#47: The Truth About Fractions

47.1  How much mathematics do you need to do genealogy? It’s up to you: a lot, a little, or none at all. You can figure out who your 3G Grandfather and 5th Cousin Once Removed are with any “cyphering” whatsoever. When it comes to deciding which of them is more closely related to you…or which of your “numbered” cousins is as closely related to you as those 2 are to themselves…well, English majors can do it, but a little math makes it a lot easier.

47.2  I hope I can explain it to you “without tears”…altho feel free to cry if you must…let it out, it’s good for you 😉 😉  But the fundamental thing you have to know about genealogical math is that it deals only with “powers of 2.” That’s the fancy term for multiplying 2 by 2, over and over. This is because we have 2 biological parents, no more, no less.

47.3  “2 to the first power” is just 2…”2 to the second power” is 2 x 2 = 4….”to the third” is 2 x 2 x 2 = 8…and so forth. Thus the fraction of genes one relative shares with another will be some combination of ½, 1/4, 1/8, 1/16, 1/32, 1/64, etc. Nobody is related to anybody by any other “even” fractions…no 1/6, 1/10, 1/12, 1/20, etc….nor by any “odd” fractions, 1/3, 1/5, 1/9, 1/15, etc. You can come close, but only as close as the “powers of 2” can approximate. For example, you cannot be 1/3 related…but you can be 11/32 or .343, which is close to 1/3 or .333.

47.4  Now you are related to your sibling by ½…which is to say, half your genes you share, half you don’t. We will see why this is so in a moment. But given the progression of fractional powers of 2, and the progression of numbered cousins, the table on the far left of Chart 158 would seem reasonable, wouldn’t it? But it is in fact incorrect…the center table is gives the true Coefficients of Relationship or CR’s. And on the right, the “gaps” are filled by the appropriate relatives.

47.5 Why is your 1st Cousin only 1/8 related to you, and not 1/4, which seems like it would be so “obvious”? I admit this puzzled me too at first, but it makes sense once you flesh it out. The short answer is: you are ½ related to your father, he is ½ related to his brother (your uncle), and his brother is ½ related to his son (your 1st cousin)…½ x ½ x ½  = 1/8.

47.6  But to see how it works in detail,  a good analogy would be to imagine decks of playing cards. And since we are dealing with powers of 2 only, our decks will have 64 cards, not 52…say there’s a 5th suit, stars, and the ace is missing. BTW, such decks exist, you can buy one here if you’re feeling adventurous.

47.7  Your father is a deck of 64 with blue backs, your mother with red backs. Since everyone has to have 64 cards, you will choose randomly 32 from your father and 32 from your mother. Genealogically, this is illustrated on the left side of Chart 159. Clearly, all the blue cards you have, your father also has…that’s where you got them from! So half of your deck of 64 matches, front and back, a card in your father’s deck…half of them don’t, since they have red backs. The CR between you and your father is thus ½…and by the same reasoning, it’s ½ with your mother as well.

47.8  Your CR with your brother is also ½, but in a different way. You took 32 cards at random from your father’s deck and he did the same. Odds are, 16 of your blue cards will match 16 of your brother’s blue cards, and 16 will be different. If you were to actually try this at home, it could come out 15 to 17, 12 to 20, or even 8 to 26…but since we have thousands of genes, and you’re picking strictly by chance, it will come out very close to 50/50, matches to non-matches. You and your brother will mark your blue cards that don’t match with an X, and leave the ones that do match unmarked. The same thing applies to the cards you both got from your mother’s deck.

47.9  As you can see on the right side of Chart 159, you and your brother share 16 blue cards and 16 red cards, for a total of 32…out of 64…so your CR is again ½. You might say that your relationship to your father is a “vertical” ½, and to your brother a “horizontal” ½, at least according to the way we diagrammed it. The important point is, the ½ comes about in 2 different ways, and that has a bearing on how you’re related to your 1st Cousin.

47.10   Now in the same way that we compared the “decks” of you and your brother, we can compare yours with your 1st Cousin’s. In Chart 160, your father and his brother (your uncle) are in the middle, blue & red, and each marry a different woman. Right off the bat, half of your cards will be different from half of your 1C’s cards…you got 16 green backs from your father and 16 orange backs from your mother, he got 16 grays and yellows from his. The only relationship you can have with your 1C is thru the 16 blues you each got, and the 16 reds. Let’s look at the blues.

47.11 When you picked 16 of your father’s blue cards, picking randomly of course, you got 8 blue X’s and 8 blue non-X’s…picking from his father, your 1C did the same. Now since the blue X’s don’t match to start with, those you got can’t match those your 1C got…so they are out. The only blue cards that can possibly match are those you picked from the non-X’s your father and his father shared. There are 8 of those cards each…and just as before, half will match and half won’t…so you and your 1C share 4 of the blue non-X’s, and 4 of them you don’t. Remember, that’s 4 out of 64 total cards, so thru the blue side of your fathers, your CR is 4/64 or 1/16. But of course the same holds true with your fathers’ red cards, so that’s another 1/16…and 1/16 + 1/16 = 1/8.

47.12  I know, it’s still complicated…but it is what it is…thank you, sexual reproduction!!  Maybe that’s why cloning has so much appeal 😉 😉 And if you’re feeling really adventurous, you can try it all again with half-siblings and half-1st Cousins…but when you count up the squares that match, you’ll ultimately find that the whole thing checks.

47.13   And it’s interesting to see how little you and your first cousin actually are related…it helps to explain why 1st Cousins marriages have been so common down thru history, and still are in many parts of the world today. Plus legal in half the states in the US…you don’t have to like it, but there you go…next time, more letters from the bulging mail-bag…take care…

Copyright © 2011 Mark John Astolfi, All Rights Reserved


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