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« on: 04 09, 2008, 10:56: AM »

We assume that Klingons use a base 10 numbering system. This is reasonable as they have 10 fingers. The fact that humans use a base 10 system instead of a base 20 system is testament that shoes were invented before arithmetic.

But is there an alternative? Not all Earth cultures used base 10. The Sumerians used a base 60 system that still lives with us today -- there are 60 seconds in a minute and 60 minutes in an hour whether you are measuring angles or time. We also have a strong base 12 following - things are measured in dozens (like months and inches in a foot) and grosses.

Why do we have 24 (12 x 2) hours? Why not 10 hours? This would make the current hour 2.4 hours long and the current number of minutes in this hour is 144 (12 x 12) but we can redefine a minute to be 1/60 of this 2.4 hour long period. Same for seconds ... Or for that matter we could have gone completely decimal with time by dividing the hour into 100 minutes and minutes into 100 seconds. We didn't; we opted for a base 12/60 solution.

Why do 12 and 60 come up? Well, 12 is evenly dividable by 1, 2, 3 and 4. 60 is evenly dividable by 5 and 6 as well.

The Klingons could also come up with some other numbering system based on nature. For example, if there were 6 prominent, visible planets in the Klingon solar system, then base 6 might have evolved to honor these gods.

Binary, octal and hexadecimal are Johnny-come-latelies in the mathematical world. Klingon computer scientists would, no doubt, come across these as well as other systems based on the powers of 2.

But then maybe they measured three states of a circuit. We recognize off (0) and on(1) they might measure off(0) on(1) and negative on (-1). In this case, there would be base 3, 9, 27 and 81 derivatives of numbering systems.

Then there is the representation of the numbers themselves. We tried a lot of systems before coming up with one where the position of the digit has as much significance as the digit itself. It is only within the last 1,000 years that the set of squiggles invented by the Arabs (Arabic Numbers) became the global standard for representing numbers.

Then there is the math itself. The same rules will apply, such as the distributive law of multiplication -- A * (B + C) = AB + AC. However, in our conventions, absent parenthesis, multiplication and division come first and addition and subtraction are then applied. That is, on Teran, 3 * 4 + 3 = 12 + 3 = 15. The Klingons might choose the opposite where 3 * 4 + 3 = 3 * 7 = 21.

If I had to bet on what numbering system Klingons use, I would pick base 10 because of the 10-finger theory. However, I wouldn't bet much.



[Edit -- changed thread title]
« Last Edit: 04 09, 2008, 10:58: AM by Kesvirit » Logged

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« Reply #1 on: 04 09, 2008, 08:54: PM »

     Actually, we don't assume.   If you take the Klingon Dictionary as canon(it is the official Klingon language after all), it specifically states the original numeration system for Klingons is trinary (0,1,2,10,11,12...), with decimalization being adopted for easier interaction with other spacefaring civilizations such as Terrans...    It is difficult to immagine a species using trinary numbers to calculate warp mechanics to get out to space to realize that everyone else worth talking to seems to use larger bases such as 8, 10, or 12.

    Particularly interesting is that while the Arabs do appear to be the best known to use a zero placeholder system, it seems that it was used in India as early as the 5th century AD.   That and the Arabs actully do not use the same "Arabic" numerals that we use!   

« Last Edit: 04 29, 2008, 12:53: AM by Klythe » Logged
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« Reply #2 on: 04 10, 2008, 10:56: AM »

I am not sure how base 3 would work out for describing warp mechanics. However the typical 10-digit Terran phone number would require 21 digits in a trinary system. While remembering one's phone number might require the mathematical skills associated with a PhD in nuclear physics, the keypad would be simple - 5 keys (including the * and #). However the display would have to be more than twice as large or it would require scrolling (not a problem with older rotary-dial systems) . What would Klingon cell phones look like? Would they come in pink?

What would a base 3 keypad look like? I am looking at my inverse keypad on my keyboard (for some reason adding machines and touch-tone dialers and their respective descendants were designed 180-degrees opposite one another). Anyone here have experience in Klingon-factors design?

What other impact does a base-not-10 have on the design of physical objects?

Finally, what is a trinary digit called? A trit?

I do have a couple more questions on the topic.
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« Reply #3 on: 04 11, 2008, 12:08: PM »

If you take the Klingon Dictionary as canon(it is the official Klingon language after all), it specifically states the original numeration system for Klingons is trinary (0,1,2,10,11,12...)   

Close, but not completely accurate. You used 0, 1, and 2 in your description. However, the old system uses the numbers 1, 2, and 3, with no 0.

TKD says the original counting system went like this: 1, 2, 3, 3+1, 3+2, 3+3, 2x3+1, 2x3+2, 2x3+3, 3x3+1, 3x3+2, 3x3+3... "and then it got complicated."  It's not trinary in the typical sense. The more proper term is "ternary". (Roger Cheesbro composed a full explanation of the system a number of years ago, but my link to it online points to a page which no longer exists.)
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« Reply #4 on: 04 13, 2008, 11:46: PM »

So let me see if I have this correct. Klingon arithmetic is sort of a reverse notation from Terran as far as numbering positions go.

The leftmost figures represent the number of "3's" that you have already amassed, while the rightmost figure is the next power up.

Hmmm, though I think it would be difficult to do advanced mathematics without the concept of a zero. The Klingon equivalent of Sir Issac Newton must have been stymied by it while inventing the Klingon equivalent of calculus.

I am still wondering what the effect of an alternative numbering system would be on devices from telephone keypads to spacecraft instrumentation.

Obviously the Klingons only need 5 symbols to describe their numbering system (1, 2, 3, + , x) - what do these look like?
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« Reply #5 on: 04 14, 2008, 12:11: AM »

    Rats.  I know I should have doublechecked before posting that. 

    Klingon Cellphones?   Why would they even have them?   Klingons are very independant and less interdependant than Terrans.   Also they perfer to to most of their communications face to face.  The need for a numerically switched telecommunications system with millions of nodes would probably never occur to Klingons, even if they had an industrial or modern technological phase.  I would assert they most likely did not, given that the times of Kahless is portrayed as a quasi-medieval level of technology.  The chronologies I've seen place the Hur'q invasion about 200 years after Kahess.  I tend to think the Hur'q didn't hand out technology to the Klingons as much as the Klingons took it from the Hur'q, jumping straight from the early steel age to the tri-tanium age.  I talked more about this in Origin of Klingon Technology.

    Klingons don't communicate the same way Hyoomins do.  Klingon communication systems would most likely have been fixed station to station with only a few mobile communications units used when necessary.  Klingons are not chatty and routine conversations are more terse than the hyoomin equivalent.  For example a Klingon might communicate at the time of departure that said Klingon is comming to visit, but probably not call again if delaying in traffic or things like "I'm 15 minutes out, want me to pick something up at the Death-in-a-box?"

     Klingon Newton?  Klingon Calculus?  There might be Klingon Newton's now, but as I explained abouve, the Klingons probably took the scientific knowledge from the Hur'q where they could and reverse engineered the rest after overthrowing the invaders and hunting them down through the galaxy.   We don't know what numerical system the Hur'q used.  But I would suspect Klingon reverse engineers would probably adapt to and adopt the Hur'q system to allow them to hunt down the Hur'q in their own ships and exterminate every last invader they could find. 

[ Split Topic - see Anatomy of Speech and Neandertals... Also added below -Klythe]


We assume that Klingons use a base 10 numbering system. This is reasonable as they have 10 fingers. The fact that humans use a base 10 system instead of a base 20 system is testament that shoes were invented before arithmetic. 

    I'm not sure this is true...  Greeks, Romans and most classical age cultures wore sandals.  But there is evidence of moccasin type footware used by Neandertals, which when you think about it, they'd have to have had...  But I think the reason is that fingers are much easier to move, so as to hide the extra fingers and hold them up to display the number of fingers to a friend.   The reason you probably have never heard anyone ask,  "How many toes am I holding up" probably has less to do with shoes and more to do with the flexibility of the toes.

    This brings up an interesting point, that Klingons have words for each of their digits, that includes four fingers and 1 thumb on one hand and 5 toes, and most have other meanings associated with them.  There is the implication that Klingon toes have a greater range of motion than Terran Homo Sapien toes.  Perhaps they might have developed a base 20 system...
« Last Edit: 04 29, 2008, 12:54: AM by Klythe » Logged
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« Reply #6 on: 06 08, 2008, 07:52: PM »

I just have this problem about Klingons changing their numerary system from a 3 numbers based to a 10 numbers based. It is so radicall that itīs nearly impossible. When during the French Revolution the metric system was established, they had developed a 10 hours day and a 100 minutes hour and a 100 seconds minute, but it was too much, was considered unpractical and never put into practice.

And about the metrical system, itīs not been completely accepted yet. Imperial and U.S systems still survive. I donīt see people in the U.S.A willing to chage their miles to kilometers, their galons to liters, an so.

I think that leads us into the historical question of how a ten digit system was set among a traditional klingon society.
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« Reply #7 on: 06 08, 2008, 08:41: PM »

...And about the metrical system, itīs not been completely accepted yet. Imperial and U.S systems still survive. I donīt see people in the U.S.A willing to chage their miles to kilometers, their galons to liters, an so.

I think that leads us into the historical question of how a ten digit system was set among a traditional klingon society.
Good question, but the metric system not being "accepted" by the US Citizenry is a case of stubbornness and megalomania, if you ask me 8-)

I think the Klingons "changed" their numeric system for one specific non-fictional reason: Humans are used to the digital system, and devising another is much more difficult than explaining why Aliens changed theirs...
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« Reply #8 on: 06 13, 2008, 05:16: PM »

I believe Marc Okrand said in either Conversational Klingon or Power Klingon that Klingons changed their additive numeral system to a positional one because it was more practical when dealing with high-tech systems ("Bekk, what's the power read-out!"  "Sir, it's three plus three plus three plus three plus three plus three..." *KA-BOOM!*).  That being said...  ...why base-10?  What not, say, base-8 or base-16, which are easy to convert into binary digits and therefore practical for a computerized society?  Why not base-9, which is base-(3+3+3), and thus a bit more in line with earlier conventions?  Well, I suppose one can always think of reasons for this:

  • As somebody mentioned, Klingons have ten fingers, so when counting with your fingers you can remember each full set of fingers as a multiple of ten.
  • Furthermore, Klingons are symmetric; they have two eyes, two arms, two legs, two livers...  Considering how important different body parts are to Klingons (eyes, arms and legs when fighting, livers when drinking...), they may have wanted to base their numeral system on an even number.
  • Perhaps the Hur'q used a base-10 system?  After defeating the Hur'q and taking their technology, Klingons may then have taken their numeral system in order to adapt to a new way of life.
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« Reply #9 on: 03 04, 2010, 02:07: AM »

for those interested in math and music:
http://www.klingon.org/smboard/index.php/topic,1810.0.html
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« Reply #10 on: 03 04, 2010, 05:08: AM »

So let me see if I have this correct. Klingon arithmetic is sort of a reverse notation from Terran as far as numbering positions go.

The leftmost figures represent the number of "3's" that you have already amassed, while the rightmost figure is the next power up.

Looking at the previous explanation no.  For that to be the case the number after 3 would be (3x(3^0))+(1x(3^1)) when instead it is ((3x1)+1).  It says something about the determination of Klingons that they were able to do any math at all with such a clunky system of counting.  This is the sort of thing that kept the ancient peoples of earth from developing thier math skills outside of geometry.  The idea that Archimedes managed to find the area under a curve seems to be widely accepted today, but he did so through exhaustive calculations instead of integration.  That's a guy widely regarded to be one of the smartest humans to ever walk the planet.  The average Klingon must be smarter than three or four genius level humans together to build an advanced society with such an unwieldly number system.
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« Reply #11 on: 07 02, 2010, 07:25: PM »

Lately I've been reading a book called Räknekonstens kulturhistoria (From One to Zero: A Universal History of Numbers in English, or Histoire Universelle des Chiffres in the original French), and I was interested to see that many ancient Earth cultures had a system very much like the Klingon one, representing numbers in groups of three or four.
It appears that the Human mind has difficulty dealing with groups larger than that, and our ability to do so is an incredible feat of abstract thinking.  If we see a group of one, two or three objects, we can immediately recognize their number.  The same is often true for four object.  For five objects or more, unless they are in some recognizable configuration which makes their number clear, we need to do some more thinking.  For instance, you can easily see that there are five dots in the image below, because you recognize that pattern.  However, if those five dots were arranged in a horizontal row, it's likely your first thought wouldn't be "five dots", but rather "three dots and two more".
(source)
One sign of use having the trouble making out numbers larger than four is the old habit of counting as follows:
|, ||, |||, ||||, ||||, |||| |, |||| ||, |||| |||, ...
Similar to the Klingon system, but with five instead of three.

In a language spoken by tribes on the west side of the Torres(!) strait, they count urapun (one), okosa (two), okosa-urapun (one-two), urapun-urapun (two-two) and then they just say ras (many).
Furthermore, in the original Roman calendar, the months were Martius, Aprilis, Maius, Junius, Quintilis, Sextilis, September, October, November and December.  Note that only the first four have proper names; the rest are named for counting words (quinque, sex, septem, octo, novem and decem; five through ten).  Also, only the first four sons of a family would recive proper first names; after that, they were named for counting words (so the fifth child was named Quintus, the sixth Sextus, the seventh Septimus, the eighth Octavius and so forth).

The book also discusses the importance of our ten fingers.  Before Humans understood the abstract notion of numbers, they understood the concepts of fewer than, more than and equally many.  So, maybe you couldn't tell somebody to go down to the market and buy six furs, but you could tell him/her to buy a fur for each finger on his/her left hand, and one for the first finger on his/her right.
Many more intricate systems existed and still exist today, which include using one's toes and elbows and so forth to count.
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« Reply #12 on: 07 06, 2010, 04:28: PM »

The original quote from Conversational Klingon is:

Quote
We are moving on to numbers. As in Federation Standard the basic count in Klingon is one to ten. Long ago Klingons counted by threes, so the number after three was literally "three plus one", for "seven" they would say "two threes plus one" and so on. Fortunately for us, over time, the system changed, probably due to interactions with other worlds and a desire to learn about advanced technology. In any event, counting should not be too hard for you.

I've always assumed, based on this and what was said in TKD, that the Klingon's switched to a base 10 system at the point that they gained knowledge of other races. To me, this implies that they switched once they met the Hurq, since that was their first interaction with "other worlds". In order to use and understand Hurq tech, they had to switch to the system the Hurq used. Prior to the Hurq invasion, Klingon's only had access to primitive technology so they never had to deal with doing warp calculations (or even advanced math) with their ternary system.
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« Reply #13 on: 07 06, 2010, 09:29: PM »

I've always assumed, based on this and what was said in TKD, that the Klingon's switched to a base 10 system at the point that they gained knowledge of other races. To me, this implies that they switched once they met the Hurq, since that was their first interaction with "other worlds". In order to use and understand Hurq tech, they had to switch to the system the Hurq used. Prior to the Hurq invasion, Klingon's only had access to primitive technology so they never had to deal with doing warp calculations (or even advanced math) with their ternary system.
Do we know that the Hur'q were the first alien lifeforms that the Klingons encountered?  In The Klingon Way, it's mentioned that the Empire conquered other worlds under Kahless' rule, and Kahless had been dead for about five-hundred years when the Hur'q stole the Sword of Kahless.

I'm thinking that maybe Klingon scientists started using a positional system long before the general populace did, and that they were able to send warriors to other solar systems at impulse speed by the 9th or 10th century (with maybe five engineers on a ship with hundreds of warriors).  Much of the Klingon population remained uninterested in or ignorant of such mathematical tools for a very long time, however, causing technology to evolve very slowly (supposedly, Klingon history is longer than Human history, allowing them to reach space flight earlier even while developing at a slower pace...  ...or perhaps the Klingon Empire was just incredibly eager to get to space and start conquering, and then grew complacent once they had developed that ability).  Then, when the Hur'q invaded in the 15th or 16th century and the Klingons bested them, Klingon technology received a huge boost.  Now, even warriors had to familiarize themselves with the workings of their ships in order to be useful in battle.  Furthermore, as they learned to travel at warp speed and to use subspace communication, interaction with other species grew more frequent and more intricate, forcing the general populace to adopt a numeral system that made sense (and chose the base-10 numeral system either because that's what scientists had been using for centuries, or because that's what the Hur'q had used, or because the Emperor at the time didn't like base-8).
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« Reply #14 on: 07 08, 2010, 09:31: AM »

Do we know that the Hur'q were the first alien lifeforms that the Klingons encountered?  In The Klingon Way, it's mentioned that the Empire conquered other worlds under Kahless' rule, and Kahless had been dead for about five-hundred years when the Hur'q stole the Sword of Kahless.


Only thing is, Klingons are generally portrayed as being more or less medieval at the time of Kahless.  conquering new worlds does not have to mean they went off planet were able to do so... Here in the UK, wines from places like California are still called "New world" 
Alot of people dont like the novel Kahless but the characters in that book very definitely did not have space travel. And if you wish to argue that it is not canon, it is no more or less canon than "the Klingon way"

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« Reply #15 on: 05 30, 2011, 03:54: PM »

Do we know that the Hur'q were the first alien lifeforms that the Klingons encountered?  In The Klingon Way, it's mentioned that the Empire conquered other worlds under Kahless' rule, and Kahless had been dead for about five-hundred years when the Hur'q stole the Sword of Kahless.


Only thing is, Klingons are generally portrayed as being more or less medieval at the time of Kahless.  conquering new worlds does not have to mean they went off planet were able to do so... Here in the UK, wines from places like California are still called "New world" 
Alot of people dont like the novel Kahless but the characters in that book very definitely did not have space travel. And if you wish to argue that it is not canon, it is no more or less canon than "the Klingon way"


Not sure how I managed to miss this post for so long, but anyhow, the exact quote from TKW is:
Quote
In time, and as more was learned about Klingon history and culture, the Federation's understanding of the role of Kahless in Klingon history changed. It is now known that Klingons consider him a great warrior who did indeed conquer other worlds, but united the Empire by giving the people the laws of honor which direct every Klingon's life.
I find it rather difficult to interpret this as referring to anything but actual alien worlds, but considering this is just a passing reference and most other soft canon and fandom seem to go against it, I suppose one had best ignore it.
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« Reply #16 on: 05 31, 2011, 01:16: PM »

That quote from the Klingon way does not in and of itself define worlds, and without our concepts of the old world and the new worlds our concept of the third world would be numbered incorrectly. However this is yet another time when we have to see the Star Trek Encyclopedia, despite being canon, as the odd man out. Before Rightful Heir Kahless was assumed to be from a space faring age. We had only ever seen General Kahless. FASA delved into his live a bit more. After Rightful Heir was when we saw the medieval Emperor Kahless. Fans started asking question and instead of writers answering them, we got a graphic designer telling us they were one and the same, and giving us that load of dung about Kirk not knowing what Kahless would have looked like. This has dug a hole for us. We either have to understand that Kahless has been the name of multiple individuals one who was an epic ruler and one who lead space fleets to conquer worlds or we have to understand that the Epic hero Kahless did indeed conquer other worlds which will require us to define what exactly a world is.

Fortunately we are not talking about having more than three Kahlesspu' as that would get us back to debating if there were four or three plus one of them.
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« Reply #17 on: 02 03, 2014, 09:48: AM »

We assume that Klingons use a base 10 numbering system. This is reasonable as they have 10 fingers. The fact that humans use a base 10 system instead of a base 20 system is testament that shoes were invented before arithmetic.

But is there an alternative? Not all Earth cultures used base 10. The Sumerians used a base 60 system that still lives with us today -- there are 60 seconds in a minute and 60 minutes in an hour whether you are measuring angles or time. We also have a strong base 12 following - things are measured in dozens (like months and inches in a foot) and grosses.

Why do we have 24 (12 x 2) hours? Why not 10 hours? This would make the current hour 2.4 hours long and the current number of minutes in this hour is 144 (12 x 12) but we can redefine a minute to be 1/60 of this 2.4 hour long period. Same for seconds ... Or for that matter we could have gone completely decimal with time by dividing the hour into 100 minutes and minutes into 100 seconds. We didn't; we opted for a base 12/60 solution.

Why do 12 and 60 come up? Well, 12 is evenly dividable by 1, 2, 3 and 4. 60 is evenly dividable by 5 and 6 as well.

The Klingons could also come up with some other numbering system based on nature. For example, if there were 6 prominent, visible planets in the Klingon solar kits, then base 6 might have evolved to honor these gods.

Binary, octal and hexadecimal are Johnny-come-latelies in the mathematical world. Klingon computer scientists would, no doubt, come across these as well as other systems based on the powers of 2.

But then maybe they measured three states of a circuit. We recognize off (0) and on(1) they might measure off(0) on(1) and negative on (-1). In this case, there would be base 3, 9, 27 and 81 derivatives of numbering systems.

Then there is the representation of the numbers themselves. We tried a lot of systems before coming up with one where the position of the digit has as much significance as the digit itself. It is only within the last 1,000 years that the set of squiggles invented by the Arabs (Arabic Numbers) became the global standard for representing numbers.

Then there is the math itself. The same rules will apply, such as the distributive law of multiplication -- A * (B + C) = AB + AC. However, in our conventions, absent parenthesis, multiplication and division come first and addition and subtraction are then applied. That is, on Teran, 3 * 4 + 3 = 12 + 3 = 15. The Klingons might choose the opposite where 3 * 4 + 3 = 3 * 7 = 21.

If I had to bet on what numbering system Klingons use, I would pick base 10 because of the 10-finger theory. However, I wouldn't bet much.


[Edit -- changed thread title] [removed excessive underlines --Klythe]

Interesting mathematic concepts. I have been searching for right topic of research for a while and being interested in maths I will concentrate on similar topic now.
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