How many ancestors do you have? At the first level, we each have two ancestors, the same as everyone else except for Dolly. Most of us have us have four grandparents and eight greatgrandparents. Let's say that for every century we have three new generations. Going back to 1700, would require (for the older readers such as myself) about seven or eight generations. If we count the generations as our parents being the number 1 generation and our grandparents as the number 2 generation and so on, then we can express the number of ancestors at any level as the number 2 raised to the number of generations back that we are evaluating. When I say "raised to," I could substitute "multiplied by itself".
For example, to go back to 1700 would be about seven generations for me, so I would expect to have 2 multiplied by itself seven times. This is 128. Suppose we went back to the time of Charlemagne around the year 800, some 1200 years ago. At three generations per hundred years, there would be 36 generations back to then for someone who was recently born. The number of ancestors in a tree at that time or level would be 2 raised to (i.e., multiplied by itself) 36 times. My calculator tells me that this would be about 68,719,000,000. For Americans, that is almost 69 billion ancestors at the time. (I said Americans because, for some other people in the world, a billion is a different amount.)
How many people were in Europe at the time of Charlemagne? Maybe one million people. So how could I have so many ancestors, say 69 billion, when there were only one million people living then? Some people had to serve as an ancestor more than once. For example, on the average, each person living in 800 AD would be an ancestor 68,719 times. The chances are quite excellent that you are a descendant of any given person that you could name from that time, assuming that the person did leave descendants.
(There is a society for descendants of Charlemagne. To be a member, you need to have a proven line of descent. This is possible for many people. However, even if the odds are excellent that you are descended from Charlemagne, you cannot join the society on this basis alone.)
The convergence within any individual's family tree has to occur. No one can escape this. As one goes back, it is a necessity that your lines will converge, i.e., become the same. To give an example, I have a few known cases in the immediate past three centuries. Theobald Crisler and Rosina Gaar are in my tree by two different paths. George Utz and Barbara Maiers are in a couple of times. Convergence in a family tree has occurred any time that one can say he or she is their own cousin. As the mathematics above proves, each of us is (must be) his or her own cousin. It is not always clear just how or when it occurs, but it must occur.
We gratefully acknowledge the work of John Blankenbaker who published over 2,500 Germanna History Notes via the Germanna-L@rootsweb.com email list from 1997 to 2008. We are equally thankful to George Durman (Sgt. George) for hosting the list and republishing the notes via rootsweb.com.