Briefly, a radioactive isotope is a kind of atom which decays into a different kind of atom: for example. one called uranium-238 turns into one called lead-206. Because we know how long this takes to happen, we can think of the isotope as a radioactive clock. Radioactive clocks are rather like the water clocks and candle clocks that people used in the days before pendulum clocks were invented. A tank of water with a hole in the bottom will drain at a measurable rate. If the tank was filled at dawn, you can tell how much of the day has passed by measuring the present level of water. Same with a candle clock. The candle burns at a fixed rate, so you can tell how long it has been burning by measuring how much candle is left. In the case of a uranium-238 clock, we know that it takes 4.5 billion years for half the uranium-238 to decay to lead-206. This is called the 'half-life' of uranium-238. So, by measuring how much lead-206 there is in a rock, compared with the amount of uranium-238, you can calculate how long it is since there was no lead-206 and only uranium-238: how long, in other words, since the clock was 'zeroed'.
And when is the clock zeroed? Well, it only happens with igneous rocks, whose clocks are all zeroed at the moment when the molten rock hardens to become solid. It doesn't work with sedimentary rock, which has no such 'zero moment', and this is a pity because fossils are found only in sedimentary rocks. So we have to find igneous rocks close by sedimentary layers and use them as our clocks. For example. if a fossil is in a sediment with 120-million-year-old igneous rock above it and 130-million-year-old igneous rock below it, you know the fossil dates from somewhere between 120 million and 130 million years ago. That's how all the dates I mention in this chapter are arrived at. They are all approximate dates, not to be taken as too precise.
Uranium-238 is not the only radioactive isotope we can use as a clock. There are plenty of others, with a wonderfully wide spread of half-lives. For example, carbon-14 has a half-life of only 5,730 years, which makes it useful for archaeologists looking at human history. It is a beautiful fact that many of the different radioactive clocks have overlapping timescales, so we can use them to check up on each other. And they always agree.
The carbon-14 clock works in a different way from the others. It doesn't involve igneous rocks but uses the remains of living bodies themselves, for example old wood. It is one of the fastest of our radioactive clocks, but 5,730 years is still much longer than a human lifetime. so you might ask how we know it is the half-life of carbon-14, let alone how we know that 4.5 billon years is the half-life of uranium-238! The f answer is easy. We don't have to wait for half of the atoms to decay. We can measure the rate of decay of only a tiny fraction of the atoms, and work out the half-life (quarter-life, hundredth-life, etc.) from that.