It would be very interesting to see how accurate of a measurment the D'ni could actually get with their "advanced" instruments...[hint hint]

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You can't use the base of the d'ni numbering system to argue about their existence of higher d'ni math - it doesn't make sense!

1. In applied math, base ten and base twenty-five have the same sorts of limitations. You can't finitely express 1/2 in base 25 "decimals". You can't finitely express 1/3 in base 10 decimals. So what? You can't finitely express pi in either base, but that doesn't stop us from using it. So I see no reason to argue that d'ni technology would have to be any more or less precise or accurate (note the difference) than ours as a result of using a different base for their number system.

2. In theoretical math, there's no reason for using a base for a number system anyway! (Except in certain branches of mathematical logic, in which base 25 would be slightly easier to use - but only just.) I do theoretical math all the time, and nothing I do ever depends on my using base 10.

I see no reason to think that the d'ni had no higher math, and plenty of evidence that they did. So I'll conclude that they did have plenty of math until more evidence comes in.

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KI# 45434

JKla wrote:

There is a presumption here that the D'ni only used base 25.

I hadn't thought of this. In computing we frequently use binary, octal and hexidecimal...perhaps the D'ni might have also used bases other than the one they were most comfortable with, strictly for technical purposes. An interesting possibility at least.

Owehn: 1. Sure, but 1/3 is a rational number. Pi isn't, but we round off when it is applied to engineering, architecture, etc. I am saying that the D'ni did not have math as its own science (supported by the fact that we have found no equations anywhere) but did have it in applied form (supported by the fact that we find architectural sketches all over the place).

2. I'm really not sure what theoretical math you are doing. First, the existance of irrational numbers is necessary to formal proof. In addition, logrythms appear time and time again in all sorts of places. Logrythms necessitate the use of a base (even irrational bases such as the highly important natural log).

Alahmnat: There are no rough estimates in pure mathematics. 1/3 is exactly 1/3. Pi is exactly Pi. The estimates only arrise when we have to physically construct something as we are limited in physical ways of doing things.

Again, I am alleging that the D'ni used ratios and the D'ni level of science is the extrapolation of Greek science.

Mrchapel, I'm a math student. I look around my room, and I don't see any equations anywhere. Does that mean I don't use them? Of course not. Why is it that just because you haven't seen any equations anywhere in d'ni, you assume that the d'ni didn't have them?

About the math I study: Let me check...algebra, geometry, topology, analysis...nope, none of them have anything to do with using 10 for the base of our number system. Irrational numbers are defined without reference to any base, 10 or otherwise. Logarithms are important, but only the natural log has any mathematical significance - the rest are just constant multiples of it.

Have you ever constructed the space of real numbers from the rationals? It can be done, and in many ways. So "the extrapolation of greek science" means nothing to me.

I am trying and failing to understand your position, mrchapel. Please tell me why you think the d'ni did not have pure math.[/i]

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KI# 45434

Owehn: I too am a math student. I find equations written in my note books all the time. Historically, looking through the notes of others has also given us random equations, even in non-mathematical notes. Fermat's Last, for example, was found in the margin of one of his notebooks.

Bases are important primarily in number theory/numerical analysis. However, as you correctly point out, analysis does not specify a base.

The 'extrapolation of Greek science' may mean nothing to you as you probably have not studied any classics:
The Greek veiw of mathematics is reflected very heavily in there society. Namely their architecture and economics. They conceived of numbers as descriptive words applied to quantities of items as opposed to entities in and of themselves. The prevailing thought amoung Greek mathematicians was that any quantity could be expressed as either a number or as the ratio of two numbers. A ration is not quite a fraction as a fraction stands alone and in Greek mathematics a number by itself was meaningless. Greeks had a very decent understanding of geometry, as apparent in some of their architecture. The drafting constructions (constructions using only a straight edge and a compass) that we have today were contributions of the Greeks. (Although they left puzzles which we later solved with more advanced techiniques such as the infamous trisection of an angle.)

Quite possibly their most well known contribution deals with the sides of a right triangle: a^2+b^2=c^2. However this theorem yeilded a fatal inconsistancy in their thinking. An isosoles right triangle with legs of measurement 1 had a hypontenuse of measurement 2^(1/2). As a student of mathematics you are well aware that this quantity cannot be expressed as a ratio between two counting numbers. Soon after this discovery 3^(1/2), 5^(1/2)... were added to the mix. Then came the cubic roots and roots of n. Patterned numbers (.565565556...) were added. As modern mathematicians we take these numbers for granted, as Greek mathematicians they were dumbfounded. By in large, however, the idea that all quantities could be expressed as ratios remained prevalent as it had already proven to work in the real world.

In addition the Greeks did not have a concept of a negative number. It was a big enough leap to say that there were 0 of some object. It was too far for them to say that there were less than 0 of some object. This might be something of a saving grace for them, actually. If they couldn't find a ratio to represent an irrational number, imagine them finding a ratio to express an imaginary number.

Pi was mentioned before as an irrational that has been in use for quite a while. Pi is humanity's attempt to 'square the circle.' Sadly it can't be done, but we do need approximates. The Greeks were able to determine the existance of Pi by defining it as the ratio between the length of the diameter and the length of the circumference. Using very precision instruments they were able to calculate Pi well enough. Hell, even the most anal of NASA minds don't use more than 15 digits of Pi. However, Pi mysteriously springs up in so many other places.

The big equation is: e^(i*Pi) + 1 = 0. You don't get this from measurement.

Why do I believe the D'ni had a similar veiw of math as the Greeks? It fits into the society. Their major acheivement is through general description, not mind numbing precision. Their theory only needed to take them as far as their saws and pens could. Hell, a lot of their architecture is even reminicent of classical or Gothic styles. Most importantly it fills plot holes. It explains why we have come across feats of engineering, architecture, chemistry, and many other sciences, but have never once seen notation for anything in mathematics. It explains why most the machinery we experience has a hand crafted feel and at times looks like a highly tested Rube Goldberg device. It explains why we don't find any notes about the speed of light or such among the D'ni ruins. It explains the question of the original post. It fits as to explain Yeesha's natural talents and why she can go so far beyond past D'ni. Finally we haven't seen anything directly to the contrary (yet).

As one (hopefully) final example I will cite the proof of 1 > 0. If a is not 0 then either a is a postive number or -a is a positive number. Then a^2 is either (a)(a) or (-a)(-a) which are both positive numbers. If we let a = 1 then a^2 > 0 becomes 1^2 = 1 > 0. To most non-math people this proof seems pointless. After explaining it to one friend she smuggly replied that she had a more elegant proof. "There are no objects in my hand, if I increase that number of objects I now have one object in my hand. Thus 1 is greater than 0." For the D'ni, the latter was proof enough.

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I'd like to make a point here (from an engineering perspective) Pi is a pretty big number and es you can take it to a high degree of accuracy but depending on the components you use will depend on how accurate you calculate things, as a rule we're taught to keep things to the lowest level of accuracy. So if I have three values and two are to the tenth place and one is to the third place you take everything to three places. Not because accuracy is a bad thing, but because your increasing inaccuracy if you don't.

The GZ works through neutrino technology (or so the DRC tell us), if your looking at creating neutrinos and measuring them in a usable way then having a firm grasp of particle physics is a must. Long range low power data communication will require an understanding of digital electronics and DSP, suggesting that they will hav edeveloped their own from of laplace, furrior and z transforms, as well as advanced binary system design.

Base 25 is just a different base, I frequently switch between base 8, 10 and 16, I can't see the difficulty in moving to 25, although I'm curious why they use 25, base 10 is believed to have been started because we have ten fingers, base 2 because of binary logic, base 8 for computer chip design and base 16 was a advancement on base 8. So where does 25 come from?

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Why do I believe the D'ni had a similar veiw of math as the Greeks? It fits into the society. Their major acheivement is through general description, not mind numbing precision.

I would hardly say that the descriptions you're talking about are general. Our best understanding of them is that they are highly precise, yet compact.

Quote:

Their theory only needed to take them as far as their saws and pens could.

Hence the examples such as the Gravitation Age, which needed extremely precise and accurate planning to work out correctly. [quote]Most importantly it fills plot holes. It explains why we have come across feats of engineering, architecture, chemistry, and many other sciences, but have never once seen notation for anything in mathematics.[/i]You don't need your assumption to fill the plot holes. We have come across the feats themselves, not the endless planning it took to create them. How many original d'ni texts have you read? Kadish's note, maybe Kenen Gor (and we suspect the latter is recently printed). Neither has anything to do with mathematics - would you have expected a mathematical graffito like Fermat's Last Theorem on one of them?

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It explains why most the machinery we experience has a hand crafted feel and at times looks like a highly tested Rube Goldberg device.

I don't get that feeling at all. The great zero is still functional hundreds of years after the d'ni left. And in what way does a neighborhood imager look like a rube goldberg device? Unless you're talking about Riven, which doesn't count. Gehn didn't have access to d'ni while he was constructing his stuff, so of course it looks handmade.

Quote:

It explains why we don't find any notes about the speed of light or such among the D'ni ruins.

You think the DRC would leave the original d'ni documents lying around for explorers to get their hands on? That's a stretch.

Quote:

It explains the question of the original post. It fits as to explain Yeesha's natural talents and why she can go so far beyond past D'ni. Finally we haven't seen anything directly to the contrary (yet).

The question of the original post was "how do the d'ni express their mathematical operations, and special numbers like pi and e?" Are you claiming that they didn't represent these at all, just because we haven't seen any notation for it? And Yeesha can go far beyond any other d'ni writer because she can do math? And has been mentioned many times: If the d'ni had no higher math, we wouldn't see it. If the d'ni had higher math, will probably still wouldn't see it. We don't see it. How can you conclude that the d'ni had no higher math?

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KI# 45434

Quote:

It seems as if at some point the D'ni people just threw up their hands and said "close enough" and called it a day. Also keep in mind that a "close enough" attitude would allow for math beyond that of simply positive real numbers just because it no longer requires the rigor of proof.

While the question of whether this is an accurate observation or not seems to have been adequately discussed in the rest of the thread, the comment itself parallels something that I thought of last night while translating the numbers over the doorways in Bevin. They translate as follows:

D'ni 6497 (base 10 = 96,482)
D'ni 7840 (base 10 = 114,475)
D'ni 4649 (base 10 = 66,359)

I have always held the completely unfounded belief that they are house or apartment numbers (Bevin itself contains no proof or disproof of this I have observed.) And while wondering whether each grouping was meant to be read as a list of its individual digits, or as the quantity it represented, and why the D'ni would label three doors right next to each other with a variance in digits in excess of 19k base 10, it suddenly occurred to me: to an entity accustomed to using base 25, larger range values of numbers probably seem "close" to one another than they do to us in base 10. The
quantities of 66,359 items and 114, 475 items might seem about as closely related to the D'ni as the quantites of 4600 items and 7800 items do to us, because both can be rendered with 4 digits in their respective systems.

Assuming that's true, it makes me wonder about their psychology. I've read the parts of the thread that discuss the relative utility of base 25 versus base 10, but it was never spelled out: is there any hypothetical basis anyone knows of for why a culture would develop base 25 as their primary system of mathematical thought? (We know the D'ni looked more or less like us-- unless they were hiding five extra digits someplace ) Were they somehow pre-disposed towards dealing with larger quantities?

Quote:

is there any hypothetical basis anyone knows of for why a culture would develop base 25 as their primary system of mathematical thought?

Sure, base ten seems natural because we have 10 fingers. But so would base 5, if you just think about one hand. (Note how common the hand motif is in D'ni art, generally just one hand.)

In fact, base 25 represented the D'ni way makes sense from just that stand point. They use base 25, but their numbering system is more base 5^2. [5] in D'ni numerals is just [1] turns 90 degrees. [10] is [2] rotated, etc. You can look at your hands and say - right hand is 1,2,3,4,5 and the left hand is 5,10,15,20,25.

The reasons for a base system used across a lay culture is generally cultural. People use a base because that is what everyone uses - ubiquity is often the most important factor, rather than efficiency (you can see that sort of thing everywhere - metric vs. standard, VHS vs. Beta, etc). All it would have taken is for that numbering to become popular enough that it became ubiquitous and in general, people would just use that.

However, specific disciplines will use their own standards for their own reasons (binary, octal and hex are used in computing because they are very useful for what computers do, linguists use IPA because it describes what they are interested in, etc). No reason the D'ni would be any different.

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The presence of the rotated symbols and number names such as "vahgahfah" ("5 and 1") suggest that the D'ni originally had a baseless, additive numbering system -- like Roman, Bahro, or Narayani numerals. Later, when their culture switched to a positional numbering system, they simply adapted their existing numerals, instead of adopting a new set of numerals as we did. Using 25 as a base might have seemed like a more practical adaptation for day-to-day use than switching to base 5, as base 5 would require far more digits to represent a quantity of even modest size.

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I'd buy that as a reasonable theory of why base 25 stuck.

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BrettM wrote:

Using 25 as a base might have seemed like a more practical adaptation for day-to-day use than switching to base 5, as base 5 would require far more digits to represent a quantity of even modest size.

Good thinking - another possible reason for base 25 to stick is that it - in the way the d'ni did it can be easily converted to base 5 in the mind.

Effectively their base 25 notation can also be considered a base 5 notation at the same time.

e.g. [20] = (4)(0) - with square brackets indicating base 25 (supply d'ni font) and round braces base 5

Personally I don't read the d'ni numbers as base 25, but rather as a base 5 system - which is perfectly possible.

So it's perfectly possible that advanced d'ni mathematis weren't done in base 25, but actually in base 5.