How do people date things from the past?

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I was in History class the other day and I asked how people are able to get “accurate” dates for how old certain objects are. He said something along the lines of there are certain elements in materials and they decay overtime, half life this so we know roughly how long ago it was made.

I’m a mathematics major and am proficient in physics. I understand the concept of exponential decay and half-life’s. My questions is how do we know how much of a material we are measuring there initially was? To me, without knowing that, we could say that something was made whenever we want it to be made. Clearly I’m missing something but I can’t quite figure it out.

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Along with the radio dating what archaeologists also note what coins are found in digs so you know the date range of objects that are found on that dig. After a few digs they can then say certain types of pottery are from a certain date when found so can then associate other items found at the being from the same general time period.

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