The stubborn arithmetic of cousins

When you get involved in genealogy, sooner or later, after days, weeks, months or years of patient research (depending on how lucky and obstinate you are) you discover that your best friend, your boss, the old lady next door, your favourite writer or singer, your loyal enemy or the latest serial killer, all are to some degree your cousin. 
Actually you knew this had to be true, in theory at least. All humans have common ancestors, somewhere in the past. But it's a completely different story to be able to identify and name them, and figure how far ago that was. It might be quite easy if you and I belong to families who have kept their genealogical records over centuries, and can proudly show their lineage tracing up to Charlemagne. No big deal, actually, since anyone tracing her ascendance thus far is likely to be in the same case. According to the genealogical database Roglo, the identified descendants of Charlemagne are more than 1,500,000, more than 20% of the database of about 7,500,000 people. But if your ancestors are, like mine, obscure and illiterate peasants, we are likely to stumble upon the lack of documents beyond the few last centuries of church and civil registries, lucky enough if we can reach as far in the past as around 1600 for some more or less reliable information about a handful of ancestors. This seems quite far away, but it's only about a dozen of generations, which means a few thousands of people. 
So, how far have you to go to find common ancestors with your best friend? Let's have a look at the harsh reality of numbers. The number of your ancestors at generation n is 2^n. You have two parents, four grand-parents, and so on. Counting thirty years for each generation (give or take a few), ten generations span three centuries. At the tenth generation you count 2^10 ancestors, which is about one thousand. Being born in the 1950's means I had around one thousand ancestors living around 1650 (under Louis XIV). Three centuries and ten generations before, it was one million around 1350 (under Jean II le Bon), and the same stubborn arithmetic leads to one billion ancestors around 1050 (under Henri Ier). Like in the famous wheat and chessboard problem, the exponential law makes figures explode beyond control at some point. Except that no one can have as many ancestors as one billion in 1050, because the entire world population by that time was less than half this figure. The curve of my theoretical number of ancestors crosses the curve of world population somewhere at the beginning of the 12th century. 
What does that mean? Any of my ancestors before 1200 is likely to be my ancestor by so many different paths, and is probably your ancestor as well. People tracing their genealogy thus far in the past know they indeed are all cousins. And all of royal ascendance, of course, since along those millions of different paths, it's highly probable to find a king or queen. But whether you know it or not, the figures are relentless. You who read those lines, you are very probably my cousin, but we'll also probably never know precisely either at which degree or the name and epoch of our last common ancestor. This is both a fascinating and frustrating conclusion.