I have found some examples but I am not sure if any of them are right. Are there any concrete examples of Tolkien numerals?
Quettar Special Publication No.1, "The Writing Systems of Middle-earth", by David Doughan and Julian Bradfield, published in 1987.
The Eldar used both a decimal and a duodecimal system, the Dwarves used a duodecimal system, and the Men of the West in the Third Age used mainly a decimal system. The digits used were as follows1:
In all systems the numbers are written with the unit digit at the left. The digits were usually marked, either by a dot over each digit, or by a line drawn above the number, for a decimal system; in a duodecimal system, the dot or line was written below.
|1.||er, min||er, min||er, min||13.||nelkea|
|6.||eneg||enque||enec||18.||tolkea / hualqe|
VT is Vinyar Tengwar, a linguistic journal published by the Elvish Linguistic Fellowship.
Ordering and names of numerals from Unicode roadmap proposal 4:
Tengwar digits CSUR encoding
|0||U+xx30||TENGWAR LETTER STEMLESS OORE (digit zero)|
|1||U+xx33||TENGWAR LETTER STEMLESS VILYA (digit one)|
|2||U+xx62||TENGWAR DIGIT TWO|
|3||U+xx63||TENGWAR DIGIT THREE|
|4||U+xx64||TENGWAR DIGIT FOUR|
|5||U+xx65||TENGWAR DIGIT FIVE|
|6||U+xx66||TENGWAR DIGIT SIX|
|7||U+xx67||TENGWAR DIGIT SEVEN|
|8||U+xx68||TENGWAR DIGIT EIGHT|
|9||U+xx69||TENGWAR DIGIT NINE|
|10||U+xx6A||TENGWAR DUODECIMAL DIGIT TEN|
|11||U+xx6B||TENGWAR DUODECIMAL DIGIT ELEVEN|
|12||U+xx6C||TENGWAR DUODECIMAL DIGIT TWELVE|
CSUR – ConScript Unicode Registry
J.R.R.Tolkien never published a runic numbering system that could be used with the Cirth. However, in the Book of Mazarbul inscriptions, some characters are specifically used to represent numbers5:
Numerals for the runes from Pictures by J. R. R. Tolkien1:
I'm not sure if the original JRR Tolkien manuscripts these were based on have been published yet, (I presume they have, I just haven't been able to locate them) but in 1981 and 1982 Christopher Tolkien mailed the editor of Quettar (a linguistic journal) some handwritten notes on how to write numbers in tengwar.
These were published in their original form in Quettar #13 and Quettar #14.
The Tengwar Numerals
For numeration, especially in lists or series, the letters up to [rune] 24 (see The Return of the King p. 396, table of Tengwar) were commonly used, with or without some sign, as [rune] [rune]
The numerals of the Fëanorian script were arranged in triads:
0 123 456 789
The smaller value was placed on the left : thus [rune][rune] (01) = 10. [rune][rune] (24) = 42
When the use of numerals as such was evident, and no confusion could arise, the dots could be omitted. Often a long series of numerals in the midst of words was marked by a line drawn above as
[rune][rune][rune][rune][rune][rune][rune] = 1780396
to be read as 'six million, nine hundred & thirty thousand, eight hundred & seventy one'.
CRT after JRRT 16 May 1981
The Eldar used two systems of numerals: one of sixes (or twelves); and one of fives (or tens). The Dwarves used for their own purposes a duodecimal system. Men of the West of the Third Age used mainly a decimal system, though their numerals showed the influence of the sixes & twelves of the Eldarin & Dwarvish use.
Where a duodecimal system was required for arithmetical purposes, or for denoting Dwarvish numerals, the dots or line were placed beneath the figures (if used). The figures [rune][rune] were then used for ten and eleven, while [rune] or [rune] could be used for twelve. The doudecade was often marked by [rune].
Thus decimal [rune][rune][rune] (441) = 144; [rune][rune] (53) = 35
duodecimal [rune][rune][rune] (001) = 144; [rune][rune] (11,2) = 35
[rune][rune][rune][rune][rune][rune][rune] = 7 0 11 2 10 3 2 = 2 3 10 2 11 0 7 = 6,930,871 decimal
CRT after JRRT 10 March 1982
Gandalf leaves a message at Weathertop for Aragorn and the hobbits by scrawling a G rune followed by three strokes, to indicate that he was there on the 3rd of the month (or at least, that’s what Aragorn surmises). Therefore, something akin to Roman numerals was definitely used occasionally, at least for small numbers.
'There seems to be a stroke, a dot, and three more strokes,' he said.
'The stroke on the left might be a G-rune with thin branches,' said Strider. 'It might be a sign left by Gandalf, though one cannot be sure. The scratches are fine, and they certainly look fresh. But the marks might mean something quite different, and have nothing to do with us. Rangers use runes, and they come here sometimes.'
'What could they mean, even if Gandalf made them?' asked Merry.
'I should say,' answered Strider, 'that they stood for G3, and were a sign that Gandalf was here on October the third: that is three days ago now.
Regarding the tengwar, we know of about six different numeral systems up to now (more might be published in the future).
From this variety of numeral systems, it would appear that Tolkien never settled for any single one of them, but kept inventing new ones. I also believe he cared less for numeral systems than for letters, seeing how his mode descriptions discuss letter uses in great detail, whereas there are hardly any explanations to the numerals.
Out of the six different numeral systems known to date, only the very early ones from the 1930s and the very late ones from the 1960s are well attested. The Lord of the Rings-era numerals from the 1950s are only completely attested in redrawings by Christopher Tolkien, but we have reasons to believe that the forms intended by Tolkien himself might have been different.
In addition to the numeral systems, the tengwar could also be used for simple enumerations, like the Latin letters A, B, C, D, ..., can be used. The problem is, though, that different sources do not use the same order for the tengwar. The most canonical one certainly is the one from Appendix E.
Overview over the different numeral systems:
- The Quettar numerals reported by Christopher Tolkien.
- These numerals are not attested by Tolkien himself, but have been redrawn by his son Christopher.
- We do not have any context about the use of these numerals. Since they resemble the DTS 49 numerals, they might be from the early 1950s.
- These numerals are the most popular on the internet because they already existed when the internet came about. It is also the system you mentioned in your question and the one described by Oni and ibid.
- Some details about the use are unclear: What is the use of a sign for 12 in a base-12 numeral system? Why are the numerals marked with a dot or a line?
- These numerals differ from all the tengwar numeral systems attested by Tolkien himself. There is not a single sign that looks like a normal tengwar (or Rúmilian) letter. There are reasons to believe that this dissimilarity came along with the way Christopher Tolkien redrew these numerals and that his father really intended them to look like normal tengwar: The disambiguation dot would only make sense if they looked like normal tengwar, and the similar DTS 49 numerals look like normal tengwar.
- The DTS 49 numerals, only partially attested (1 short carrier with a dot above, 3 nwalme with a dot above, 4 halla, 6 mirrored unque with a dot above
- These numerals are similar to the Quettar numerals, but they look like normal tengwar.
- This could mean that these are the forms Tolkien really intended for the Quettar numerals.
- The context of these numerals are a rejected epilogue for the Lord of the Rings. It had been worked out much detail, but Tolkien decided against its publication by early 1954. This means the numerals are probably from the early 1950s.
- See also Per Lindberg’s analysis Tengwar numerals in the King’s Letter.
- “dwarven numerals (Fëanorian)”, DTS 87 (1 parma, 2 tinco, 3 calma, 4 quesse, 5 umbar, 6 ando, 7 anga, 8 ungwe, 9 unque, 0 stemless vilya with a stroke inside, all said to be used “with a dot”)
- These numerals are from the mid 1960s, concomitant to explanations about Dwarvish runes and a nice illustration of Bilbo’s contract.
- The ordering of parmatéma before tincotéma makes sense phonetically, but it differs from the earlier ordering seen e.g. in Appendix E to the Lord of the Rings. Note that the mode descriptions found in the same documents also have this ordering with parmatéma before tincotéma.
- Tolkien explains that the numerals were used “with a dot”, except for the numerals for 9 which had no use (in this phonemic English mode) and 0. Also, Tolkien says that these were really “no numerals”, but “letters of the alphabet”.
- “dwarven numerals (Rúmilian)”, also DTS 87
- This numeral system is from the same mid 1960s source as the “dwarven numerals (Fëanorian)”. The Rúmilian numerals are mentioned after the Feanorian numerals, and Tolkien said that the Rúmilian numerals were “sometimes used”, so it would appear the Feanorian numerals were preferred.
- These Rúmilian numerals are different from any other Rúmilian numeral system published so far, cf. Helios’s analysis Rúmilian numerals.
- PE 20 Q10h, Q11j numerals (1 long carrier with dot above, 2 tinco, 3 ando, 4 vala, 5 esse, 6 silme, 7 calma, 8 anga, 9 rómen, 0 úre)
- These numerals are from the 1930s.
- Note that the tengwar used in this system resemble our own Western Arabic numerals.
- Seeing that the later tengwar numerals do not resemble Arabic numerals, it appears that Tolkien abandoned that idea. Of course, these numerals would still be adequate in a tengwar document written in a 1930s style “Qenya alphabet” mode, which is really a phonemic English mode and by far the best attested tengwar mode of them all.
- PE 20 Q10h, Q11j Arabic numerals in tengwar style (the numerals that look the most like tengwar are: 1 short carrier with a dot above, 3 alda, 6 silme, 9 rómen, 0 úre)
- These are really just Arabic numerals with a few stylistic modifications so the fit the tengwar better.
- They are from the same mode descriptions as the other PE 20 numeral system. There is no explanation about how the two systems would be used, but both sources mention this one after the other.