How does carbon dating work?
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Any radioactive isotope has a half life (5730 years in carbin 14). For a given quantity of that isotope, after one half life, half of that isotope would have decayed into its children. In the case of carbon 14, it would have beta decayed, resulting in nitrogen 14 (normal nitrogen) and an electron. We know how much carbon 14 there is in relation to carbon 12 (normal carbon) here on Earth, and any sample of (non isolated) carbon you take randomly should contain that same ratio of carbon 12 to carbon 14.
All living things ingest carbon in some form, so naturally something that’s constantly ingesting carbon from the environment, should have the same ratio of carbon 12 to carbon 14 in their bodies as that environment. Once that organism dies, it stops taking in carbon, so that carbon is now isolated. After 5730 years, half of the carbon 14 in that organism has decayed, so the ratio of carbon 12 to carbon 14 should be different, showing half as much carbon 14 as in the environment, so we know that the organism died 5730 years ago. Different ratios allow us to find different amounts of time, but after about 10 half lives or so, the ratio becomes too hard to measure, but then you can often use something of a longer half life like uranium.
The same can also be applied to say pottery. The clay out in the environment constantly has things dying in it, introducing carbon, until a human takes the clay out of the environment and fires it, isolating that carbon and allowing us to figure out when it was made.
Half lives are also not a fundamental law of how radioactivity works, but rather a result of statistics. A radioactive atom, after one half life, has a 50% of decaying. It isn’t communicating with all of its neighbors to work out which atoms decay when, there’s just so many atoms that after one half life, half of them have decayed, but if it’s off by a couple hundred atoms, no one will notice among the billions of atoms in the sample you’re taking. The sample size is so large, that we can always assume it behaves that way.
The formula:
Remaining isotope = initial isotope * e^(-[ln2/half life] * time)
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