Definition
Radioactivity is the property of an unstable atomic nucleus that releases radiation on its own and settles into a more stable state, usually by turning into a different atom. Half-life is the fixed amount of time it takes for exactly half of a batch of those unstable atoms to decay.
Why do some materials give off energy on their own?
Picture a Jenga tower stacked so high and so shakily that even a tiny breeze could bring it down. Among the atoms that make up everything around us, some have a nucleus at the center that is too heavy or out of balance, so it wobbles on the edge.
An unstable nucleus can't just hold still. It flings out a small particle or throws off energy in the form of powerful light. The waves or particles that shoot out are called radiation, and the ability to give off this energy by itself is called radioactivity.
Once the atom has thrown that energy away, it settles into a more stable state, usually as a different kind of atom. If the new atom is unstable too, it keeps decaying through several steps until it finally becomes stable. It's just like a shaky Jenga tower: a few blocks fall off, and the tower finally settles into a low, solid, safe stack.
A regular timetable of cutting in half
Imagine you drop 100 popcorn kernels into a hot frying pan. Nobody can tell which kernel will pop first after one second. But with lots of kernels in the pan, the time it takes for half of them, 50, to pop is amazingly consistent.
Radioactive atoms behave a lot like those kernels. There's one difference. Popcorn pops faster as it heats up, but an atom's chance of decaying stays the same no matter how much time passes. You can't predict exactly when one particular atom will change. But in a huge clump of atoms, the time it takes for half to decay is as precise as a math formula.
The time it takes for a radioactive substance to shrink to exactly half is called its half-life. Say you have 100 g of a radioactive element with a half-life of 10 years. After 10 years, only 50 g is still the original element. After 20 years, 25 g is left. The rest doesn't vanish. It has turned into another element. Doubling the time doesn't bring you to 0 g. The amount just keeps getting cut in half, step by step.
A little more precisely
The half-life of a nucleus barely changes whether you boil the material in a furnace or freeze it solid in Antarctic ice. Changes inside the nucleus are a built-in property that heat, high pressure, and chemical reactions can hardly alter. Some kinds of decay, such as electron capture, involve the electrons around the nucleus, so they can shift very slightly (around 1%) depending on the chemical environment.
Scientists use this nearly unshakable natural clock to uncover puzzles from the deep past. The best-known example is radiocarbon dating. Living things absorb radioactive carbon while they're alive, and after death it declines at a steady rate (half as much after each half-life). Measuring what's left lets scientists work backward.
Carbon can date ancient sites tens of thousands of years old. Other radioactive elements with much longer half-lives, like uranium, can reveal when the Earth was born. Beyond that, people safely put these natural rules to work across industry, from managing nuclear power fuel to non-destructive testing that checks for flaws inside thick steel plates.
Common misconceptions
A radioactive substance with a 10-year half-life disappears completely after 20 years.
After two half-lives, it doesn't hit zero. Half of the remaining 50%, which is 25%, is still there. It keeps getting cut in half forever, heading toward zero without ever quite arriving. And atoms that have decayed don't vanish. They stay around as a different element.
Where you meet it
Wood or bones dug up from ancient sites thousands of years old are dated using carbon's half-life.
Airplane wings and building steel frames are checked for tiny cracks using radiation imaging, without taking anything apart.
In one sentence
Radioactivity is how unstable atoms shed energy to become stable, and half-life is the steady amount of time it takes for a substance to shrink by half.