Hello, everyone, and welcome to the D’ni numbers and math lecture. This lecture was requested by Sabine, the organizer of the anniversary celebration, who thought it might be helpful if I put together an introduction to the D’ni numbering system. Since that would not have taken very long, I expanded it to include everything we know about D’ni mathematics as well.
The goal today is to help you recognize D'ni numbers, and to help you understand how those numbers are used.

I’ve prepared some tables for the lecture, but because of limitations in the game, I can’t display them in the viewers. Instead, I’ve set up a section of my Myst web site so you can see them.
Please open a web browser and go to www.florestica.com/hpotd/math. I'll wait for you all to come back. Please let me know when you have returned to the game.

Introduction to D'ni Numbers

Click on the button at the bottom of the page to see example #1. Let me know when you're ready.

The first thing one needs to know about D'ni numbers is that they are mostly based on only four symbols, which represent the numerals one, two, three, and four. These four numerals are rotated counterclockwise and compounded to make higher numbers.

Below the main table, there is an illustration that shows an easy way to remember the base numbers. It's coincidental, but the the D'ni number one looks like the vertical line of the Arabic numeral one, the D'ni number two resembles the curve in the Arabic two, D'ni three resembles the connection between the upper and lower arcs in the Arabic three, and the number four resembles the angle in the lower right corner of the Arabic numeral four. As a bonus, the D'ni five resembles the horizontal line at the top of the Arabic five.

Going back to the main table, once you've memorized the numbers one, two, three and four, the rest of the numbers are just variations of them.

The numeral for five is a numeral one which has been rotated 90° counterclockwise, and represents 1 x 5. The numeral for six is the number five with a number one added to it, seven is a the number five with a two added to it, eight is the number five with a three added to it, and nine is the number five with a four added to it.

To get ten, the number two is rotated 90 degrees counterclockwise, representing 2 x 5. Fifteen is a rotated number three, and twenty is a rotated number four.

There are only four numbers which do not follow that system. They are zero, twenty-five, cyclic zero and infinity, because infinity is written as a numeral, rather than a mathematical symbol.

The symbol for zero is a dot. Twenty-five can be written two ways: as an x when it's being shown as a single digit, or one-zero when it's part of a numerical progression. The symbol for infinity is an x over a cross, and the symbol for cyclic zero is a diagonal line. Cyclic zero is a special number that is used when a numeric sequence is repetitive, and takes the place of both zero and the highest number.. It's most often seen on D'ni timepieces.

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, etcetera. 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.

Are there any questions?

Correlation of D'ni Letters and Numbers

Click on the button at the bottom of the page to see example #2. Let me know when you're ready.

The D'ni had thirty-five letters in their alphabet. There were twenty-four base letters, and eleven letters with alternate pronunciations. The letters are made from stems that can have tops and bases added to them.

An explorer named Christopher Gilson was the first to notice that there is a connection between D'ni numerals and D'ni letters. He discovered that the the shapes of the glottal stop and the four stems used in D'ni letters looked very much like the symbols for the numerals zero, one, two, three, and four if you removed the boxes. This was later confirmed by Richard Watson, who said that he'd wondered if anyone would ever notice it.

That caused him to re-examine the other numbers, and he noticed that they seemed to have a relationship to D'ni letters with bases and tops. This lead in turn to the discovery that D'ni alphabetical order is, in fact, numerical order. The letters are arranged according to their corresponding number, from zero to twenty-four. The eleven letters with alternate pronunciations are kept slightly separated and are also in numeric order.

This means that instead of having a separate alphabetic order, D'ni letters are arranged in aphanumberic order.

A question that the Myst community has had for a long time was whether the D'ni letters were dirived from the numbers, or if the numbers were dirived from the letters. The answer is that the letters came first, and the numbers were created by simplifying their shapes. Richard says that he'd intended for the numbers to just be the base shapes. It was Rand Miller who wanted to put them in boxes, and since he was the boss, that's what they did.

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Are there any questions?

Base 10 versus Base 25

Click on the button at the bottom of the page to see example #3. Let me know when you're ready.

This chart shows some conversions from base 10 to base 25 and how the numbers appear in D'ni. The first four numbers are places in D'ni mathematics, and the final three rows show some of the differences in how base 10 numbers are expressed in base 25.

The fifth line is how Gehn wrote "233rd Age". In one of his journals, he wrote in English and mentioned the "98rd Age". The "rd" doesn't make sense until you realize that he was simply directly transliterating the base 25 figure without converting it to base 10. Because of that, many explorers mistook the name as Age Ninety-Eight, which is wrong. It's really his two hundredth and thirty-third Age, which explains the "rd" in the journal entry.

The curly brackets in the seventh line are used because a single place in D'ni notation can be from 0 to 24, and that cannot be expressed as a single digit in Arabic notation. The brackets 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.

Are there any questions?

Words for D'ni Numbers

Click on the button at the bottom of the page to see example #4. Let me know when you're ready.

This chart shows the basic D'ni numbers. In D'ni, one does not say "six". Instead, the term 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 that 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 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 chart, the words for the numbers used for D'ni months are written below the regular ones.

Are there any questions?

D'ni Numeric Places

Click on the button at the bottom of the page to see example #5. Let me know when you're ready.

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.)

In practice, a suffix is attached to a number indicating its place and value, and then single place numbers are 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).

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

Below that example is an explanation of the known place value suffixes.

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D'ni Mathematics

Click on the button at the bottom of the page to see example #6. Let me know when you're ready.

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 we have no idea what D'ni mathematic symbols look like, I'm going to fake a few for the purpose of these 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 might have used a system of notation in which the operands preceded the operators. As an example, take this simple problem, 5 + 5 = 10. Richard told me that when they were discussing D'ni numbers in the planning stages, they toyed with the idea of D'ni mathematics and Rand was the one who proposed using this system. However, it never got any farther than that one discussion because they never needed to show math equations in any of the games.

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 may 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.

Are there any questions about any of the material I've covered today?

Thank you for attending my lecture today. I hope this has helped you and that you found it fun and interesting!