The Basics

D'ni numbers are based on characters for one, two, three, and four. These four digits are rotated counterclockwise and compounded to make higher numbers¹.

The numeral for five is a numeral one rotated 90° counterclockwise (in other words, 1x5). Six is 5+1, seven is 5+2, eight is 5+3, and nine is 5+4. Ten is a rotated numeral two (2x5). Fifteen is a rotated numeral three (3x5), and twenty is a rotated numeral four (4x5). The only numbers which do not follow that system are zero and twenty-five. Zero is a dot, and twenty-five can be written two ways: as an X when it's being shown as a single numeral, or one-zero (10) when part of a numerical progression.

The reason why twenty-five can be written as 10 is because D'ni used base twenty-five mathematics, rather than base ten. In base ten, the places from right to left represent 1s, 10s, 100s, 1,000s, etcetera. In base twenty-five, from right to left, the places represent 1s, 25s, 625s, 15,625s, etcetera². As with Arabic based math, the D'ni wrote their numbers from right to left, with the far right place before the decimal point representing single digits, the next place to the left representing twenty-fives, and so on.

Richard A. Watson (a.k.a. Rawa) mentioned that D'ni numbers might have been derived from a system of counting on one's fingers. Imagine counting on the fingers of your right hand, while using the fingers of the left hand to keep track of how many times you've reached five, and that appears to be the basis of D'ni numerical progression.

This chart shows some of the conversions from base 10 to base 25, and the D'ni notation for them. The first four numbers are places in D'ni mathematics, and the final three rows show some of the differences between how base 10 numbers are expressed in base 25. The fifth line is how Gehn wrote "233rd Age" in one of his journals. He used English for the journal and wrote "98rd Age". That doesn't make sense until you realize that he was simply transliterating the base 25 figure instead of converting it to base 10. Because of that, it appeared at first glance to be Age Ninety-Eight, which is wrong. It's really Age Two Hundred and Thirty-Three.

The strange notation in the seventh line is used because a single place in D'ni math can be from 0 to 24, which can't be expressed in Arabic notation. The parentheses are used to show that the two digits of 15 are both in the twenty-fives place. Without them, the number would appear to be 1,150, instead of 1 in the 625s, 15 in the 25s, and 0 in the ones places.

¹To the D'ni, the rotation would have been clockwise, since their clocks ran in the opposite direction from ours.
²It worked for them, but for us, D'ni math is more than enough to induce headaches. At least for now, I can't even imagine trying to tackle D'ni fractions! This site has a working copy of Simon Riedl's D'ni calculator, so if you want to convert from Arabic notation to D'ni or vice-versa, it can do it for you in a flash, with no mental trauma required.

The Numerals

This chart shows the basic D'ni numbers. We do not know any of their mathematic symbols, although we do have the symbols for period and and en-dash, which can be used for a point and a minus sign. The chart is read right to left, top to bottom, beginning with zero. The two additional numbers after the character for twenty-five are the cyclic zero symbol, and the infinity symbol. Cyclic zero is used in places where zero and another number overlap. An example of that would be a d'ni compass or timepiece.

The written or spoken words for D'ni numbers are a bit like Old English and German. In German, one does not say "twenty-one"; instead, the number is "one and twenty". The main differences are that the D'ni said "twenty and one" and they began using that pattern after the number five. "Six" in D'ni is "five and one", or vagafa, and twelve is "ten and two", or nāgabrē. As you can guess, ga is the D'ni word for "and".

The names of D'ni months use a different system of numbers than normal. My own speculation is that they may be from a much older form of the Ronay language than the D'ni used, the way that month names in English use Latin numbers in some of their names: Sept-ember, Oct-ober, Nov-ember, Dec-ember are seven, eight, nine and ten respectively. When asked about it, Rawa refused to confirm or deny the theory. The second set of number names are only used in the calendar.

There are only ten months in the D'ni year, but I was able to extrapolate the words for eleven through fourteen from them. For fifteen through nineteen, I guessed the word for fifteen based on the observation that a and ē are converted to o in the regular words for one and two, and that the suffix for ten remains the same as the regular version. That means that the adjectives for fifteenth through nineteenth are conjectural. Since there is no example of how i would have been converted, I cannot guess the words for twenty through twenty-four. I colored all of my conjectural numbers blue to separate them from the confirmed numbers.

In the following chart, the words for the numbers in D'ni months are written below the regular number words.

The names of places in written or spoken numbers

Just as we have words for the places in our base ten numerical notation, such as tens, hundreds, thousands, ten thousands, and so on, the D'ni had names for the places in their notation and they used suffixes to represent them.

Currently, we know words for up to six places, and so we can write the names of complex numbers at least up to 244,140,624. (Which — and I hope I get this right — should be rigatorblo, rigatormel, rigatorlan, rigatora, rigatorsērigator, and is written }}}}}} in D'ni numerals.)

To form a number, a suffix was attached to a root number, and single place numbers were added to bring the sum up to the desired value. As an example, 26 is fasēfa (25+1). 628 is farasen (625+3). 31,258 is brēlanvagasen (31,250+5+3). The highest value that can be expressed in any place is 24, so 624 is rigatorsērigator (600+24), which is }} in D'ni numerals.

Explorer Korov'ev of the Guild of Messengers suggested this example of how a complex number using D'ni notation would be laid out.

-sē : Combining form for multiples of 25.

fasē = 25. In D'ni notation, 1 x 25. Written as 10
brēsē = 50. In D'ni notation, 2 x 25. Written as 20
sensē = 75. In D'ni notation, 3 x 25. Written as 30

-ra : Combining form for multiples of 625.

fara = 625. In D'ni notation, 1 x 625. Written as 100
brēra = 1,250. In D'ni notation, 2 x 625. Written as 200
senra = 1,875. In D'ni notation, 3 x 625. Written as 300

-lan : Combining form for multiples of 15,625.

falan = 15,625. In D'ni notation, 1 x 15,625. Written as 1000
brēlan = 31,250. In D'ni notation, 2 x 15,625. Written as 2000
senlan = 46,875. In D'ni notation, 3 x 15,625. Written as 3000

-mel : Combining form for multiples of 390,625.

famel = 390,625. In D'ni notation, 1 x 390,625. Written as 10000
brēmel = 781,250. In D'ni notation, 2 x 390,625. Written as 20000
senmel = 1,171,875. In D'ni notation, 3 x 390,625. Written as 30000

-blo : Combining form for multiples of 9,765,625.

fablo = 9,765,625. In D'ni notation, 1 x 9,765,625. Written as 100000.
brēblo = 19,531,250. In D'ni notation, 2 x 9,765,625. Written as 200000.
senblo = 29,296,875. In D'ni notation, 3 x 9,765,625. Written as 300000.

Explorer Davide has made a Microsoft Excel script that converts base 10 numbers into base 25 and gives you the name of the result in D'ni. With a specific D'ni TrueType Font pack installed, it also shows the result in D'ni numbers. This is the second version of the file, and now includes a D'ni numeral to base 10 Arabic number converter.

Explorer Korov'ev made a similar file called DniNumbers2 in ODS format with more functions. His version also has a section to convert base 25 numbers into base 10. Of the two, Korov'ev's has more functions, and Davide's is easier to use. Both are excellently done.

D'ni Mathematics

Virtually nothing is known about D'ni mathematics. The DRC never released any documents that contained equations, or that dealt with the subject. All we have is semi-official speculation.

Because of the lack of D'ni mathematic symbols, I'm going to fake a few for the purpose of examples. The math signs you will see are in no way official, and are just placeholders until such time as authentic ones are found.

I've spoken with Richard A. Watson about the subject briefly, and by his recollection, the D'ni may have used a system of notation in which the operands preceded the operators. The closest surface system to it is known as "Reverse Polish Notation." As an example, take this simple problem, 5 + 5 = 10.

Supposedly, the D'ni would have written it 5 5+, and we don't know yet how they would have displayed the result. Provisionally, I do so by displaying the result followed by an equality sign: 5 5+ 10=.

So, the problem might have looked something like this: 5 5 )

Let's take a slightly more complex problem, (9+9) x (12-4) = 144. The supposed D'ni system would not use brackets, because you apply the operands in the order they appear. The curly brackets in my example are just there to represent that numbers which are two digits in Arabic numerals are one digit in D'ni.

So, the problem would be written 9 9+ {12} 4-x 5{19}=. It might look something like this: 9 9 @ 4 5(

In that example, nine and nine are followed by a sign to add the second nine, and then twelve and four are followed by a sign to subtract the four. Then comes a sign to multiply, and that gets applied to the results of the first two operations.

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