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If you know the rate that carbon-14 decays at, and how much of the carbon in a shroud, iceman or piece of old wood or bone is radioactive, you can work out how long ago they stopped breathing or photosynthesising. We know that on average it takes an atom of carbon-14 a little over 8,000 years to decay to nitrogen (although you never know when an individual atom is going to decay — it's completely random). But the value that's used to calculate the age of an object isn't an absolute figure, it's a statistical term called half-life.
We even know that in a gram of carbon, 14 carbon-14 atoms turn into nitrogen every minute. The half-life of a radioactive isotope is the amount of time it takes for half of the atoms in a sample to decay. That means that no matter how many carbon-14 atoms were present when something died, after 5,730 years only half of them are left — the rest have decayed to nitrogen.
Read more on calibration and accuracy of radiocarbon dating.
Different cultures around the world record time in different fashions.
It's not that the radioactive carbon in air or food doesn't decay, it does.