In the earliest days of mankind, 13 was written as "............." The number of dots represented the number. Later the Egyptians had a different hieroglyph for 10, so 13 could be written as "#..." where "#" means 10 and "." means 1. Much shorter. 33 was written as "###...". Nice. Then the 0 was invented. And nowadays, we have "hieroglyphs" for all numbers up to 9 and we have this notion that every number is multiplied by 10^its position. Is that the end? Or will this look as ancient as counting dots in a million years from now?
I think everyone is misinterpreting the question. This isn't about the fact that we're using base-10. This is about the fact that we're using the Arabic "symbol-valued cardinal exponential" notation:<p><pre><code> ABC = (val[A] × base^2) + (val[B] × base^1) + (val[C] × base^0).
</code></pre>
Examples of other systems, as the OP said, are tally-marks (uniform-valued ordinal additive) and Roman numerals (symbol-valued ordinal additive). The question is, is arabic notation optimal for doing simple math quickly? It might not be, given that e.g. mathematical savants seem to be doing something involving geometric/visual computation.
I'm going to take the contrarian view: yes. Inertia is very powerful. We'll stay with base-10.<p>Scientific notation is, not, btw, fundamentally a different number system; it only provides for approximation of (most) very large or very small numbers, unless you want to spell out all the digits before the exponent, which of course would defeat the purpose. It's still really base-10.
There are already other notations, like scientific notation which only needs a few digits to represent e.g. 2.3 * 10^8. Not to mention hexadecimal or even base-32 which is used in Bittorrent magnet links <a href="https://en.wikipedia.org/wiki/Magnet_URI_scheme#URN.2C_containing_hash_.28xt.29" rel="nofollow">https://en.wikipedia.org/wiki/Magnet_URI_scheme#URN.2C_conta...</a>
Your question contains a very teleological narrative, in that you assume/recognize a natural evolution from the first to the last, with cause and result. Partially because you take all of mankind as one group.<p>Then, to look at your question: who's your "we"? The Chinese already use another system (in many contexts). African cultures use their own systems. And who knows what might happen in the future.
I can vaguely imagine a more advanced numeral system based on geometric visualizations in 2 or 3 dimensions (rather than the existing 1-dimensional digit string), that is also more suited to probabilistic representations than fractions or decimals.<p>Relational reasoning is a key use case to be considered for any popular numeral system (How many do I have? How do I signal that amount to others? Do I have more or less?). For small, whole quantities (<100), alternate numeral systems could likely reach a similar learning curve as Arabic numerals. Very large and complex quantities, and things of a number theoretic nature are probably areas where a future numeral system will be differentiated.
Base-10 isn't the only thing cultures around the world came up with. I vaguely recollect that a handful used base-5, base-6 (spaces between knuckles plus each side), base-12 (with two hands) and base 20 (two hands, two feet; or two sides for each finger) -- and probably others I forgot. We still use base 60 (which we inherited from babylonians, and are still using to count time and angles). As well as base-2, base-8 and base-16 in computer science.<p>Whether we stick to base-10 or collectively decide to use something saner in the future (base-12? base-60?) is anyone's guess, but methinks inertia will spell doom to efforts to part from it, much like efforts to bring sanity to the calendar never took off in the 19th century.<p>This much is probably sure, though: we won't go back to colorful subdivisions. Nobody except the US (and Liberia) uses anything but the metric system nowadays.
I think we, i.e. those who already use base 10 and neglecting the possibility of us getting invaded in some way, will continue to use base 10 unless our civilisations fall apart. When we handed off the basics of our system to machines so that the low level tasks could be done quickly, the need for a more efficient system decreased. Out to a ridiculously large number, the basic components of our system are, when coupled with machines, essentially instantaneous.<p>What's 395847593874382754238754987 * 389756987476347629845 ?<p>1.5428437e+47<p>What could any system of numbers give me that would make that operation faster than typing it in?<p>My old math teacher's objection to this was:<p>"But what if you don't have a calculator?"<p>And I was not smart enough at the time to realise there'd be more serious problems in a world where I didn't have a calculator and wanted to multiply large numbers (hey, I was only six.)<p>But what she ought to have said, when I was objecting to learning the tables, was that it makes sense to be reasonably fast with the basics of a system, so that you can do things beyond linear algebra quickly. If you need to stop and work out the very basics of a system every time you do algebra you're probably not going to get very far. The low level tools you have available influence what you can build on top of them.<p>However, the faster you are at the low level the less the practical gains are. The value of reducing an operation that takes a minute to one that takes seconds is likely to be enormous - but the value of reducing that second to a half second is not likely to be as significant.<p>It's similar to handwriting. We had a superior system of handwriting that we used to teach: Shorthand. It was more efficient even than most people's typing. But the additional value of that speed over typing was not sufficient for it to remain.<p>Unless there's an argument that, say, our being a half second or so faster at the low level will allow us access to some new high-level concepts, I think we're likely to stick with it.
If the metric of the relevance is the % of numbers represented or the amount of calculations done with it, base 10 has effectively been dead for many years. It's only used to occasionally communicate numbers over extremely narrow bandwidth channels to or between organics.
Apart from a different base, how about something <i>completely</i> new? Nobody seems to expect that. Do we really have reached the be all and end all with the positional notation? Or will something come up that we cannot imagine by now?
I think Knuth's up-arrow notation (see <a href="http://en.wikipedia.org/wiki/Knuth%27s_up-arrow_notation" rel="nofollow">http://en.wikipedia.org/wiki/Knuth%27s_up-arrow_notation</a>) can be considered a newer way of representing numbers, and it continues the progression you describe in that each the older notations are usable for small numbers, but the newer notations are better for large ones: Knuth's up-arrow notation is only really required for truly gigantic numbers and as a result I don't think it would be generally useful. (As generally useful numbers aren't that huge.)
A bit late to the party, but here's what I wrote in my JavaScript book:<p><pre><code> Asides: Number Encoding
This may be a bit of a strange concept to discuss in a book about JavaScript,
but the numbers that we’re all familiar with is only one of many different
kinds of encoding that exsits in the world.
The numerical system that we encode our numbers with today came from ancient
India, and was popularized by the Persians, hence the name Arabic-Hindu numbers.
It is a positional system. For example, the number 15 represents 1 unit in
the tens position, and 5 in the singular position - essentially 1 × 10 + 5 × 1.
Likewise, 314 is 3 × 100 + 1 × 10 + 4 × 1. This is read as 3 units in hundreds
position, 1 unit in the tens position, and 4 units in the singular position
There also exists numerical systems which are non-positoinal. Perhaps
the most famous example are Roman numerals. It’s also positional-ish, since
the position of the numerals are somewhat important for specific cases -
VI and IV mean very different things. The Mayan numeral system is another
example of a non-positional system, intermixed with a unary-ish system.
Most positional numerical systems have somewhat evolved into the same state,
despite having different runes and conventions to represent the same thing.
But perhaps the greatest innovation to numerical systems is the representation
of fractional numbers in a positional numerical system. It allowed us
to do really much fancier mathematics. However, as can be seen in the example
with 1/3 above, representing a fraction in a positional numerical system is
somewhat difficult.
The reason why this section is even here is to function as a reminder to the
reader that binary numerical systems used by modern computers are also
just another system - imagine it to be from another civilization, if you will
- and not be intimidated by it.
Speaking of fractions, the ancient Egyptians were one of the first civilizations
to use a fraction system (the Chinese were the other). Especially by modern day
standards, it was a very interesting fraction system. The fractions used by ancient
Egyptians are expressed only in terms of unit fractions
- i.e. fractions with 1 as the numerator.
</code></pre>
There was quite a lot of junk cut out from my book too about ancient chinese numerical systems - they had different numerical systems for different classes of people, and different numerical systems for different bases. And even had negative numbers!<p>Your question is actually a question of positional and nonpositional notation. I'm quite sure we'll stick with positional notations for some time to come, but uh, you never know about the future. For all we know, there could be a superior nonpositional system out there.
I don't think anyone knows what the world will look like in a million years. Will we be around as a species? Given the tendency toward exponential progress in all things we do, I tend to think that within 10,000 years, we'll either be off the planet or extinct. (That's not to say that we're destined to make the Earth uninhabitable, although that could happen. I just think we're "up or out" as a species. We'll either end economic scarcity or kill ourselves off within <i>500</i> years, I'd say, and the former means we're mining asteroids and, over the millennia, moving to other planets.)<p>If we get off the planet, it's unclear what we'll "look like" in many ways. We could be cyborgs. We could have enormous lifespans (millions of years). Unless we achieve such an immortality, we <i>will</i> continue to evolve (physically and culturally) and after a million years on different planets, we'll probably see all sorts of variation in terms of number of digits, representation of knowledge, and language.<p>So, looking a million years out, the answer is probably "no". Our system may still be alive, but if we're alive in a million years, I'd bet that we're off the planet and human culture will have forked, making the question of what is "final" unclear.<p>For the next 500 years (even 2500) I don't think we'll see another numeral system. Arabic Base-10 works, and there isn't much ti be gained in changing it . The glyphs themselves may evolve (our "Arabic" numbers look nothing like the original Arabic digits) but the concept will be the same.