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Most people use base 10; it's obviously the common notation in the modern world.

However, if we could change what became the common notation, would there be a better choice?

I'm aware that it very well may be that there is no intrinsically superior base, but for the purposes of humans, is there a better one?

I've heard from sources such as this and this that base 12 is better, from here that base 8 is better, and, being into computer science, I would say that base 16 is the most handy.

Base 12 does seem to be the most supported non-base 10 number system, mainly due to the following reason pointed out by George Dvorsky:

First and foremost, 12 is a highly composite number — the smallest number with exactly four divisors: 2, 3, 4, and 6 (six if you count 1 and 12). As noted, 10 has only two. Consequently, 12 is much more practical when using fractions — it's easier to divide units of weights and measures into 12 parts, namely halves, thirds, and quarters.

And, on top of that, previous societies considered very advanced used other systems, such as the Mayans using base 20, and the Babylonians using base 60.

So, summarized, my question is: Is there an intrinsically superior base? If not, is there one that would be best for society's purposes? Or does the best base depend on the context it is being used in?

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    $\begingroup$ The usefulness of base-$12$ is certainly being pushed by the Dozenal Society of America. FWIW, if you can get your hands on a copy of Dudley's Mathematical Cranks, there is a delightful discussion on the people who tirelessly push for the more widespread use of other bases. $\endgroup$ – J. M. isn't a mathematician May 6 '13 at 10:57
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    $\begingroup$ But if we used base 12, "The Dirty Dozen" would have been called "The Dirty Ten", which is not as nearly as good title. $\endgroup$ – Asaf Karagila♦ May 6 '13 at 11:04
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    $\begingroup$ @Asaf, I'm sure y'know that $10_{12}$ isn't read as "ten", right? ;) $\endgroup$ – J. M. isn't a mathematician May 6 '13 at 17:39
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    $\begingroup$ Probably the biggest qualitative difference in usefulness that I've heard of is that from a practical computational perspective highly composite bases are ideal, whereas prime bases are more suitable for pure mathematical purposes. If there was a base that could reap both benefits simultaneously, there would probably be more of a push to switch to such a system. But of course, there isn't $\endgroup$ – David H May 12 '13 at 3:27
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    $\begingroup$ It's not a XKCD, but I think this comic is mandatory here... cowbirdsinlove.com/43 $\endgroup$ – woliveirajr May 14 '13 at 12:12

12 Answers 12

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I like the factorial base, where the integer part of a real number is written as $\sum_{i=2}^n a_i i!$ where the $a_i$ are integers such that $0 \le a_i < i$ and the fractional part is written as $\sum_{i=2}^{\infty} \frac{b_i}{i!}$ where the $b_i$ are integers such that$0 \le b_i < i$.

The nice thing about this is that the integer part has a unique representation and the fractional part terminates if and only if the number if rational (except for the case corresponding to $\frac1{n!} = \sum_{i=n+1}^{\infty} \frac{i-1}{i!}$, the same as 1 = .99999...).

This is a special case of the following result: If $(B_i)_{i=0}^{\infty}$ is an increasing series of positive integers with $B_0 = 1$, we can represent all positive integers in the form $N=\sum_{i=1}^m a_i B_i$ where $0 \le a_i < B_{i}/B_{i-1}$ and $N < B_m$. This representation is unique if and only if $B_{i}/B_{i-1}$ is an integer for all $i$.

The usual decimal, binary, and hexadecimal bases have $B_i = 2^i, 10^i$, or $16^i$. The factorial base has $B_i = (i+1)!$.

I worked this out over 40 years ago and found it quite interesting. I am sure the result is several hundered years old.

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    $\begingroup$ Um... call me dumb, but in what base do you represent each $a_i$, considering that no constant finite base will suffice once $i!$ is greater than it? $\endgroup$ – Lily Chung Aug 28 '14 at 21:32
  • $\begingroup$ Instead of each $a_i$ satisfying $0 \le a_i \le B-1$, they satisfy $0 \le a_i \le i$. $\endgroup$ – marty cohen Sep 1 '14 at 4:18
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    $\begingroup$ Doesn't that imply that you need a potentially unbounded number of symbols to represent large numbers? If a number requires $n$ factorial digits, then you need $n$ symbols to represent the most significant digit. In a classic base $B$, you only ever need $B$ symbols. $\endgroup$ – Lily Chung Sep 1 '14 at 5:29
  • $\begingroup$ Yep. That's why we have the integers. $\endgroup$ – marty cohen Sep 1 '14 at 14:53
  • $\begingroup$ @Marty Cohen: But how do you represent those integers? Seems you'd still need a "regular" base type system in there as a "sub-digit" system, like how the Babylonian base-60 system used a base-10 "inner" base. $\endgroup$ – The_Sympathizer Jul 26 '16 at 3:42
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I think base $6$ would make counting on our hands particularly convenient, we would have a $1$'s hand and a $6$'s hand and would be able to count up to $35$.

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    $\begingroup$ There is a Korean technique that uses the thumb to represent 5 and so can count up to 99 on two hands. This seems rather more convenient to me than your suggestion. $\endgroup$ – MJD May 15 '13 at 15:40
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    $\begingroup$ Base 6 also has the advantage of trivial divisibility tests for 2 and 3, and easy tests for 5 (sum of digits) and 7 (sum of digits with alternating sign). $\endgroup$ – Dan May 16 '13 at 3:40
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    $\begingroup$ @MJD If you want to use your hands and are allowed to distinguish between fingers on each hand, then binary is clearly optimal, allowing you to count from 0 to 1024 (half-open). $\endgroup$ – Lily Chung Aug 28 '14 at 21:34
  • $\begingroup$ Multiplication tables would be greatly simplified up to $36$. It would also be handy to use the $36=100_{6}$ characters $0,1,\dotsc,9,a,b,\dotsc,z$, ready available on any keyboard, to represent numbers. I think this is the reason why $6$ is called perfect. (Beware the Dozenal Society though, they may hijack your cause for theirs...) $\endgroup$ – Oskar Limka Mar 23 '18 at 7:02
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In order to answer this question, it is first necessary to ask: What makes one base “better” than other? Some reasonable things to consider are:

Size

There is a tradeoff between the number of distinct digit characters used in a base (Base $b$ has exactly $b$ of these, from $0$ to $b - 1$, inclusive) and the length of the numeral required to represent a given number (which is $O(1/\log{b})$).

If the base is too small, then numbers explode into cumbersome long strings of digits. For example, in binary, the current year is 111 1101 1101, and the population of China (according to its 2010 census) was 100 1111 1101 1010 1001 0100 0011 0100. Modern computers can easily work with 32-bit or 64-bit binary numbers, but humans can't, which is why programmers have developed more compact encodings of binary, such as hexadecimal.

On the other hand, if we picked a very large base, like 2520, then you would need only 3 characters to represent the population of China, but typing them would be just as challenging as typing Chinese. And forget about learning the mulitplication table, whose size is $O(b^2)$. The only practical way to use such a large base is to split it into sub-bases, the way base-60 is represented as a mixture of base-6 and base-10.

So, what we want is a happy medium.

Fraction-friendliness

This is the main argument advanced in favor of base-12 or other highly composite bases (2, 4, 6, 12, 24, 36, 48, 60, 120, ...).

If a base has a lot of factors, it makes fractions easier to work with. For example, in base ten, 1/3 is represented as the infinitely repeating 0.333 333 333... (often rounded to 0.33 or 0.333), and this awkwardness crops up in deals like “3 for $5” or +/- grading systems. But in base-12, 1/3 is a nice simple 0.4.

Of course, because there are an infinite number of primes, it's impossible to completely avoid repeating “decimals”. And base-12's simplicity for the fractions 1/3 (0.4), 1/4 (0.3), 1/6 (0.2), 1/8 (0.16) and 1/9 (0.14) comes at the price of making 1/5 (0.24972497...) and 1/10 (0.124972497...) recurring dozenal fractions. But 1/3 is more common than 1/5.

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Brian Hayes in his American Scientist article Third Base argues that "When base 2 is too small and base 10 is too big, base 3 is just right."

Figure 1 has the caption

Most economical radix for a numbering system is $e$ (about $2.718$) when economy is measured as the product of the radix and the width, or number of digits, needed to express a given range of values. Here both the radix and the width are treated as continuous variables.

Figure 2 has the caption

Most economical integer radix is almost always 3, the integer closest to $e$. If the capacity of a numbering system is $r^w$, and the cost of a representation is $rw$, then $r=3$ is the best integer radix for all but a finite set of capacities. Specifically, ternary is inferior to binary only for 8,487 values of $r^w$; ternary is superior for infinitely many values.

Figure 3 has the caption

Ternary structure may offer the quickest path through a telephone menu system. Putting eight choices (assumed to be equally likely) in a single octonary menu (left) forces the caller to listen to 4.5 menu items on average. A binary structure (middle) has the same performance, but the ternary tree (right) reduces the average to 3.75.

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    $\begingroup$ Sure, base 3 may minimize the number of beads on an abacus, but would you really want 1/2 to be 0.111111111... recurring? $\endgroup$ – Dan May 16 '13 at 3:47
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    $\begingroup$ 1/3 is 0.3333333 recurring in base 10... $\endgroup$ – Michael Aug 6 '13 at 17:15
  • $\begingroup$ @michael, that's still tolerable. Dan has a point. Doubling and halving is the most common mental arithmetic people do. I wouldn't want 1/2 to be recurring but 1/3 is more OK. $\endgroup$ – nawfal May 26 '18 at 7:48
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Balanced Nonary (base 9) would probably be really good. The digits go from -4 to 4, so taking the negative of a number would just be taking the negative of each digit, so subtraction is easy. Multiplication and division are particularly easy too if you make the easy conversion to balanced ternary first. Then there's no carrying when multiplying single digits (like in binary), and division is just testing inequalities (if you can divide by 2). Of course, if you want to do things faster, learning a balanced nonary times table would be easier than learning a regular nonary times table since you only really need to know the table for 1,2,3,4 and then handle negatives (and zero) appropriately.

There have even been computers based on balanced ternary.

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For computer applications, bases like 2, 8 and 16 are obviously the best. Given that a large percentage of numerical data is stored in and processed by computers, these days, one could argue that what's good for computers is good for society.

Of the three I mentioned, I suppose that 8 or 16 would be better than base 2. Having the price of bananas as a binary number in the supermarket wouldn't work too well. Binary numbers are too long, and they all tend to look alike, so they're hard for people to read.

In the world at large (as opposed to the narrower world of mathematics and computers), reading numbers is probably just as important as doing arithmetic with them. Think of speed limit signs on roads, distances of journeys, prices in stores, or temperatures in weather forecasts. These numbers need to be read and understood quickly (by human beings), and I doubt that this would be possible if they were written in binary. We'd no longer be taking advantage of the wonderful human ability to quickly recognize symbols, and it would be a pity to waste that ability just so that we can make computing easier (in my opinion).

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  • $\begingroup$ Not sure why $4$ was left out of $2, 8, 16$. $\endgroup$ – Lance Pollard Jan 13 '19 at 16:41
  • $\begingroup$ Because bases 2, 8, 16 are already widely used in computing, whereas 4 is not (as far as I know). But, I suppose it would work OK. $\endgroup$ – bubba Jan 13 '19 at 23:34
  • $\begingroup$ Okay that makes sense. 4 is not as used b/c computers are 8-bit at least, so 4 is inefficient. But 2 is there because everything is just 1's and 0's. $\endgroup$ – Lance Pollard Jan 14 '19 at 0:04
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The quater-imaginary base $2i$ is quite amusing, being able to express every complex number using only digits in $\{0, 1, 2, 3\}$. Being the only base yet proposed here that includes the elements of this extended system, it is clearly the best one for this purpose :)

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While bubba raises valid points about base 2 from a practical standpoint, I myself would defend the choice of base 2 for the following reason: it makes addition and multiplication incredibly easy. This is, in fact, the way computers do these basic operations.

Addition in binary operates under the following rules:

$0 \oplus 0 = 0$

$1 \oplus 0 = 1$

$0 \oplus 1 = 1$

$1 \oplus 1 = 0$ (carry a 1)

Therefore when you do long addition in binary, the algorithm is particularly simple: if there are 2 $0$'s in the column, you put down $0$, if there is one $0$ and one $1$ you put down $1$, if there are 2 $1$'s you put down $0$ and carry $1$ over to the next place value. Imagine the time we could save by teaching kids to add this way. We could start teaching actual mathematics instead!

Long multiplication is just as easy: for every place value you're multiplying by either $0$ or $1$, which makes the computation very simple. I invite you to try out a few simple sums and products in binary to see what I mean.

See http://en.wikipedia.org/wiki/Binary_arithmetic#Addition and http://en.wikipedia.org/wiki/Binary_arithmetic#Multiplication for more on these two operations; the article has details on subtraction, division, and square roots as well.

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  • $\begingroup$ Binary makes arithmetic easy on paper. But how would it affect mental arithmetic? $\endgroup$ – Jack M May 24 '13 at 14:09
  • $\begingroup$ I'm not really sure, but mental arithmetic is no easier in any other base - the only difference is the number of digits for larger numbers. I say let people use a calculator. Arithmetic is the job they were born to do, and the job we were born to automate. $\endgroup$ – Gyu Eun Lee May 24 '13 at 18:00
  • $\begingroup$ It makes numbers a bit longer. $\endgroup$ – PyRulez Jul 1 '13 at 14:42
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    $\begingroup$ Sure, base 2 is convenient; however, some numbers that have a finite decimal expansion (e.g. $(0.2)_{10}$) but an infinite binary expansion ($(0.00\overline{11})_2$) is problematic, as it is a source of roundoff error in floating-point-number systems. $\endgroup$ – jub0bs Sep 30 '13 at 15:23
  • $\begingroup$ @Jubobs Same with $(\frac{1}{3})_{10}$: $0.\overline{3}$. $\endgroup$ – Cole Johnson May 25 '14 at 21:49
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Not all bases are 10 in their own notation. There are a group of alternating bases where the base is not 'ten' but a 'hundred'. The most elegant of these is the long-hundred of the proto-germanics and their decendents. Reckoning in the six-score long-hundred (ie 120), was still common enough in 1350 to pass without comment.

Yes, i have used this base for some thirty years. It's truly elegant, being more efficient than either 10 or 12. It's the first base, for which the (number of proper divisors)/(ln base) is greater than 3.

Also 120 is the smallest multiply perfect number, and has the same features as the perfect numbers. For example, 120 = 1+2+4+8+15+30+60 = 3+5+6+10+12+20+24+40, all of these numbers make the total divisors of 120. The second set corresponds to a set of weights, eg

  • 1 oz, 2 oz, 4 oz, 8 oz, 1 lb, 2 lb, 4 lb. 15 oz = 1 lb , 120 oz = 1 clove.
  • 1 ct, 2 ct, 4 ct, 8 ct, 1 dr, 2 dr, 4 dr : 15 ct = 1dr , 120 ct = 1 oz
  • 1 lb, 2 lb, 4 lb, 8 lb, 1 st, 2 st, 4 st : 15 lb = 1 st, 120 lb = 1 cwt

When one considers not just integers, but also fractions x/y and y/z, arranged by xy, one finds that the expressions of these are very short for the first sixty or so, even in things like 56 (8/7 = 1:17.17.17, vs 7/8 = V5), and this unusual pair at 96 (3/32 = 11:30, 32/3 = 10:80.

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It is not necessarily true that the multiplication is of the order of $O(b^2)$, since this is a particular implementation of the base, rather than the base itself. The mayans divided their score into four sticks of five dots, and had a true zero in the dot-position (eg fifteen is "3-fives-zero").

One should remember that counting (multiples) and division are separate operations, and that it is possible to use different number-systems for them. Historically, the sixty-wise system is one of divisions: the first column is that of units, and later places are divisions by sixty. Likewise, the romans multiplied by 10's, and divided into 12's.

An alternating base like 60 or 120, supposes two rows on each column of the abacus, where the unit (in the bottom row), is counted to 10 to make one carry up, but divides into 12 to be borrowed into the top row of the lower column. Since one can start in either the top row or bottom row (for counting), the use of tens by twelves or twelves by tens, automatically produces an alternating base.

Using alternating arithmetic then reduces the size of the tables to the order of $O(b)$.

It should be noted that the sumerian system is a division system to avoid division. We see this from recriprocal tables (eg 2 <=> 30 ), and tables of reckoners of multiples of the recriprocals (eg multiples of 4.26.40). Even in their reckoners, multiples are supplied for 1 to 20, and 40. Neugebauer even gives reference to a paper on the seven brothers, ie what is 1/7. It is concluded it lies between 0.8.34.16 and 0.8.34.18.

But i wrangle base 120 for nearly 30 years, and never felt the need to go past 12*12.

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Base 10 is a positional fixed base (The value of the digit depends on its position in the number as well as on its value) and it is neither to large nor to small which is best suited for most purposes. This has many advantages and the andvantage this has over factorial base is, that we need only 10 symbols to represent every possible number. Factorial number system is appealing, but it lacks this practical part. As for other fixed number bases: well,we have 10 digits and are used to think in terms of that. It probbably has something to do with the way we learn our first numbers as children. It is similar to measuring an angle in degrees or radians: we are just so much more used to it being expressed in degrees. It is just the matter of everyday practicality: base 10 and degrees are just so much more practical in everyday life than anything else and mathematics is for everyday live as well as for advanced mathematicians! And the sciences that need other number bases, just make best use of it (like i.e. the case of computer science). I guess there is NO number base that is wasty superior to others, so fixed base that is neither to small neither to large best suits ALL purposes and if in some particular base we need a better suited base,then we just make use of one. The ancient Greeks didn't even use either positional number system, nor a decimal one. They didn't even have a standard number system. Their system was similar to that of Romans (not positional) and Babilonians (not decimal but base 60).

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I personally like base 6 a lot. It's the largest number with the property that all positive integers up to half of it are a factor of it. That gives it a very simple single digit multiplication table. So simple is the multiplication table that I think the students will actually train their brains better if it's not written for them at all and being taught to use the written single digit multiplication table is replaced with telling them to know how to figure out the product of any two single digit numbers all on their own. After they use that ability a few times in a problem that requires it, they might notice simple patterns in the multiplication table and then know it by heart. That also makes it easier to mentally compute the quotient and remainder of a division problem of positive integers. Maybe binary is even easier to mentally compute long division on numbers of the same number of digits but I think I prefer base 6 because it also uses fewer digits to express the same number which probably ends up making a division problem of any two positive integers in general easier.

However, I'm not sure that it's actually better in my opinion to use base 6 than base two if introducing the notation comes so early. I read on the internet that in China, people split large number written in decimal notation into groups of 4 digits and they perceive very large numbers as so small. As mortals, that doesn't feel like a very nice way to perceive things. I think most preferable of all is to teach students only the bijective unary numeral system while they are so young and then give them the inductive definions of natural number addition and multiplication

  • $\forall x \in \mathbb{N} x + 0 = x$
  • $\forall x \in \mathbb{N}\forall y \in \mathbb{N} x + S(y) = S(x + y)$
  • $\forall x \in \mathbb{N} x \times 0 = 0$
  • $\forall x \in \mathbb{N}\forall y \in \mathbb{N} x \times S(y) = (x \times y) + x$

with no reference to sets because I think defining natural numbers as cardinal numbers is the reason some students are struggling so much to figure out how to add and subtract, and guide them to teach themselves how to prove the 5 laws of natural number addition and multiplication

  • $\forall x \in \mathbb{N}\forall y \in \mathbb{N}\forall z \in \mathbb{N} (x + y) + z = x + (y + z)$
  • $\forall x \in \mathbb{N}\forall y \in \mathbb{N} x + y = y + x$
  • $\forall x \in \mathbb{N}\forall y \in \mathbb{N}\forall z \in \mathbb{N} (x \times y) \times z = x \times (y \times z)$
  • $\forall x \in \mathbb{N}\forall y \in \mathbb{N} x \times y = y \times x$
  • $\forall x \in \mathbb{N}\forall y \in \mathbb{N}\forall z \in \mathbb{N} x \times (y + z) = (x \times y) + (x \times z)$

After reading https://www.inc.com/bill-murphy-jr/science-says-were-sending-our-kids-to-school-much-too-early-and-that-can-hurt-th.html, I think that they should continue getting them to teach themselves math results before they introduce the base 6 system because by the time they introduce it, they will be old and smart enough that it will not cause them to miss interpret numbers that are actually very large as being so small. However, if the choice was to either teach them base 2 to base 6 very early, I would probably prefer that they be taught base 2 because the same number will not seem as small if they use base 2 as if they use base 6.

Although this doesn't change the fact that base 6 is the best base to use in my opinion, after they get taught base 6, I still think it would be better if they're guided to teach themselves how to compute the quotient and remainder of any division problem instead of being told how to do long division because it helps make them smarter and more able to learn other material later.

I think it can be taught as follows. They will literally define the base 6 notation of any positive integer by the method of getting to that number by starting at 0 and performing a sequence of operations each of which is of the form of multiplying by 6 then adding a number between 0 and 5. How to you then divide a given large number by 14 which is 22 in base 6. Then maybe they will be guided to recognize that the number is the number the string of all digits but the last digit represents times 6 plus the last digit. Then they will be like maybe if I first compute how to express the number the string of all digits but the last digit represents in the form a 14 times a natural number plus a number from 0 to 13, then I can use that to figure out how to express the original number that way.

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