My Relations, Your Relations

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Let's start with the thesis.
We are all related.
Here is some (hopefully straightforward) proof, using math.

But first, a couple of things you need to know, and believe.
- the two times table
- a crude sense of the world's population growth over the last few thousand years.

The 2 times table is easy with single digits. 2 x 8 =16 sort of thing.
It is not that difficult with more digits, 2 x 512 = 1024
You get the idea.

Next, World population growth over the last 2000 years - say since the time of the historical Jesus - may be seen at a chart here...
http://www.ecology.com/population-estimates-year-2050/
For our purposes, a key figure is that in year 1250 AD,  the high estimate for the world's population is 450 million people. And that figure has been growing steadily ever since. Now we are almost at 7.5 billion folks, or 7,500 million people.

OK, so much for background.
Now you need to consider your own ancestral lines. In numerical terms.
1 generation back - you had 2 parents.
2 generations back you had 4 grandparents (typically).
3 generations back 8 great grandparents.
4 generations back 16 great great grandparents.
The pattern may be obvious.
The number of direct lineal ancestors grows x 2 with each preceding generation (going back in time).

Now a generation is usually 25 years or less.
So if we imagine going back 750 years about, to 1235 AD, that would be going back 30 generations.

Now all we have to do is multiply 2 30 times to find out how many direct lineal ancestors that would be. 2 x 2 x 2 x2 x2 x2 x2 etc. 30 times.
Fortunately most smart phones have a calculator that can do this with a few keystrokes.
Drum roll...
And the answer is 1,073,741,824.
That's 1 billion, 73 million, 741 thousand, 824    great to the 28th ancestors.

Hold on a sec.
This does not compute.
How could you have over a billion (1,000 million) ancestors in the year 1250, if the total number of people in the world was only 450 million?
Say that is a good question.
Well, the answer has something to do with a paradox labelled the Pedigree Collapse.
In turn the answer is inbreeding. People mating with their own ancestral line.

But the core idea is still alive.
Going 30 generations back, I too, would conceptually have a billion direct ancestors.
And if there were only 450 million people around then, the overwhelming proportion of them  were likely related to me.
And without putting too fine a point on it, these are the same 450 million ancestors from that time who are your direct lineal ancestors.

Kinda mind blowing.
The take-away:
We're all related.