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Number Base Comparisons 
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#1
Jul504, 11:20 AM

P: 44

I am a member of the Dozenal Society of Great Britain ( www.dsgb.orbix.co.uk ) . We maintain that base twelve is better than ten, and, in our own small way, try to examine and explore this point.
Anyway, my point is I was wondering if anyone had any interesting thoughts or ideas on comparisons between the main number bases (by 'main', I mean likely to ever be adopted or of potential use i.e. bases 2, 4, 6, 8, 10, 12, 16, 20, 24, 60); any interesting essays on this point; sidebyside comparisons of differing number bases and so forth. In short, I want this thread to be THE numberbases thread. 


#2
Jul504, 04:43 PM

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The drawback of small bases is that numbers have long representations; the drawback of large bases is many symbols to remember.
But it's irrelevant. The choice of base is largely arbitrary, and has no effect whatsoever on the mechanics of arithmetic or higher mathematics. In short, IMO, there's no reason at all to be concerned with it.  Warren 


#3
Jul504, 05:29 PM

P: 44

I shall therefore start with a basic list of divisions in octal, decimal and dozenal. Octal 1/2 = 4 1/3 = 2.52 52 52 1/4 = 2 1/5 = 1.4631 4631 4631 1/6 = 1.25 25 25 Decimal 1/2 = 5 1/3 = 3.333 1/4 = 2.5 1/5 = 2 1/6 = 1.666 1/7 = 0.142857 142857 142857 1/8 = 1.125 Dozenal 1/2 = 6 1/3 = 4 1/4 = 3 1/5 = 2.4972 4972 4972 1/6 = 2 1/7 = 1.86A351 86A351 86A351 1/8 = 1.6 1/9 = 1.4 1/A = 1.2497 2497 2497 Note that: useful fractions (i.e. 1/2, 1/3 and related fractions (1/6, for instance)) are more neatly expressed in dozenal. These fractions are obviously extremely useful in the real world, whether we are talking cutting up cakes to building cabinets. Dozenal has three fractions running to more than just one number repeated in the decimal parts, as compared to one for both octal and binary. However, also note that these fractions could be argued of not much use (when is 1/7 of any use?). Furthermore, a fair approximation of 1/5 in dozenal is 0.25, which is far closer to 1/5 than the decimal approximated figure to 1/3 of 0.33 (onethird in itself probably a far more useful fraction, inherently, than onefifth) Factors and Prime Factors: octal has two factors, one prime; decimal and two factors, both prime; dozenal has four factors, two prime. We must also consider that binary multiplication and division is the simplest and most easily attempted form of all; easier than tenery or decimal or any other kind. We've also to consider that arrangement by six is actually the most efficient method; That sixtyfour  the square of eight is the first cubic and square number; That twelve is the first abundant number; And a host of other things (hopefully this has gotten the ball rolling...) Bryan 


#4
Jul504, 05:49 PM

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Number Base Comparisons
It's still pointless. Sure, it's easier to represent the decimal expansion of 1/3 in "dozenal" than in decimal. Does it matter? No. Anyone with any sense would just call it 1/3 anyway. And when it comes to cutting up cakes or building cabinets, you're going to be limited by how accurately your saw can cut a line, not by how many digits are in your favored expansion of 1/3. Give me a break. If you're so worried about the number of factors in your numerical base, go back to base 60 like the Babylonians.
The only reason to ever worry about numerical bases is when you're building a machine, because the complexity of the switches depends strongly on the numer of positions it must represent.  Warren 


#5
Jul604, 02:17 AM

P: 44

No, I disagree. Again.
What about weights and measures? Basic fractions cannot be expressed in decimal system properly, yet there is an argument for dovetailing numerical base and the W&M system. Anyway, you still haven't addressed must of my post. Also, we have already dealt with why base sixty is no good (too many numbers) 


#6
Jul604, 02:20 AM

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The expansion of fractions is irrelevant if you keep the numbers as fractions, so your argument is limp.
As far as weights and measures, I know you're going to try to sell us the stupid and cumbersome British system. I, for one, don't care for your proselytizing. Millions of scientists, engineers, and others have already made the decision for you.  Warren 


#7
Jul604, 05:02 AM

P: 44

Your ignorance here demonstrated is supreme. For one, the decision was made for us by bureaucrats, politicians and revolutionaries. Secondly, I am not trying to sell the immensly practical Imperial System to you, I am here trying to discuss number bases. This is a maths board, after all, is it not? And yet you, despite being administrator, seem not to want to discuss a topic of mathematics. I find that incredible.
As to fractions, I see you've not addressed the point. I will with an example: frequently in metric countries goods are sold in binary or tenery multiples because they are simply more practical (4800mm lengths of wood, binary fractions of 1000g and multiples of 25g). You see, decimal just doesn't work, as it is incapable of expressing basic fractions properly. Thus, if you are to have a measurement system aligned to base, then the worth of that base being ten is highly doubtful. Judging from your hostility, I can see that you clearly are one of those hideous metricphiles I hear so much about ( ;) :p ). I would start on practical anthropocentric systems of weights and measures, but this thread is abut number bases. Thank you :) 


#8
Jul604, 05:23 AM

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1) The decision is made every single day by countless scientists and engineers around the world. We certainly still have the choice to use the imperial system  and some companies and some industries still do. No revolutionary figure has ever walked into a university and declared that the metric system is to be used in physics classes for... revolutionary reasons. Certainly I've never heard any political figure order me to talk in terms of kilograms, nor has any warrior ever put a gun to my head and told me to use meters. Engineers and scientists have many choices of unitary systems  even within the metric system, we have to choose between mks and cgs, for example. We choose units simply because they are the best for the job. If you're going to make the ridiculous claim that somehow the metric system has been forcibly imposed upon me, I'd like for you to support it. Who was this masked man?
2) You're the one who brought up weights and measures (and thus the imperial system), not me. I'm certainly welcome to laugh at is as much as I like, even if you wish I wouldn't. 3) Decimal is capable of expressing any fraction that duodecimal or hexadecimal can express. The difference is largely moot, and it's really ridiculous to spend your time trying to change something as utterly pointless as arithmetic base, particularly in a society where almost all arithmetic (past grade 4) is done by machines anyway. If you really think the convience of being able to express a couple of smallnumber fractions with whole numbers is so important that we should discard the arithmetic system used by every single culture on the planet for thousands of years to use it, you're out of your goddamn gourd. And I'm proud to say it.  Warren 


#9
Jul604, 12:17 PM

P: 44

Or not. Metric has almost always been imposed on craftsmen, ordinary people, and engineers. Soem of the most notable actos of imposition were: *1871 Germany Imperial Decree of Wilhelm I *1793 France Drastic compulsory law *1823 French Compulsion reimposed *Latin america compulsion in the late 19th Century *Plus the current business with the 'metric martyrs' in Britain, where traders are prosecuted for using English/Imperial units.... ...So people have a choice, do they? Not when the Government(s) compel them otherwise, they don't. No revolutionary figure... for revolutionary reasons France. The Terror. They wanted to get rid of all history they even abolished the Christian calendar and introduced decimal time. Why? Because they hated everything old and wanted a completely new order. For a few years after the revolution, before the calendar was dropped due to its absurdity, years were reckoned in years since the revolution. As already demonstrated, metric has been forced on the people whether they wanted it or not. I brought Imperial into it, as it is an example of how number bases are relevant. In fact, if we had used a dozenal or binary basis for attaining the metre, the metre would have been 11.7355" (divided into 16 units of a digit size binary) or 10.98966" (divided into 12 inchsized units base twelve). In short, anthropocentric measures were attained. Likewise, if we say, has 5/8" as the basic size, decimal multiples of this would all be anthropocentric (6.25 " shaftment/handwidth, 62.5" pace, 62,500" mile) Who said I was in favour of change? I thought this thread about debating the relative merits comparing different number bases. SIlly me. By the way, many cultures have not used base ten. Some have been sixty, soem have been eight, some have been twenty. So, back to number bases then... 


#10
Jul604, 12:53 PM

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The problem with moving to a different base is although more talented mathematicians may be to easily switch it will take a few generations before the confusion finally goes. Furthermore a change base would provide no actual advantage for mathematicians.



#11
Jul604, 01:02 PM

P: 44

Absolutely, the confusions resulting if we did switch would almost certainly be almost insurmountable. However, many mathematicians and philosophers favour/ed the change to base twelve, eight, sixteen sixtyfour, four or two (Leibniz, Charles XII, Spencer, Bernard Shaw and a bunch of others)



#12
Jul604, 03:35 PM

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The metric system does not shine because the kilometer is really a better unit than a mile for people driving their cars. They're comparable, of the same order of magnitude, and certainly equally good. The metric system does not shine because people's heights are much better expressed in kilograms than in pounds  once again, for the same reasons. The metric system shines in its ability to adapt to anything from picoseconds to nanometers to exabytes without confusing anyone. Because you're 19 and don't seem to be studying any technical fields, I have to conclude that you've never actually had to do any real engineering or scientific work  and no, solving a couple of firstyear physics problems doesn't count. I really hate to say this, but I feel I must: you almost surely have no idea how much better the metric system is than the imperial system, because your experience with them both is only in your head.  Warren 


#13
Jul604, 03:40 PM

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We want to replace decimal numeration by dozenal. Now, since you're a member of this organization, I would expect that you adhere to its beliefs. If you do not adhere to its beliefs, why would you be a member?  Warren 


#14
Jul604, 05:19 PM

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Here's a fun fact; there's a nice conversion between powers of two and powers of ten: 2^10 = 10^3 with an error of 2.4%.
There is no such simple conversion between powers of two and powers of twelve; the smallest conversion with similar accuracy is 12^12 = 2^43. 


#15
Jul604, 06:11 PM

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I can see 2 valid points for abandoning decimal representation.
.1_{10}= .00011001100... _{2} So if you are doing high precision arithmetic on a computer you MUST round off .1. If you ever find yourself using a counter in .1 steps do NOT ADD .1 to the counter at each step, instead multiply .1 by the step number, this way the round off error will not accumulate. Base 12 and base 16 (my candidate for a better representation) have more prime divisors then base 10. This means you have more ways evenly of dividing intervals. 10 with only 2 is about as bad as you can get in this respect. While these may not be extremely critical issues they are valid. 


#16
Jul604, 11:36 PM

P: 678

And a lot of the floatingpoint numbers we use on a computer arise from division and not from hardcoded decimal numbers; regardless of what base you're using, you're almost always going to introduce error when you use floatingpoint numbers for division. 


#17
Jul704, 08:06 AM

P: 44

Metric is now established in most countries, therefore a change back would be like a change back to speaking Latin it has nothing to do with Latin somehow being "inferior". The metre is the size it is as it was intended to be 1/40,000,000 the earth's circumference. I was making the point that if we has used a binarytype fractio nof the earth, or a dozenal one, we would have ended up with an anthropocentric length unit. In fact, the base units of Imperial, by the way, excluding the pound, can be expressed EXACTLY in temrs of fundamental phenomona and constants (the foot, second and rankine etc) 


#18
Jul704, 08:10 AM

P: 44

Warren's world philosophy It is not possible for a member of an organisation political or otherwise to hold slightly different views to those expressed in the organisation's manifesto. Actually, the DSGB beleives dozenal should replace decimal EVENTUALLY, and in the mean time wishes to conduct research into number bases. That is why I am a member. 


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