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Basically, we have learned through observation. When people first began working with radioactive materials, scientists didn't understand radioactive decay, and reports of illness were scattered. 

As the use of radioactive materials and reports of illness became more frequent, scientists began to notice patterns in the illnesses. People working with radioactive materials and x-rays developed particular types of uncommon medical conditions. For example, scientists recognized as early at 1910 that radiation caused skin cancer. Scientists began to keep track of the health effects, and soon set up careful scientific studies of groups of people who had been exposed. 

Among the best known long-term studies are those of Japanese atomic bomb blast survivors, other populations exposed to nuclear testing fallout (for example, natives of the Marshall Islands), and uranium miners.