Longcat is long. So's this explanation.
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Justification and introductionClick here to view the secret text
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This system exploits one of the suspected features of DROD numbers: they seem to be running serial numbers, assigned strictly in ascending order.
If this is as true as it seems (and from here on in I'll assume that it is), then any set of DROD players who are alive at the same time will have DROD numbers within a few hundred billion of each other. In other words, once the DROD numbers go into the trillions and quadrillions, the "most significant" digits will be the least distinctive in daily intercourse, because they'll be the same for every living drodder.
This is why this system is written and read "back to front", from least to most significant digit.
Overall shape and segmentation of the numbersClick here to view the secret text
×The digits are written along an imaginary spiral, starting at the top and going clockwise and inward. (Actually, starting at the top is just a convenient convention -- the numbers can be rotated withouth losing information.)
Each digit occupies a 60° segment along the spiral. Yes, yes, I know -- I could have made it 45° as in "eighth", but there are two reasons to use a sixth. First, 45° segments would have been a bit cramped, forcing you to write the numbers larger. Second ... I'll have to explain later once you know more. Remind me.
The digitsClick here to view the secret text
×Digits are in a sort of hybrid base. They're base-100, but clearly composed of a tens and a ones part; making them more like base 10^2. Furthermore, the shape of the tens part is based on base-3, with 0 tens as a special case. Confused? Great. This is where you turn to the attached image for enlightenment.
The top two rows show how to construct any base-100 digit. The examples are given in the orientation of a first (i.e. least significant) digit: they are written from the top to the bottom right. The centre of the spiral is towards the bottom left. A second, more significant digit would be attached at the bottom right, rotated -60° from the samples, and run straight down etc. -- as announced, clockwise along an inward spiral.
To construct any base-100 digit, take the tens shape from the top row and add the ones strokes from the second row, following the tens line. (The example ones are all on a "zero tens" curve; don't try to add that as well. And forgive me for making the curve in "07" and "08" too straight so that they are confusingly close to "17" and "18"; but I can't easily rescan a corrected drawing right now.)
The intention is that you write each full base-100 digit and then move to the next. Don't do the tens first and then go back to add in the ones; that would be just sick.
Remebering the tensClick here to view the secret text
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Zero tens is the special case, being just an arc. The remaining tens shapes are constructed using a base-3-ish system:
straight-straight = 10
straight-outward = 20
straight-inward = 30
outward-straight = 40
outward-outward = 50
outward-inward = 60
inward-straight =70
inward-outward = 80
inward-inward =90
Remembering the onesClick here to view the secret text
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Just zig and zag, starting from the end of the tens stroke (where the next digit will be attached later) and starting outward. Count all the parts of the line between angles and places where it crosses the tens line. Odd numbers end after crossing the tens line (and it's the crossings you'll be counting while zigzagging towards your goal: "one three five seven ... eight.") Even numbers end after angling back towards the tens line, but before crossing it. Curving the last stroke of an even number back away from the tens line is in fact an optional (although recommended) clarification.
Zero ones is not usually indicated. You can optionally do a single zig going inward. This is not usually necessary, but can serve as a clear marker between digits in those cases where you expect a number to be read in random rotation, so that it becomes more difficult to keep track of where a new digit begins (usually you know the boundaries are at 60°, 120°, 180° ...)
Large numbers ending in (i.e. in this system beginning with) lots of zeroesClick here to view the secret text
×As you see, 00 is usually just an arc (0 tens, and leaving out the optional zig for 0 ones). This is convenient when writing large rounded (heh) numbers: Simply start with a spiral to quickly add the noughts you need. See the examples in the image.
And here comes the second reason to use 60° rather than 45° per digit: It's really easy to keep track of where you are, particularly if you use long-scale naming: 180° means "... million" (of whatever digit follows), 360° is "... billion" etc. pp. With short-scale naming, the half-circles are "... million", "... trillion", "... quintillion" etc., which is slightly less elegant but on the other hand lines up nicely with the "one, three, five ..." from the ones strokes.
The third line of numbers in the imageClick here to view the secret text
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Just examples. The first two are the mandatory one and twentyninepointfivebillion, followed by my own DROD number, and finally Erik's figure for "maximum number of people between now and ever" from the Illumination.
Finally: Laziness, or: ShorthandClick here to view the secret text
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The fourth and last line in the image shows another optional extension to the system: In those cases where you only need the least significant few digits of a number, like for instance to talk about a living drodder, you can use this "squiggle" to indicate "the number goes on, but I've stopped caring". The two examples both show the number of a drodder from towards the end of this planet's life expectancy, once written in full and once abbreviated. (Obviously an abbreviated number is no longer unique, but the more digits you include, the less likely it is that any of the other carriers of identically-ending DROD numbers survive.)
The four testsClick here to view the secret text
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OK, this is really your job, but here are my thoughts.
Tattoo: Middling; OK if you like barbed wire. Maybe there's something you can do with colours, since they're not significant within the system.
Drawn in blood (speaking of colour): Don't really know how this compares. For what it's worth, any base-100 digit can be drawn in a maximum of 7 strokes (1 or 2 for the tens part, 0 to 5 for the ones). In the actual writing-in-your-own-blood situation, the fact that it's written "backwards" (least to most significant, i.e. most to least distinctive) means that no matter when you run out of blood (or your brain runs out of oxygen), you'll be giving useful information even if you don't manage to finish the number. You could obviously start at the end in any system, but with this one it'd be ingrained. Downside: if you're writing with a finger-width stroke, you'll need to make it quite large, requiring both more blood and a larger blood-free canvas area. A more general ease-of-writing problem is that you'll have to be able to write each base-100 digit in 6 different orientations, which probably screws with muscle memory or at least uses unnecessarily much of it.
Cocktail napkin: As long as people are using base-10 in daily life, it's trivial to encode two base-10 digits into one digit of this system at a time. You need to memorize 20 symbols (10 for tens and 10 for ones, and c'mon, the ones are really easy). No napkins necessary for the actual encoding; perhaps one to write out the number in your everyday base-10 system so that you can work in least-to-most significant direction without losing track.
Hey-I-know-you: Decoding is similarly easy. Recognition without detailed decoding: so-so again. The digits are distinctive, but not massively so. Once you start getting into more than one loop, the most useful digits for recognition are on the outermost loop and thus written largest. It is fairly easy to get an idea of order of magnitude, but then I've been arguing all the time that this is not very useful for recognizing someone who is alive at the same time as the person doing the recognising.
I actually had ways to group squares into larger glyphs so you only needed a few of them. It always ended up being just a bit too sprawling for my liking though, not to mention that ripping off D'ni numbers like that is fairly unoriginal.
. I mean your system simply requires to read the number, no decoding at all, it has to be the easiest right? 
I've tried to read bits and pieces of the conversation in this thread and much of it is just so far over my head, so I'm feeling a bit exposed, but reckoning that the worst you can do is tell me that what I've come up with is nonsense ... 
/dy