To state the absolutely obvious, every person living today is directly descended from the very first cells that lived billions of years ago. However, to make the time scale easier to grasp, let’s just state that every human alive today is necessarily directly descended from someone who was alive a thousand years ago, when, according to one estimate, the human population of the world was about 265 million.
Of those 265 million, we can assume not everyone managed to produce offspring, who, in turn, produced offspring, over forty-one generations (assuming four generations per century) between the years 1000 and the year 2000. Families were wiped about by wars, plagues, other natural disasters, and some people would have been celibate or infertile. However, a certain, evidently substantial fraction of those 265 million did manage to breed successfully, and today they have an estimated seven billion descendants.
From the perspective of a person concerned with genealogy, the important question is: which of the 265 million humans, who were alive a thousand years ago, are my ancestors? If you make a simple but fallacious calculation, you might be led to the conclusion that you are descended from them all. A person born in the the year 2000 has two parents who were born in 1975, four grandparents who were born in 1950, eight great-grandparents born in 1925, and sixteen great-great-grandparents who were born in 1900. Taking it back to the year 1000 would mean 241 ancestors, which is 219,902,325,552,
or more than thirty times the present human population of the world. We could not possibly be not descended from two-hundred, twenty billion different people. Hence, many of our ancestors must, in fact, sit in more than once place on our family trees.
For example, suppose two of your great-grandparents, not necessarily a man and wife, were first cousins (let’s say your mother’s grandmother and your father’s grandmother). That would mean that two of your great-great-grandparents were siblings, and those siblings would, by definition, have the same parents. So instead of thirty-two different descendants in that generation, you would only have thirty. That’s the process that whittles down the number of our actual ancestors from the impossible billions that the arithmetic offers. In the villages where most of our ancestors probably lived, in times when people tended to live and breed close to the places where they were born, the pool of available spouses could not have been very large, so many of our ancestors were related to each other. Instead of fanning out more or less infinitely, our lines of direct descent overlap and tangle. Not only are we directly descended from someone who lived a thousand years ago, we are descended from that person (and from many of our other actual ancestors in that generation) along multiple paths.
Some people I know are very proud that they can trace their ancestry back several centuries, but, obviously, that usually means they can name one person out of the many from whom they are descended in each previous generation. I would be interested in my own genealogy only if I could discover more than a name and place of residence. If I knew what one of my ancestors did with her life in the twelfth century, that might be interesting.