Counting the Elvish Way – Duodecimal System

elvish duodecimal 2012 in TengwarMost races and cultures are known to use a decimal system (base-10) incorporating the numbers zero through ten. Though the elves were also known to use the decimal system, they preferred to use a system of counting in sixes and twelves. This is known as a duodecimal, or base-12 or dozenal, system, and I will try my best to explain how they used this system as simply as possible.

The duodecimal, or base-12 or dozenal, system consists of twelve (12) numbers. It is identical to the decimal (base-10) system of counting for numbers zero through nine, but the numbers 10, 11, and 12 have different notations. In the duodecimal system, 10 = A, 11 = B, and 12 = 10! The numbers 10 and 11 are assigned the letters A and B because they are two-digit numbers. This is so it is understood that 10 and 11 are each complete numbers and not two separate numbers written together (i.e. 1 & 0, 1 & 1).

That leaves how the number 12 = 10. Remember that the duodecimal system is also known as a base-12 system, or dozenal. In this system instead of counting in multiples of ten (10, 100, 100) as in the decimal, base-10 system, you instead count in multiples of 12 giving you 12, 144, 1728. The number 10 in duodecimal means “1 dozen and 0 units,” or 12 & 0 units which equals 12 in the decimal, base-10, system. The number 12 in duodecimal (“1 dozen and 2 units,” or 12 & 2) equals  14 in decimal!

Here is a chart showing the decimal and duodecimal systems as well as the associated elvish tengwa for each number.

Elvish duodecimal number chart

You may have noticed that the tengwar for the number 12 (10 in duodecimal) appears to be backwards, written as 01. That is because it is written backwards. The elves write and read their numbers from right-to-left. Also, to indicate that we are using the duodecimal system, a small circle is drawn under the left-most digit in the number. Small dots, or closed circles, can be drawn under each additional digit in the number.

To better understand this system, let’s write the year 2012 using the duodecimal system and elvish tengwar. First you need to convert the decimal number 2012 to duodecimal. Think only in multiples of 12. Above I mentioned that in the duodecimal system the multiples of 12 were 12, 144, 1728. That is 12¹, 12², and 12³. For this example we should also include one as we need to have a 4-digit significant number. So now we have 12³ 12² 12¹ 1  as the place holders in the number 2012. Determine how many multiples of 12 you need for each digit. For instance, the first digit or place holder signifies 12³ or 1728. 2012 is only divisible by 1728 once, so we have (1×12³) as our first digit. Subtracting 1728 from 2012 we are left with 284 and in the second digit position or 12², which is 144. 284 is only divisible by 144 once, so we now have (1×12²) as our second digit. This leaves us with 140 (by subtracting 144 from 284) and in the third digit position or 12¹. 140 is divisible by 12 eleven (11) times giving us (11×12¹) as our third digit. We are left with 8 for the last digit and since it is a single digit and not divisible by 12 we leave it as it is. I realize this may be confusing, so let me show it to you this way…

2012 = (1×12³) + (1×12²) + (11×12¹) + 8 , remove the plus signs and the powers of 12 you have (1) (1) (11) 8 = 1 1 B 8

Hopefully that will help to clarify how we go from decimal 2012 to duodecimal 11B8.

Now, to write the number using tengwar we first have to write the number backwards, giving us 8B11.

Change each digit to the corresponding tengwa shown in the table above and you have:

The elvish duodecimal year 2012 in tengwarTry it yourself with some other numbers, just remember that what would be called tens, hundreds, thousands (10, 100, 1000) in the decimal system of counting are called dozen, gross, great-gross in the duodecimal (12, 144, 1728). I have made a PDF of all the numbers from 0 to 72 in preparation for the elvish calendar I am making, which may help in understanding the duodecimal system. It is available to view/download here. When looking at the list, remember that the tengwar are written in reverse!

This is by far not a complete teaching of how the elves count, I have only focused on the duodecimal system here. More information can be found on Thorsten Renk’s site Parma Tyelpelassiva – The book of silver leaves on the page The Eldarin Numerals.

(The tengwar font used in this post is Tengwar Sindarin.)

34 Comments

  1. Myrtonos says:

    You may have noticed that the tengwar for the number 12 (10 in duodecimal) appears to be backwards, written as 01. That is because it is written backwards.

    Actually, it is written forwards in relation to their language.

    The elves write and read their numbers from right-to-left. Also, to indicate that we are using the duodecimal system, a small circle is drawn under the left-most digit in the number. Small dots, or closed circles, can be drawn under each additional digit in the number.

    I remember reading that in Sindarin, one puts the units before tens, and tens before hundreds. They count: [all letters pronounced as in Wesh] min; tâd; nêl, neledd; canad; leben; eneg; odog; toloth; neder; caer or pae; minib; ýneg

    It sholud be mentioned that ýneg is the lowest common multpile of tâd, nêl and neledd, no fractional notation is needed for halves, thirds and quarters of a foot or troy pound. When using the Elvish dozenal digit system, the three most elemetary fractions can be reperesented using the dozenal point notation, 0;6, 0;4 and 0;3 respectively. The international standard railway gauge could be either as 4;86′ or 4’8;6”. A long playing record spins at 29;4 rpm.

    I should also mention the Fibonnaci sequcence, where the first two numbers are 0,1, each number plus the previous equal the next one, this sequence, together with the Lucas one, occurs in nature. Every third Fibonnaci number is even, and every forth geos into three, and every sixth goes into four. So we know the twelfth must go into three fours, in other words twelve. Every fifth one goes into five and every seventh goes into seven. It turns out that Fib-twelve is also the square of twelve and the only non-trivial square fibonnaci number. The only other non-trivial prefect power in this sequence is eight.

  2. Lorna says:

    This is very interesting but I cannot for the life of me figure out what 1984 would be! Help a sister out? 🙂

    • alatariel says:

      Thank you for visiting my site. The duodecimal equivalent of 1984 is 1194, gotten by (1×12^3)+(1×12^2)+(9×12^1)+4, then removing the powers of 12 to leave us with 1 1 9 4. Reverse the number to write it in elvish, so you have 4911. You can reference the chart above for the tengwar to use for each numeral. Hope this helps!

  3. Kath says:

    I have had a lot of fun since a came across your blog! But I have a problem and I was wondering if you might be able to help me out?
    My roommate wanted me to figure out what 131313 would be. My guess is 158130 but I am not sure if it is correct.

    • alatariel says:

      131313 converted into duodecimal would be 63BA9. Start off by figuring out which 12th power is the closest to our starting number. 12^4 = 20736 while 12^5 = 248832, which is much larger than 131313. Taking 12^4 = 20736 we see that it can go into 131313 six times (131313 / 20736 = 6.326). We are only interested in whole numbers so ignore everything after the decimal. 6 is the first number in our duodecimal conversion. Next take the answer from 6×12^4 (powers get worked first, so you have 6 x 20736 = 124416) and subtract this from our starting number 131313 like so, 131313-124416 = 6897. Your next step is to once again figure out how many powers of 12 can go into 6897. We just used 12^4 so we need to use the next one down, 12^3. You repeat this process until you are left with a number less than 12. The full equation for the conversion should be (6×12^4)+(3×12^3)+(11×12^2)+(10×12^1)+9. Removing the powers of 12, the plus signs, and parentheses you are left with 6 3 11 10 9. In duodecimal 10 and 11 are written as A and B respectively, giving you 63BA9 for your conversion answer. Hope you have found this helpful.

  4. Toni, says:

    How would I write the date 20-07-2014?

  5. hennia says:

    what is 13 in elvish?

  6. Jackie says:

    Hi there,

    I’m having trouble finding out the number 425 would look like written out this way. I’ve been researching for a potential tattoo and I’m not sure what I have is correct, I’ve seen several ways of doing this and don’t want to put a permenantly incorrect tribute on me! Please help!

    • Alatariel says:

      Hi Jackie. Sorry for the really late reply. 425 in duodecimal is 2B5. The elves read/write their numbers from right to left so this would become 5B2. Referencing the chart included in the article you can see what Tengwar you need for each character.

  7. Alatariel says:

    This was posted by reader Pete two months ago, but sadly due to a website restore the comment was lost.

    Pete “This is an excellent article. I just want to point out a typing error. In the clause “This leaves us with 140 (by subtracting 140 from 284) and…”
    It should be “…140 (by subtracting 144 from 284)…”, since 284-144=140, not 284-140=140.”

    Thank you reader Pete for catching this. I hope to have it updated soon!

  8. Jess says:

    I’ve looked every where and cannot find a translator that will help me write 15 and 2015. I’ve tried figuring it out using this but duodecimals are not my strong point! Could someone please help?

  9. Sarah says:

    This is a really great explanation for this! I just wanted to check and see if I was right: is 1990 A911? And do you know what order dates are written in? Would it be the like the British dd/mm/yyyy ?? Thank you!

    • Alatariel says:

      Hi Sarah! Yes, 1990 would be A911 in elvish duodecimal. The only date I have ever seen written out was in the Kings Letter, a letter Aragorn wrote to Sam. It followed the British way of writing the date as Day/Month/Year. Since Tolkien was British this would make sense. For the month, the name of the month was used rather than the number.

  10. Lalaith says:

    Great article! I just wanted to inform you that the link has changed, and you can now find the book of silver leaves at: http://www.science-and-fiction.org/elvish/index.html

  11. cepheus says:

    Hi, a great site you have and a very instructive article too. Though I don’t understand the number 12. Why there is no sign for it since it’s a duodecimal system? I can’t understand the difference between tengwar 12 (for me, 1 number) and tengwar 13 (12+1)
    Thanks.

    • Alatariel says:

      Hello Cepheus. I’m not sure why Tolkien did not create a unique sign for 12. I have looked around and cannot find anything definitive regarding it. I have read that regardless of the counting system being used, and this is for counting systems used today, (i.e. base-2, base-10, base-12), unique identifiers are only needed up to the base unit. In the decimal or base-10 system that we mostly use, the number 10 could also be considered not unique as it is essentially the digits 1 and 0 put together. So essentially, any number past 9 in the decimal system is not unique but a combination of the digits 0 through 9. This was probably the way Tolkien thought about it when deriving the tengwar for the elvish duodecimal system. In a duodecimal system only the digits 0 through B are unique. What you see as the number 12 is seen as 10 in a duodecimal system, meaning 1 dozen and 0 units. Decimal number 13 is seen as 11 in duodecimal, meaning 1 dozen and 1 unit. I hope some of this helps and thank you for visiting the site.

  12. Emma dalziel says:

    Hi! So glad I fount this page 🙂 I’m trying to translate two birth dates, as I’m looking to get them as a memorial for my mum and dad and incorporate them into a tattoo. The dates are: 11/05/1950 and 06/01/1949
    I hope you can help as I’m not sure if I’m managing to translate it properly! 🙂
    Thank you!
    Emma x

  13. Max says:

    I noticed that all of the cast members from the Fellowship of the ring have a different “9” tattooed than what I see you have here. Any explanation? Thanks!

    • Alatariel says:

      Hi Max. The Fellowship cast members who were in the group of nine to take the One Ring to Mordor to destroy it have the word “nine” in tengwar tattooed rather than the actual number. All except John Rhys Davies who had his stunt double get the tattoo instead. Another interesting tidbit is that Peter Jackson chose to have the word “ten” tattooed on himself.

  14. David S. says:

    In terms of writing numbers out, there seems to be a decimal system bias.
    That is 20 is written as “taphae” (‘tad’+’pae’ or two tens), thirty as “nelphae” (three tens), forty as “canaphae” (four tens), etc.
    When using a duodecimal, shouldn’t it use a written form of 12 “uiug” as a suffix?
    Say perhaps “tadiug” (two twelves), “neliug” (three twelves), etc.
    The final one would be “miniug (eleven twelves) followed by a word for the english equivalent for gross (144) or Twelve Twelves.
    I looked on other sites and did find a Quenya word “tuska'” which meant 144.
    It also eluded to some numbers being pronounced in quenya with a base 12 suffix (#36 & #84).
    Or did the more modern Sindarin adopt a base 10 mathmatic system?

    • Alatariel says:

      Hi David, sorry for the late response. The article I link to (link has been fixed!) at the end of mine above regarding Eldarin Numerals might be able to answer your question (http://ambar-eldaron.com/thorsten/numerals.html). The following written in the conclusion of the article seems to sum up which system is used : “we are to assume that Elves did use duodecimal counting in the reckoning of time and when dealing with mathematical problems, but not in everyday language, but some words of everyday language were influenced by the duodecimal forms.” So it would appear that the decimal system is indeed more common but some of the terms for duodecimal numbers (such as 144) would still be used.

  15. Alexandra says:

    Hi! I was wondering what 2014 would be? Because there’s a 0 left. So is it 10 1 1? Or do you include the zero in front of the ten? Or is my math just completely wrong?

    • Alatariel says:

      Hi Alexandra. Decimal 2014 converts to duodecimal 11BA, which in reversed format would be AB11. The following is a quick run through of how you get it and I hope it’s not confusing. You will need to reference the article above. 2014 is divided by 1728 (which is 12 to the third power) = 1.1 but we only care about the digit preceding the decimal so it’s 1 ( I write it to the side of my equations as I go along to create the duodecimal number). Because 1728 can only go into 2014 once we want to now subtract that, so 2014-1728=286. Repeat the same steps with 286 but now with 12 to the second power (144). 286 divided by 144=1.9. Write the 1 to the side. Then 286-144=142. 142 divided by 12 = 11.8. Convert the 11 to the letter B for duodecimal and write the B to the side. 12 was able to go into 142 eleven times so multiply 12 x 11 to get 132. subtract this result from 142 to get your remaining number. 12 x 11 = 132. 142-132 = 10. Again check the chart at the top to convert 10 to duodecimal letter A. Your final result for the date is 11BA.

  16. Jindra Rotscheid says:

    Im wondering what 2016 is in Elvish. 1728 fits in once and 144 twice, so that would be 1 and 2, written as 21. However, this could also be read as 146. Should there be a 0 somewhere?

  17. Swati says:

    Hi! Could you please help me write 1916 in Elvish numerals? I tried calculating it but it’s for a tattoo and I don’t want to get it wrong! I’m planning to write November 1, 1916 and I’m assuming I can use 1 and 11 directly from the chart above. Thank you very much!

  18. Becky says:

    Hello there would you please please help me to convert some dates of birth?
    11/05/2105
    28/01/2012
    28/02/2010
    27/11/2000
    09/07/1993

    I would be so so grateful

  19. Mila Chemdi says:

    Hi how would you write a date? How would you separate between month/day/year? Ktz

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