbon 14 we started with? This process is constantly occurring, and has been for a very long time, so there is a fairly constant ratio of carbon 14 atoms to carbon 12 atoms in the atmosphere.We have devices to measure the radioactivity of a sample, and the ratio described above translates into a rate of 15.6 decays/min per gram of carbon in a living sample.And if you play with the exponential decay equations, you can come up with the nice formula (1/2)=(current decay rate)/(initial decay rate), where n is the number of half lives that have passed.Now living plants 'breathe' CO indiscriminately (they don't care about isotopes one way or the other), and so (while they are living) they have the same ratio of carbon 14 in them as the atmosphere.Animals, including humans, consume plants a lot (and animals that consume plants), and thus they also tend to have the same ratio of carbon 14 to carbon 12 atoms.