Yeah, mathematically the only thing base 10 has to recommend it is its accidental correspondence to the number of digits most of us have. 8, 12, 16 or 60 would all be better from a strictly mathematical viewpoint.
I think we need to clarify what “base” actually means.
One can make up all sorts of ways to package groups of numbers by words, but for me, a base has no other meaning than how many unique digits there are before the numbers are expressed as combinations of those unique digits.
So, given that, do we have any examples of older cultures using more or less than 10 unique digits?
The Greeks had 27 numerals (1-9, 10-90 and 100-900); I think the Egyptians had something similar. The Romans had i,v,x,l,c etc. The Mayans had three symbols (1, 5 and zero).
Since tens were symbolized differently than units, the Babylonian sequence of factors could be viewed as (…, 6, 10, 6, 10, 6, 10) rather than (… 60, 60, 60). Similarly Mayan numbers might be viewed as (4, 5, 4, 5, 4, 5).
Similarly, the short hundred (five-score or 100), long hundred (six-score or 120) and even gross (twelve dozen or 144) were all in use in Britain up until the 17th century, so reckoning with long hundreds implied a (10, 6, 20) system. (The long thousand was 1200.) Our number 327 might have been written as ii[sup]c[/sup] iiii[sup]xx[/sup] vii — two long hundreds, four score and seven. It was the long-overdue adoption of Hindu-Arabic numerals that finally spelled the end of this quaint and ambiguous accounting.
Well, we use octal and hex to represent numbers in programming because they allow you to directly read off the bit pattern, but for use in everyday life, you are missing that factor of 3. 1/3 still comes out repeating. 6 would provide a factor of 2 and 3, and is sometimes brought up in these discussions. Kind of small, though, and you wind up with awful long representations for things. How many digits would our street addresses, phone numbers, credit card numbers, etc, be?
BTW, IS there a term for expansion of real numbers in “decimal” type format irrespective of base?
Well, you will note that I considered “rules” as distinct from “flash cards”, the point being that you don’t need to carry pairs around in your head for the 0 and 1 rows and columns in your “times” table. You still have to remember that 1 times anything is the thing, and zero times anything is zero (and that they commute). Not quite an “infinite” improvement.
Side question, are there any systems where 1 is not equal to 1? Like “1” is equal to 1/2 in base 10.
A ‘Fibonacci base’ is sometimes used in coding. Ignoring the terminal ‘1’, the places, left-to-right, have values 1, 2, 3, 5, 8, 13, 21, etc.:
12358
11 = 1
011 = 2
0011 = 3
1011 = 4
00011 = 5
10011 = 6
01011 = 7
000011 = 8
100011 = 9
010011 = 10
001011 = 11
101011 = 12
0000011 = 13
Adjacent '1’s needn’t arise in the representations, so ‘11’ denotes end-of-number and avoids the need for a separate comma symbol. Thus 111100110000011 unambiguously denotes 1,1,3,13.
From a “strictly mathematical viewpoint” this radix or base stuff is irrelevant; how many mathematical theorems do you know that refer to specific bases? That is what the OP was asking about.
Similarly, people have always picked up whatever is convenient or traditional: programmers have their octal, sailors have their Babylonian degrees, bankers have their LSD, grocers have their pounds and stones, etc. 10 is no newcomer, but it has always existed alongside other magic numbers. Based on this, none of them are strictly special.
The Chinese are always complaining about those silly Western alphabets, without enough letters to represent all the different sounds and syllables, and those awfully long words wasting so much space on the page 
Other than describing it as a positional numeral system without mentioning a specific base? In mathematics, there are power-series expansions, but that is not precisely what we are discussing (they may be infinite, or involve a variable)
I can think of an example - the construction of the Cantor ternary set, which is an uncountable set of real numbers with zero measure:
You can construct the thing by considering the representation of the numbers on the unit interval in ternary (base 3) expansion.
Not quite what I had in mind. I was looking for a culture that had, say 8 distinct numerals and to show “9” they would use their ‘10’ as the expression. Or imagine what we call hex being a system used by an older culture where instead of A-F for 10-15, there would be 6 other distinct numerals, but used to the same effect. Did any such numbering systems exist. I’m not looking for systems where multiple instances of digits were used together, such as Roman numerals or the Mayan system. You could not call their system base-anything.
I don’t think the term “base” means anything specific outside of modern positional notation. Certainly I don’t believe Greek or Roman numerals correspond to anything we would consider a base at all.
For what it’s worth on ‘natural’, there’s one example showing how much things depend on how you were raised.
In the West, we tend to group around 10^3, so thousands, millions, billions, trillions, etc.
But in China, Japan, Korean, and possibly a few other countries, numbers are grouped around 10^4. It’s more natural to separate zeros that way, too, i.e. instead of 100,000,000 they would tend to think 1,0000,0000 instead.
Neither system is preferable to the other but it does cause some cognitive dissonance switching between them. My parents often trip over thousands or ten-thousands when translating because there’s a single word for ten-thousand for them but two words for the same number in English.
You could - the Romans basically used base-ten, though their notational system had some quirks.
For the most part, positional notation itself wasn’t common historically, so historical examples of positional notation in other bases don’t really exist. The Babylonians sort of had a version of this with their number system based around 60 but it did not feature notation purely based on powers of 60.
But this is a strange notion of ‘base’. The base doesn’t have to depend on a positional notation purely derived from powers of a base.
Our currency is sort of an example of this. If you have to deal strictly with coins, counting out change is an example of how quickly we can adapt to different bases if given sufficient practice.
In the US, our metal currency is a bit of a mix between base-5 and base-10. Pennies, nickels, and quarters are the first 3 powers of 5. So, if you only had pennies, nickels, and quarters, counting out 87 cents (using the fewest possible coins) would be 3 quarters, 2 nickels, and 2 pennies, i.e. 87 in base-10 is 322 in base-5. I’m not sure about the younger generations, but you can be sure many, if not most, people over 40 years old could figure this out without much thought, meaning base-5 thinking to some extent is already not difficult for many adults, at least if you couch it in familiar terms instead of as an arithmetic problem.
True.
That isn’t what base means, here.
In math, a base has a simple meaning: If your numbers are represented such that each digit stands for s*b[sup]e[/sup], where s is the significand and e is the exponent, the base is b. That’s the basic concept behind place-value notation, and place-value notation is the only context where it makes sense to talk about numerals being written in a specific base.
For example, if you take the log[sub]10[/sub] of 10, you get exactly 1, no matter what base your “10” happens to be written in: log[sub]ten[/sub] of ten is 1, as is log[sub]two[/sub] two and log[sub]e[/sub] e. More generally, the function log[sub]b[/sub] grows proportionally to how many digits it would take to write a number in base b in place-value notation. That’s a pretty good clue that the concept of base depends on place-value notation.
Perhaps it can be defined that way, but it’s pretty different from how base is used in the rest of the thread.
The use of various counting/arithmetic devices, including the abacus (which is implicitly positional even if the cultures originally using it had not adopted positional notation otherwise), predates positional notation and was actively in use across a wide swath of the world. And in most cases, used to do base-10 arithmetic.
The words used to represent numbers and historical documents clearly show many cultures adopting base-10 arithmetic, even before they adopted the Hindu-Arabic numeral system and positional notation.
Perhaps they did not explicitly call it a number ‘base’ but that’s what it was.
Thanks to all contributors to this thread. I’ve learned a great deal more than I expected as responses to my vastly oversimplified OP. Please do go on!
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Mounted on the outside of the Pioneer spacecraft is a “plaque” that was designed to to carry information about humanity to any possible extraterrestrial life sufficiently intelligent to gently rendezvous with it and try to understand what is inscribed on it.
Link: https://en.wikipedia.org/wiki/Pioneer_plaque
Cleverly, the scientists make no assumption about what math base the aliens might be using, because anyone sufficiently versed in math knows that the numbers are the same, regardless of how they are written down. For example, you can convert any number in any base to a simpler base, such as binary, which is pretty easy to interpret.
In the wikipedia article linked to above, the diagram introduces a new number unit, a distance metric related to light waves and the behavior of hydrogen. This is then used to explain where we come from relative to a few obvious “landmarks” in space, given direction and a binary representation of distance based on that given unit. Interesting!
Aside: Those plaques (and the phonograph records on the Voyagers) were indeed intended to convey messages to intelligent life, but not aliens. The creatures the messages were intended for are here on Earth, and the message was received before the spacecraft were even launched. The motivation was not actually to contact aliens (which would be far more likely with an artifact here on Earth than on a deep-space probe), but to get humans excited about the possibility and so to approve the missions (which had numerous other real, legitimate purposes).
Did Carl Sagan (who was a major factor in the golden disk) agree with you, or is this just your post-event opinion?
I can buy this. It certainly fits into the Sagans’ wheelhouse on popularizing / selling science to Earthlings.
Still, it’s a fun exercise in devising an attempted means of communicating with an extraterrestrial intelligence, as devoid as practically possible of Earth-centric ways of thinking. One of the criticisms I saw on the wikipedia article was on the use of an arrow to indicate trajectory of the spacecraft…what if, asked a critic, the aliens have no idea what an arrow is?
I don’t know if Sagan ever said anything about it, but I thought it was a pretty obvious ploy to get more people interested and involved in space exploration. The chances of any of those spacecraft being found by some alien civilization are so low they may as well be zero. So their real target audience must be right here on Earth.