It rounds up to none, but getting there is fun.
A note before I get started: I'm going to use the word "blood" a lot, and very imprecisely. I don't literally mean blood, or genetic markers, or anything like that. I just mean ancestry, the way I would call myself 1/4 Irish, because my grandmother was ethnically Irish. "Blood" is just a slightly simpler way to express that idea.
Aragorn's parentage diverges from the Elves with Elros, brother of Elrond, who made the choice of the Edain at the end of the First Age. There is no other intermarriage with Elves along Aragorn's line until we get to Aragorn himself, so we can simply count the Kings (and Queens) of Númenor, Gondor, and Arnor, and the Chieftains of the Dúnedain, until we get down to Aragorn. We start with part i of Appendix A.I, the kings of Númenor.
"The Thain's Book" (note: defunct and not archived) helpfully provided a family tree of the Kings of Númenor, which goes down as far as Isildur before skipping to Aragorn, so you can see how these different people relate to one another:
For the sake of accessibility, and also citing some of the more surprising statements, let's go through the chain:
- Elros Tar-Minyatur
At this point Aragorn's line diverges from the Kings. The next King is Tar-Elendil's son Tar-Meneldur, but Appendix A tells us that Aragorn is descended from Tar-Elendil's daughter Silmariën:
In the days of Tar-Elendil the first ships of Númenóreans came to Middle-earth. His elder child was a daughter, Silmariën. Her son was Valandil, first of the Lords of the Andúnië in the west of the land, renowned for their friendship with the Eldar. From him were descended Amandil, the last lord, and his son Elendil the Tall.
Return of the King Appendix A "Annals of the Kings and Rulers" (i) "Númenor"
So let's continue the list
Unfortunately here we don't know any more until Eärendur, but he's identified in Unfinished Tales as the Fifteenth Lord of the Andúnië, so there are 13 unaccounted for names.
- Eärendur's unnamed son (or daughter)
From here we turn to Part ii of Appendix A, "The Realms in Exile". "The Thain's Book" again compiled a helpful family tree of the Rulers of Arnor, and the Chieftains of the Dúnedain, all the way from Elendil to Aragorn:
For accessibility purposes:
- Amlaith of Fornost
- Argeleb I
- Arveleg I
- Argeleb II
- Arveleg II
- Aragorn I
- Arahad I
- Arahad II
- Arathorn I
- Arathorn II
- Aragorn II
So, what we have is that Aragorn is the 64th descendant of Elros. The Elven-blood gets a little more diluted with every generation. Unfortunately we can't nail down an exact percentage, because we don't know the lineage of these 64 wives; when you get a tree this deep, it's almost inevitable that you're going to end up marrying your 30th cousin or something (Arvedui married his 23rd cousin 6 times removed)2.
But we can put a lower bound on things by assuming that the spouses were all 100% human, so the bloodline dilutes by half each generation. Believe it or not, there's a formula for calculating this: for generation
i, the amount of Elven blood is
n is the amount of blood in generation
0 (which, in this case, is Elros).
Now, Elros was not a full Elf. His parents were both half-Elven, and his grandmother was even more half-Elven.
- Elros' father was Eärendil, who was the son of Tuor (a Man) and Idril (an Elf), so Eärendil was
- Elros' mother was Elwing, the daughter of Dior (son of Beren and Lúthien3) and Nimloth, a full-Elf. Dior was
1/4 Elven, making Elwing
This means that Elros, genetically, was
Finally, we can arrive at an answer, using the formula from earlier4:
(9/16)/(2^63) = (9/16)/9223372036854775808 = 6.1x10^-20 = 0.000000000000000000061 = 0.0000000000000000061%
That tiny number is the smallest quantity of Elven blood that can be in Aragorn's veins. Absent his complete family tree, rather than just the line of Kings and Queens, this is the best we can do. However, considering how tiny that number is, I highly doubt that there was enough Elven blood in all of Númenor to bring that up significantly higher; you might gain a couple of decimal places, but in the end it's still going to be effectively nil.
To give you an idea of how miniscule of an amount this is, if I mixed a single drop of my blood, a volume of 0.0357 mL, into Lake Erie, with a water volume of
4.8x10^14 L, Lake Erie would have about as much "me" in it as Aragorn has "Elf" in him.
1 That Amandil was the son of Númendil is repeated frequently throughout History of Middle-earth XII.
2 It's nearly impossible that there wasn't some marriage between (distant) cousins in Aragorn's family tree. A 2004 paper estimated that the most recent common ancestor of all humans lived about 3000 years ago, and Aragorn and Elros are separated by much, much more than that5 In fact, we know this happened at least twice: Arvedui's wife was a princess of Gondor descended from Elendil, and Aragorn's mother Gilraen was said to be related to the Chieftains of the Dúnedain.
3 Since Lúthien's mother Melian was a Maia, we can use this same formula to determine that Aragorn is
4 Because my formula assumes Elros is generation 0, but my list counts from 1, I'm going to use the exponent 63 instead of the 64 I gave earlier for Aragorn's generation number.
5 From Appendix B, Elros died in S.A. 442, and Aragorn was born in T.A. 2931; so Aragorn was born nearly 6000 years after Elros' death.