Does the base-10 (decimal) system have a natural advantage over other bases?

Question

Note: This question arguably spans mathematics and linguistics, and possibly other disciplines, but I am posting it here because I think that mathematicians are best placed to answer it.

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The base-10 (decimal) number system is the most common symbolic counting system used by people today. Scholarly literature on number systems state that this is derived from the use of fingers in the representation of numbers (see e.g., Ores 1948, p. 1-2). However, there are also cases of some cultures using other bases, up to base-20. It seems evident from this history that humans can handle number systems with various bases.

The use of binary numbers in electronic computing occurs because it is efficient to manufacture components that exist in a binary state. This suggests that the most efficient base is the smallest base, which is the binary system. Since this is (arguably) the most efficient system for electronic computation, it is arguably desirable for humans to be able to easily and rapidly convert between their own common number system, and the binary system.

This suggests that it would be more efficient for humans to adopt a number system that is base-2$^k$ for some $k \in \mathbb{N}$. The obvious choices, operating within an appropriate range of symbols and digits, are the octal system (base-8) or the hexadecimal system (base-16). These systems can be trivially converted back and forth with binary numbers, since they merely require the user to know the binary representation of each base element, and string numbers together with the place method.

Question: Aside from its derivation from finger counting, is there any property of the base-10 number system that gives it a natural advantage over other bases for human use? What are the pros and cons of adopting a base-2$^k$ number system (e.g., base-8 or base-16) in place of the presently common decimal system? Would conversion to one of these systems be desirable?

Answer 1

No, base 10 doesn't have any natural advantage over other bases

There is no, 'one size fits all'. Choosing the best base will need to take into account the collections of things you will be counting and/or keeping track of.

• base 2 - works well with electricity and computing (8 digits = 256 values)
• base 8 - less repetition of symbols than base 2 (3 digits = 512 values)
• base 16 - less repetition of symbols than base 8 (2 digits = 256 values)
• base 10 - works nicely with human hands
• base 26 (the English alphabet) - can be used in Microsoft Excel for counting items divided into A-Z categories (A=1, Z=26, AA=27, ZZ=26*26=676)
• base N - great when you need to count items divided into N categories

Regarding Natural Bases:

The history of electronics engineering has deemed base-2 as naturally superior since it is often easier & cheaper to build components that operate at either a high=1 or low=0 voltage state.

Humans mostly have 10 fingers, so this is a natural choice as a base. Ten is also very close enough to 7+2=9, for which Psychology has found to be the average number of items that most people can remember. (Search: "7 plus or minus 2")

This final example different in a subtle way because "counting items" is different than "counting combinations". If you are keeping track of combinations of different things, say apples, oranges, and limes, then you can record (and count) all the combinations naturally with base-3. However, I want to point out this is mostly just a way to track and count combinations. In this example, to use this method as a general way to count fruit means the order of the fruit has to match the order of the counting symbols (O, A, L) every time you count.

• Counting combinations = good
• Counting total fruit = bad

i.e. Combinations of 3 items with : O = orange, A = apple, L = lime

   This is a way to "count combinations" using base-3

• OOO OAO OLO -- AOO AAO ALO -- LOO LAO LLO
• OOA OAA OLA -- AOA AAA ALA -- LOA LAA LLA
• OOL OAL OLL -- AOL AAL ALL -- LOL LAL LLL

Answer 2

Some preliminary thoughts on the pros and cons of the decimal number system (hopefully this can be supplemented by other users):

• Finger counting: As noted in the question, decimal corresponds to finger counting, so it is very simple for people to count decimal numbers on their fingers. It would be more difficult (but not impossible) to count on other bases on the fingers. Bases higher than ten become more difficult, which militates against a base-16 system.

• Division by simple primes: It is desirable for the base of the number system to be a product of simple prime factors, so that the base is divisible by simple primes. The decimal number system uses base $2 \cdot 5 = 10$, which uses the first and third prime. This makes it easy to divide by these two simple primes. This is arguably less simple than a base-2$^k$ number system, which can easily be divided by the smallest prime multiple times. Nevertheless, the decimal system allows easy division by five, whereas the base-2$^k$ systems do not.

• Inertia/conversion costs: Given that the decimal system is in such widespread usage there would be substantial costs to converting numbers to a base-2$^k$ system. There would be costs in time and frustration for people to learn and get used to the new system, and there would be monetary costs to the translation of documents, etc., which represent numbers in the decimal system. These costs would be very large, which militates in favour of using the existing decimal system. On the other hand, the conversion costs are likely to grow exponentially over time (as new documents are generated in the decimal system) and so if it is desirable to convert, then it is probably preferable to convert earlier rather than later.

Answer 3

This is, at most, amusing speculation. I don't think that there is any realistic chance that we will move away from the decimal system for normal day to day use. The upheaval would be far greater than any standards change attempted so far. Most countries have achieved the transition to the metric system but there are a few hold outs. However, attempts to decimalise time failed: Decimal time (Wikikpedia).

The benefits seem to be few. Mostly it would just be easier to talk to computers. However, computers should work for us not us for them. If it is convenient for computers to work in a different base then they should do the conversion to and from the human's preferred format. I am old enough to remember a time when it was useful to be proficient in octal (a day before even 8 bit bytes were standard) but that is ancient history. Even that illustrates the important point that computers may change. Today, 8 bit bytes are standard and hence hexadecimal format is convenient but that could change. It would be highly wasteful to switch to hexadecimal and then find in a few years that computers had moved away from it. Some early computers worked in decimal, and even today, some business orientated computers use BCD (binary coded decimal) to emulate decimal in the hardware.

I see just one non-computer benefit of a power of 2 base: spigot algorithms. These often work in a power of 2 base. It would be nice if they worked in our preferred base. Spigot algorithm (Wikipedia)

A disadvantage of a power of 2 base is that even fewer fractions would have a terminating expression in the equivalent of decimal notation. In the familiar base $10$, the decimal notation terminates if the denominator is of the form $2^m 5^n$. For a power of 2 base, that would be just $2^m$.

For the benefit of humans rather than computers, I can only see that bases with more factors rather than fewer would be attractive. Base $12$ would allow easy division by 3 as well as 2 which would often be convenient ( we would lose easy division by 5 but that may be less often required than 3). Base $30$ would add easy division by 3 and retain easy division by 5 but would require a lot of symbols. Base $210$ would add easy division by 7 but the required number of symbols would probably be impractical.