Jim Wilson, of Dunedin, asks :-
As I trace my family tree I was aware that my ancestors doubled with every generation (two parents, four grandparents etc) I then came to the realisation that the number of ancestors would become a very large number relatively quickly. If I assume that each woman had a child when they were 20 years old. then, working backwards, the number of my ancestors far exceeds the "official" world population of about 200,000 in the year 7BC.
While I can understand that disease, war, famine and natural disaster must have accounted for many, I can not understand the significant difference between the two figures?
Bram Evans, a retired mathematician from the University of Otago, responded.
On the face of it, by the 10th generation before ours, each of us has 2 multiplied by itself 10 times = 1024 ancestors. For the 20th generation before us, squaring 1024 gives more than a million. This takes us back perhaps 500 years, and if our ancestors lived in a well-populated region their number still looks plausible.
However, in the 30th generation (multiplying our ancestor tally by another factor of 1024), we apparently have over a billion forebears – exceeding Earth's total population in medieval times.
But these billion-plus ancestors in the 30th generation were not all different people! (Nor were they all necessarily contemporaries, though this is only a side issue.) Increasingly, as we trace the generations back, to reach the theoretical (power-of-two) number we must count some of our ancestors more than once, because we are their descendants via two or more paths. For example: when first cousins marry, their children usually (cf. "double cousins") have not eight but six great-grandparents, of whom one pair must be counted twice.
In some present-day societies (such as ours), marriage partners are typically related so distantly that their family connections are unknown to them. But if we had a complete family tree covering, say, 60 generations, we would find many past intersections of its branches. Following the tree back, we eventually reach a stage when, in a given region, the number of our distinct ancestors is limited by the total number of people living there at that time. Beyond this point, kinships between marriage partners tend to become closer. Close-cousin marriages (with associated inbreeding risks) were historically very common everywhere, and are still prevalent in some societies.
Sometimes it is helpful to think of distant forebears as communities rather than individuals. Topics to google include ancestor paradox; double cousins; and pedigree collapse.