# Number Base Comparisons

by Bryan Parry
Tags: base, comparisons, number
 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; side-by-side comparisons of differing number bases and so forth. In short, I want this thread to be THE number-bases thread.
 Emeritus Sci Advisor PF Gold P: 10,427 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
P: 44
 Quote by chroot The drawback of small bases is that numbers have long representations; the drawback of large bases is many symbols to remember.
Indeed, hence why I am concerned primarily with comparisons on the octal, decimal and duodecimal/dozenal bases.

 Quote by chroot 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
I disagree (that it is irrelevant). I do not wish entirely to address your point as this topic will become a debate as to the validity of this topic, as opposed to actually assuming validity and then addressing the topic. However, I will say this much: if patterns in other numbers bases are found (as they have been), which make mathematics more "natural" or maths skills easier to aquire, if the other number bases display certain patterns or properties more overtly, if other number bases make more practically possibly everyday and not so every day tasks, then we must know. It is important we examine other number bases.

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 (one-third in itself probably a far more useful fraction, inherently, than one-fifth)

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 sixty-four - 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

 Emeritus Sci Advisor PF Gold P: 10,427 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
 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)
 Emeritus Sci Advisor PF Gold P: 10,427 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
 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 :)
P: 44
 Quote by chroot 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?
The decision is made every single day by countless scientists and engineers around the world.

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.

 Quote by chroot 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.
What an incredibly intelligent piece of arguing.

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 inch-sized 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/hand-width, 62.5" pace, 62,500" mile)

 Quote by chrook 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 small-number 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

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...
 Sci Advisor HW Helper P: 1,123 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.
 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 sixty-four, four or two (Leibniz, Charles XII, Spencer, Bernard Shaw and a bunch of others)
Emeritus
PF Gold
P: 10,427
 Quote by Bryan Parry *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....
Do you really think laws that were passed in the 19th century have any bearings on what units scientists and engineers use today? I'll answer for you: no.
 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.
And of course, what happened in France drastically affected everyone on the planet, yes?
 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
How do you use an arithmetic system to attain a unit? That doesn't even make sense. Units are defined by reference standards. The meter used to be defined by some marks on a platinum bar. The meter is now defined with respect to the speed of light and a specific number of wavelengths of a specific atomic emission line.
 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/hand-width, 62.5" pace, 62,500" mile)
Binary means base two, not base sixteen. You seem to be keen on anthropocentric measurements -- a philosophy that has been out of date for hundreds of years. Which part of your body would you like to use as a reference for measuring the wavelength of light? How about the distance between galaxies? How big is YOUR forearm?

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 first-year 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
Emeritus
PF Gold
P: 10,427
 Quote by Bryan Parry Who said I was in favour of change?
You linked us to the Dozenal Society of Great Britain, and explained that you are a memeber. The Society's page makes its position very clear:

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
 Emeritus Sci Advisor PF Gold P: 16,091 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.
 Mentor P: 7,320 I can see 2 valid points for abandoning decimal representation. .110= .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.
P: 678
 Quote by Integral I can see 2 valid points for abandoning decimal representation. .110= .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.
The only problem with this argument is that switching to a different base doesn't really make the problem go away. If you really need to represent 1/10 on a computer, then the fact that you're now calling it 1/A doesn't help.

And a lot of the floating-point numbers we use on a computer arise from division and not from hard-coded decimal numbers; regardless of what base you're using, you're almost always going to introduce error when you use floating-point numbers for division.
P: 44
 Quote by chroot Do you really think laws that were passed in the 19th century have any bearings on what units scientists and engineers use today? I'll answer for you: no.
You are really naive, you know that. Laws passed in the nineteenth century outlawing customary systems of measurement and replacing them with metric of course still have relevance to-day. Look: These customary systems were outlawed and abolished, and all standards switched to metric. It is almost impossible to switch back. That metric is perfectly servicable is why people have not converted back (given the massive effort of converting back), not because other systems are so rubbish there would be no reason to switch back.
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".

 Quote by Chroot And of course, what happened in France drastically affected everyone on the planet, yes?
Read some history books. But back on topic, of course it affected other nations- when Napoleon invaded half of Europe (a direct result of the Revolution), metric was spread to most of those nations. Likewise, the revolutions in Russia, China and the unification of Germany were the reasons for metrication in those countries. Given that all of the main European nations plus China had converted to metric for these reasons, that means almost the whole world (3/4 of it) would effectively automatically become metric too (due to colonies and spheres of infulence etc). The incredible expense of converting means it is unlikely to ever be reversed.

 Quote by chroot How do you use an arithmetic system to attain a unit? That doesn't even make sense. Units are defined by reference standards. The meter used to be defined by some marks on a platinum bar. The meter is now defined with respect to the speed of light and a specific number of wavelengths of a specific atomic emission line.
Thank you for intentionally misinterpretting my slightly loose language; YOU HAVE WON THE ARGUMENT, WELL DONE!! If you had LISTENED to me, instead of trying to score semantical points and thereby claim victory, when it is obvious what I meant, you would have realised that what I was saying was....

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 binary-type 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)

 Quote by chroot Binary means base two, not base sixteen.
I know that. Sometimes 'binary' is used to refer to all powers of two, and you know that.

 Quote by chroot You seem to be keen on anthropocentric measurements -- a philosophy that has been out of date for hundreds of years.
why is it out-of-date? If a man- be he a tailor, carpenter, bricklayer or man buying his fruit at a market stall- cannot make good approximations in his measuring systems without reference to rulers or standard weights, then the system is rubbish. Anthropocentric measures are better than totally arbitrary units (like the centimetre and metre)

 Which part of your body would you like to use as a reference for measuring the wavelength of light? How about the distance between galaxies? How big is YOUR forearm?
Yes, because metric expresses natural constants so nice and evenly, doesn't it? And English/Imperial cannot express large quantities well? Of course it can. For sure, for large things- distances between galaxies- anthropocentric measures are not much better than metric, but most uses of the measuring system are NOT to calculate the distance 'tween galaxies.

 Quote by chroot 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.
Or not. Prefixes can be used in English too (and are). Microinches, picoseconds, exabytes anybody? (you are claiming seconds and bytes are metric now!?)

 Quote by chroot 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 first-year physics problems doesn't count.
Riiiight, insult me. Nice arguing point. Because I am not an engineer, I have no right to debate or discuss matters of metrology or mathematics, right? And because you obviously are involved in it somehow means your views are correct? And the fact I know many engineers who prefer English units too is irrelevant?

 Quote by chroot 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.
All in my head!? Are you for real or what?? I sue both systems extensively every day.
P: 44
 Quote by chroot You linked us to the Dozenal Society of Great Britain, and explained that you are a memeber. The Society's page makes its position very clear: 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

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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