(SciShow intro music plays)

Michael: If you want to know how many carbon atoms there are in your body, you can write down eight.
Next, write down twenty-six zero's.
That is, eighty trillion trillion.
But about eighty trillion of those eighty trillion trillion are super-ultra-carbon atoms, an isotope of carbon known as carbon-14. These atoms are important because we can use them figure out how old something is, a process known as radiocarbon dating, as long as it's dead and less than 50,000 years old.
Normally, carbon has six protons plus six, maybe seven neutrons. Carbon-14, with eight neutrons, can form high in the Earth's atmosphere when high powered cosmic rays start a reaction that ends up giving an extra neutron to an atom of nitrogen. This neutron-infused nitrogen isn't stable, so it quickly dumps a proton, turning it into more-stable carbon-14. Now, all life on Earth needs carbon, but it isn't picky about what kind of carbon, so organisms happily take up carbon-14 to make the compounds they need. When plants turn carbon dioxide into glucose, for example, some of that CO2 contains carbon-14, and eventually, that carbon makes it all the way up the food chain, which is why all living things have some carbon-14 in their molecules.
But: that carbon doesn't stick around forever, because it's not completely stable. At some point, it undergoes a process known as beta decay, which basically -- it loses an electron, turns one of its neutrons into a proton, and becomes nitrogen-14. And luckily for scientists who specialize in studying old biological material, like archaeologists, this decay process happens at a fixed rate.
We know that it takes about 5700 years for half of any given sample of carbon-14 to decay into nitrogen-14. This timespan is known as its half-life.
So after one half-life, you'll have half of the C-14 that you started with. After two half-lives, you'll have one quarter. After three, you'll have an eighth, and so on. This is useful for dating dead stuff because while living things are constantly taking in carbon about as fast as it decays inside them, after death, there's no new carbon coming in. So when the C-14 decays inside the dead organism, the proportion decreases. More importantly, it decreases at the same rate no matter what the dead thing is.
So let's say you find an old piece of a dead tree. Since we know that about a trillionth of the carbon atoms in living things are carbon-14, all you have to do is measure how much C-14 the dead thing has and do some calculations.
Of course, figuring out how much C-14 a sample has turns out to be a little bit complicated. The first scientist to try it was American chemist Willard Libby in the 1940's, who used a Geiger counter to measure how many beta particles a sample spat out over time. He could then calculate the proportion of carbon-14 and figure out the age of the sample. Libby ended up winning the Nobel Prize in chemistry for figuring this out.
But since the late 1970's, scientists have been using a better method, called accelerator mass spectrometry, or AMS. In this rather long and expensive process, a sample is bombarded by ions, shot through a particle accelerator, and finally separated by magnets to isolate individual carbon atoms that can then by measured.
AMS is a lot more accurate than the beta decay method, but it's range is only about 50,000 years, because at that point, there's so little C-14 left that we can't reliably detect it. So to date samples that are older than that, like fossils, scientists track other elements, like uranium or potassium, which have half-lives in the millions of years.
But there are still other problems. It turns out that the amount of C-14 in the atmosphere has changed occasionally over time. Several decades worth of nuclear bomb tests in the 20th century, for example, managed to create a big spike in the amount of C-14 in the atmosphere. But at the same time, all of the carbon being vaulted into our atmosphere by burning fossil fuels may actually be diluting the concentrations of the isotope. As a result, scientists often have to combine radiocarbon dating with other methods, like tree-ring dating to corroborate their findings.
I mean, you probably already know this, but dating is hard. Especially when what you're dating is dead.