Reverse octal
and other numberings of ancestors

Sosa numbering

Genealogists who compile a list of a person's ancestors, called an ahnentafel or ahnenreihe, most often use Sosa numbering. The "root" individual whose ancestors are being listed is numbered 1, and at each generation the father and then the mother of each person in the previous generation is listed, and numbered, in order.

Thus here are the first few generations:

1. Person.
2. Person's father
3. Person's mother
4. Person's father's father
5. Person's father's mother
6. Person's mother's father
7. Person's mother's mother
8. Person's father's father's father
9. Person's father's father's mother
10. Person's father's mother's father
11. Person's father's mother's mother
12. Person's mother's father's father
13. Person's mother's father's mother
14. Person's mother's mother's father
15. Person's mother's mother's mother

This system has these features:

• Ancestors are numbered sequentially starting from 1. Gaps are visible as missing numbers.
• If an ancestor is numbered n, his or her father is 2×n and his or her mother is 2×n+1.
• With the possible exception of the root, all males have even numbers and all females have odd numbers.
• Generations are numbered by powers of two. Ancestors g steps removed from the root are numbered starting at 2^g and up to but not including 2^g+1.
• It is widely known and understood.

Nevertheless, Sosa is not always easy to work with. For instance, tracing an ancestral line, or seeing who is an ancestor (or even parent) of who, can require considerable arithmetic. It takes some crunching to get that the descent from #7399 goes 3699, 1849, 924, 462, 231, 115, 57, 28, 14, 7, 3, 1. Which of the sixteen of the root's great-great-grandparents, 16 through 31, is #889 an ancestor of? (Answer: 27.)

And these are fairly small numbers. Many genealogies sport far more generations. Here are two extreme examples, where the largest numbers appearing are 18484050353669565969914629455873 and 147793783161630549847572480, respectively. Can you guess which of the root's great-great-grandparents each of these is alleged to be the ancestor of? These examples are of course rather incredible, but many ahnentafels with two or three dozen generations are not, such as long royal descents. It can be quite hard to see the relationships in the numbers.

Binary

Today the whole world uses a positional number system with a base of 10, also known as decimal. Starting from the right, each position to be filled with a digit in a written number has a value ten times the previous one. So, "4253" represents four thousands plus two hundreds plus five tens plus three ones.

The choice of ten is arbitrary, stemming from the number of fingers humans have. A positional system based on another number, say seven, works just the same. Then "4253" represents four seven-cubeds (343 in decimal) plus two seven-squareds (49 in decimal) plus five sevens plus and three ones, or what in decimal we would call 1508.

Other bases are mostly a mathematical curiosity, but do have applications. For instance, computers internally use base 2, or binary, since it is easier to make an electronic switch with two positions than with ten. Thus the stereotype that machines think in ones and zeros, or bits. "11001001" is 201 in decimal.

And ancestor numbering is really binary. Analogous to bits of 0 or 1, each ancestor is either a mother or a father. As an example, take the Sosa number above of 8109. In binary, this is 1110101110111, and the line of descent goes 111010111011, 11101011101, 1110101110, 111010111, 11101011, 1110101, 111010, 11101, 1110, 111, 11, 1, in other words we just drop the bits off the end. Similarly, we readily see that #1101111001 (889) is an ancestor of #11011 (27).

Binary ancestor numbering has many nice features:

• As noted, descent can be traced by dropping bits off the end.
• Ancestors of a person are numbered by adding bits. The great-grandparents of 11001 are 11001xxx, with each x a zero or one.
• The generation number, counting the root as "generation one", is equal to the number of bits in each person's number. Thus, generation five will be all five-bit binary numbers.
• Each bit has a meaning, 0 is "...'s father", and 1 is "...'s mother", except the initial 1 means the root person. Thus 110010 is the root's mother's father's father's mother's father.
• Other basic information inscrutable in base 10 is readily available. For instance, the root's paternal ancestors all start with 10, and maternal ancestors start with 11.

However, it has one glaring defect: it is quite cumbersome. Even eight- or nine-bit numbers are an eyesore, and hearing one spoken may just be hypnotic.

Octal

Computer programmers who must work with binary also find it cumbersome, so a common technique is to group bits by threes or by fours. This is accomplished by using a base of a power of two, in this case 8 or 16 respectively. These bases are better known as octal and hexadecimal, and are both widely used.

Thus "10101001" could be grouped "10 101 001", and thus be "251" in octal, or as "1010 1001", and be "a9" in hexadecimal (since hexadecimal requires sixteen distinct digits, the letters a through f are added with the values ten through fifteen).

We do the same with Sosa numbering. I do not recommend hexadecimal, since mixing letters and numbers is confusing, and letters are often used to label descendants. Octal has only two fewer digits than decimal, and the average number is about 11% longer written in octal versus decimal¹, which is not too burdensome.

For comparison, the last generation in the above list would instead read as follows:

10. Person's father's father's father
11. Person's father's father's mother
12. Person's father's mother's father
13. Person's father's mother's mother
14. Person's mother's father's father
15. Person's mother's father's mother
16. Person's mother's mother's father
17. Person's mother's mother's mother

The next generation would be numbered 20–37.

However, some benefits of binary are weakened with octal:

• Instead of dropping bits to follow the line of descent, there is a cycle of three. If we start with 16347 (in decimal the familiar 7399), the line goes 7163, 3471, 1634, 716, 347, 163, 71, 34, 16, 7, 3, 1. Thus, arithmetic is still needed, although less than in decimal.
• Ancestral relationships are easier to see, but again not as trivial. It may not be obvious that 1571 (in decimal 889) is the ancestor of 33 (decimal 27), but once you understand octal it is easy to see that 1571 is the ancestor of 157, whose child is 67, whose child is 33, or that 1571 is the ancestor of 15, whose parents are 32 and 33.
• All people in one generation have numbers with the same number of digits, but the converse is not true. Instead, groups of three generations have the same number of digits. E.g., generations one through three have one digit, four through six have two digits, and so on. However, the first digit tells you which of the three you're in: 1 means the first of the group, 2–3 means the second, and 4–7 means the third.
• Each digit except the first represents sequences of three "...'s mother" and "...'s father" steps rather than one. "5" is for instance "...'s mother's father's mother". Surely with practice could can simply read them off, and write them down, which may make arithmetic easier.
• Ancestors of the root's father start with 10–14, 2, 4, or 5. Maternal ancestors start with 15–17, 3, 6, or 7.

Using octal for Sosa numbering is certainly still appealing, but I propose a further alteration.

Reverse octal

Different bases can be used not just with whole numbers (integers), but with any real number. For instance, in decimal 0.43 means four tenths plus three hundredths. "0.43" in base 7 would thus be four sevenths plus 3 forty-ninths, which is about 0.633 decimal. The "." (or a "," in some countries) is often called a "decimal point"; this can be confusing if we're not using decimal, so I will call it a binary point, an octal point, or whatever is appropriate.

Reverse binary numbering works just like binary numbering, except we put the bits after the binary point (in the same order). Thus, we number ancestors with fractions rather than integers, starting with 0.1 instead of 1. I will explain in a moment why we would do this, but first a technical point: In binary Sosa numbering there is a placeholder 1 at the start. The placeholder is needed because 110, 0110, and 00110 are all the same number, while 1110, 10110, and 100110 are not. In reverse binary we put the placeholder bit at the end, because 0.011, 0.0110, and 0.01100 are all the same number, but 0.0111, 0.01101, and 0.011001 are not. We thus start with 0.1 (the root); 0.01, 0.11 (the parents); 0.001, 0.011, 0.101, 0.111 (the grandparents); and so on. 0.001101 is the root's father's father's mother's mother's father.

Reverse binary offers no real advantage over just binary, but reverse octal does offer an advantage over octal, namely that the bits are not shifted through the groups of three when a generation is added. For instance, if a person's reverse binary number is 0.1011100111, then her (it is a her!) parents are 0.10111001101 and 0.10111001111. Grouping by threes, these numbers are 0.101 110 011 1, 0.101 110 011 01, and 0.101 110 011 11, or in octal 0.5634, 0.5632, and 0.5636. No arithmetic is needed at all! The only confusing part is the disposition of the last digit, but for that you just need to remember "the basic pattern":

The Basic Pattern

All ancestors of x4 start with x, and this pattern iterates indefinitely.

I use the distinct symbol ° as the octal point. Also, the initial zero before the octal point is redundant and so is dropped, so the person above would be labelled °5634. The octal point itself may be optionally dispensed with in tables and lists.

The root person is thus °4. The Sosa decimal example of 7399 translates to °63474, and the line of descent is 6347, 6346, 6344, 635, 636, 634, 63, 62, 64, 7, 6, 4. It requires no arithmetic to see that °5714 (decimal 889) is the ancestor of °56 (decimal 27), though you do need to know that any x7y (for y not empty) descends through x74, x7, x6, x4. Someone's direct patrilineal line goes 4, 2, 1, 04, 02, 01, 004, 002, 001, etc., and the matrilineal line goes 4, 6, 7, 74, 76, 77, 774, and so on.

A summary of the properties of reverse octal:

• Lines of descent and long-term relationships are almost trivial to see.
• As in octal, groups of three generations have the same number of digits. The first of any three has all numbers ending in 4, the second in 2 or 6, and the third in 1, 3, 5, or 7.
• Ancestors are not numbered sequentially, but through an odd-looking sequence, albeit one that becomes familiar with practice, allowing you to see gaps.
• A person is male who has a number ending in 1, 2, 5, or x4 where x is an even digit. A person is female who has a number ending in 3, 6, 7, or x4 where x is an odd digit. Note that the root °4 is in neither category.
• The root's paternal ancestors start with 0–3, and maternal ancestors start with 4–7.

Mathematicians may appreciate another interpretation of this system: The root "owns" the interval (0,1), where all his or her ancestors are. The root's father owns (0,1/2) and the root's mother owns (1/2,1), and so on, with each person's two parents splitting the interval that person owns. A person is labelled with the midpoint of their interval.

A long example

I illustrate this system with Stewart Baldwin's table of ancestors of Llywelyn the Great. This work, though impressive, is hard to follow via the Sosa numbers, the largest of which is 25763840. Here are the same ancestors labelled with reverse octal (for full details see Baldwin's page):

Note that for octal strings of six digits or more I group the digits in fours, similar to how decimal digits are traditionally grouped in threes.

With this labelling, one can "read off" interesting facts. For instance, the last entry, Agricola °4222 0000 4, is a direct patrilineal ancestor of the woman Tangwystyl ferch Owain °4223. The line founder Dúnlaing, who appears three times in the genealogy, is the direct patrilineal ancestor of three women—Máelcorcre ingen Dúnlaing °07, her distant ancestor Condadil ingen Crundmáel °0600 06, and her husband's grandmother Gormlaith ingen Murchada °047. There are four "maternal" steps out to Agricola—and six (out of 17) to Failend ingen Suibne °0600 37.

The only tricky part is the gymnastics with the last digit, but that just takes practice.

Twisted octal (a tangent)

Reverse octal has a few unsatisfactory properties, the largest one in my view being that it uses funny fractions instead of a simple sequence of integers starting with 1. What I call twisted octal is a variation designed to address this. I don't really intend it as a serious proposal, just an interesting one. Who knows though?

It's twisted because the order of the bits is reversed but the placeholder bit is at the start, as is necessary for integers. The numbering is done at each generation by sequentially numbering first the fathers of everyone in the previous generation, and then the mothers. Hence, it starts off

1. Person.
2. Person's father
3. Person's mother
4. Person's father's father
5. Person's mother's father
6. Person's father's mother
7. Person's mother's mother
10. Person's father's father's father
11. Person's mother's father's father
12. Person's father's mother's father
13. Person's mother's mother's father
14. Person's father's father's mother
15. Person's mother's father's mother
16. Person's father's mother's mother
17. Person's mother's mother's mother

Or, keep the "natural" order and number them

1. Person.
2. Person's father
3. Person's mother
4. Person's father's father
6. Person's father's mother
5. Person's mother's father
7. Person's mother's mother
10. Person's father's father's father
14. Person's father's father's mother
12. Person's father's mother's father
16. Person's father's mother's mother
11. Person's mother's father's father
15. Person's mother's father's mother
13. Person's mother's mother's father
17. Person's mother's mother's mother

This system's structure is similar to reverse octal. However, as in "forward" octal, the first digit rather than the last is the exceptional one. The ancestors of 14 in this scheme are just those whose number ends in 4. Ancestors within each generation are no longer in numerical order, in contrast with Sosa numbering (of any base) and reverse octal.

I leave it to the reader to work out the other properties of this system, many of which are direct analogs of those in reverse octal.

Other operations

Sometimes one wants to computate relations relative to another root.

In Sosa numbering, the #76 of #123 is... #7884. In octal, the #114 of #173 is #17314. In reverse octal, the °144 of °734 is °73144. In these cases, octal can be conveniently applied without arithmetic. However, this is just "lucky"; both numbers denote six steps from the root, a multiple of three.

In other cases, a "shift" operation is needed. Consider in Sosa that #204 of #247 is #31692. In octal, #314 of #367 is #75714. In reverse octal, °462 of °736 is °73631. In these cases, one of the two numbers—the second in the case of "forward" octal, the first in the case of reverse octal—must be multiplied or divided respectively by two and attached to the other number, with due accommodation for the transitional digit. This is still far simpler than decimal.

Simpler examples: My °13 is my father's °26, and my °704 is my mother's °61. My °251 will be my children's °1244 or °5244, depending on my (hypothetical) sex.

Conclusion

The proposed numbering schemes trade some advantages for others. For me, the disadvantages of Sosa numbering led me to develop and use this altenative, despite Sosa's established position.

Whether reverse octal is a good idea or a crazy one, I have been using it for several years now. Perhaps others will also find it useful.