
Ask HN: Are we using the final numeral system? - itry
In the earliest days of mankind, 13 was written as &quot;.............&quot; The number of dots represented the number. Later the Egyptians had a different hieroglyph for 10, so 13 could be written as &quot;#...&quot; where &quot;#&quot; means 10 and &quot;.&quot; means 1. Much shorter. 33 was written as &quot;###...&quot;. Nice. Then the 0 was invented. And nowadays, we have &quot;hieroglyphs&quot; 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?
======
derefr
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:

    
    
        ABC = (val[A] × base^2) + (val[B] × base^1) + (val[C] × base^0).
    

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.

~~~
brudgers
The Babylonians used a base 60 [sexagesimal] floating point system with the
exponent implied by the context. Knuth discusses in in TAoCP volume 2, some of
it referenced from _The Exact Sciences in Antiquity._

[https://en.m.wikipedia.org/wiki/Babylonian_numerals](https://en.m.wikipedia.org/wiki/Babylonian_numerals)

[http://store.doverpublications.com/0486223329.html](http://store.doverpublications.com/0486223329.html)

------
Bud
I'm going to take the contrarian view: yes. Inertia is very powerful. We'll
stay with base-10.

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.

------
sp332
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
[https://en.wikipedia.org/wiki/Magnet_URI_scheme#URN.2C_conta...](https://en.wikipedia.org/wiki/Magnet_URI_scheme#URN.2C_containing_hash_.28xt.29)

~~~
itry
Good idea to mention the scientific notation. As it really _is_ different from
the positional notation. You can express _some_ numbers in a shorter way then
in the positional notation. Thats pretty nice. A variable length encoding with
compression.

As for different bases: I consider these very similar to base 10. Maybe we
will use base X in a million years, but that wouldnt be a big surprise. I was
more thinking about a revolution like from "............." to "13".

~~~
sp332
Well the notation lets you tell how specific you are being about a number.
Instead of saying "about 1,000" you can say 1 * 10^3 or 1.00 * 10^3 which give
different amounts of information. So it's not exactly the same as base-10. And
if you're talking about really big numbers, it's nice not to drag all those
0's around. It's just as big a step from .......... to 10 as from 10000000000
to 10^10.

------
computer
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.

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.

~~~
itry
What do chinese and africans use?

~~~
vbuterin
Chinese:

零一二三四五六七八九

Africans are a diverse group so I expect that they have many different
systems.

~~~
chc
Well, those are the numerals 0–9. You're missing 10, 100, 1000, 10000,
1000000… Chinese has a lot of number characters.

They use a system where a power-of-10 number is potentially preceded by a
smaller count and followed by a smaller number. So, for example, 241 is
represented as [2][100][4][10][1].

~~~
vbuterin
Right, they also have:

十 = 10 白 = 10^2 千 = 10^3 万 = 10^4 亿 = 10^8

This leads to the non-Western-like feature that the word for a million is 一白万
(1 10^2 10^4) whereas ten million is 一千万 (1 10^3 10^4) instead of 十白万 (10 10^2
10^4) which you might expect if you think using Western number systems and
just internalize the rule that "白万 = million".

------
pirateking
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.

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.

------
ddebernardy
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.

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.

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.

~~~
aflinik
I wouldn't say we use base-60 for time or angles. We still use just 10 digits
to write them down, so it's still in base-10.

~~~
sp332
It's definitely base-60. Carrying from one place to the next happens on powers
of 60, even if you write the value of each place with base-10 numbers. And if
you have an analog clock, you don't need base 10 at all.

------
6d0debc071
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.

What's 395847593874382754238754987 * 389756987476347629845 ?

1.5428437e+47

What could any system of numbers give me that would make that operation faster
than typing it in?

My old math teacher's objection to this was:

"But what if you don't have a calculator?"

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.)

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.

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.

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.

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.

------
stevedekorte
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.

~~~
itry
What base are numbers represented inside a computer? Are there really little
"things" holding 0s and 1s in DRAM for example? Or is it technically a
different base?

~~~
jerf
"Are there really little "things" holding 0s and 1s in DRAM for example?"

At the moment, close enough to yes to make no difference. There's various ways
of trying to score internet pedant points by fiddling around the _very_ edges
of that statement, but they're not relevant to a casual question. Yes, there
is somewhere where you can find a physical thing that we might call a "one"
and a "zero", even if it's not quite the same exact 1 or 0 you might get in
your CPU due to parity or forward error correction or whatever other crazy
thing you may have.

I mention those fiddly details to get them out of the way so I can bring up
the bare handful of devices that actually functioned on true, no-kidding
ternary:
[http://en.wikipedia.org/wiki/Ternary_computer](http://en.wikipedia.org/wiki/Ternary_computer)
Per Knuth's note in that article, as we continue to optimize the heck out of
our silicon and as we start running out of ability to simply shrink, it's not
inconceivable that ternary computing could make a comeback in the future. It
is often casually assumed in current sci-fi and such that binary is the true
"final" base of the future, but ternary is not out of the running.

On that note, see also:
[http://www.americanscientist.org/issues/issue.aspx?id=3268&y...](http://www.americanscientist.org/issues/issue.aspx?id=3268&y=0&no=&content=true&page=5&css=print)

This would have next to no impact on anything else, though... in fact the
first ternary computers would certainly simply run current programs, possibly
recompiled and certainly at an efficiency penalty, but with no other end-user-
visible effect.

(And let me warn you away from various speculations that ternary computers
could somehow compute something binary computers can't. Turing Completeness,
along with frankly obvious common (programmer) sense, precludes that. We
already never truly work with bits... if we want bits we actually have to go
out of our way to extract them from things at least the size of bytes, if not
larger. Arguably modern computers are already in many ways Base256, as it is
effectively impossible to manipulate anything smaller than an octet, which
also obviously encompasses anything Base3 can do.)

------
itry
Apart from a different base, how about something _completely_ 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?

~~~
sp332
How about quarter-imaginary? [https://en.wikipedia.org/wiki/Quater-
imaginary_base](https://en.wikipedia.org/wiki/Quater-imaginary_base)

------
abeld
I think Knuth's up-arrow notation (see
[http://en.wikipedia.org/wiki/Knuth%27s_up-
arrow_notation](http://en.wikipedia.org/wiki/Knuth%27s_up-arrow_notation)) 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.)

~~~
pmelendez
I thought Knuth's up-arrow notation was actually an operator rather than a
numeral system. Just in the same way that multiplication is a short for sums,
the arrow operator would be a short for exponentiation.

------
stcredzero
Bases are more energy efficient to compute with as they get closer to _e_ , so
in the future trinary numbers might be the way to count.

~~~
tedsanders
Interesting! I dint understand why, though. Do you have a reference for this?

------
beachstartup
probably not.

also, your description is incomplete: we also use binary, hexadecimal, octal
and other numbering systems on a regular basis.

------
zdw
Probably, until our descendants are genetically engineered to have more/fewer
than 10 digits...

~~~
chronial
That statement seems to imply that base 10 is a good system because we have
ten fingers. That is probably the reason why it was created, but we don’t use
it because it’s the best – we use it because everybody does.

Base 12 is actually a way more useful system since it is divisible by 2, 3, 4,
5 and 6 (vs. 2 and 5 for base 10), so 1/3 is not 0.3333, but 0.4.

~~~
lucio
1/3 is 0.333 in base 10 and base 12. 10/3=4 in base 12.

~~~
drdeca
Wouldn't 1/3 be 0.4 in base 12?

4 _12^-1 4 /12 1/3

Whereas .33333... In base 12 would be 3_.111... In base 12, =3*1/B =3/B

------
chewxy
A bit late to the party, but here's what I wrote in my JavaScript book:

    
    
        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.
    
    

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!

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.

------
michaelochurch
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 _500_ years, I'd say, and the former means we're mining asteroids and,
over the millennia, moving to other planets.)

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 _will_ 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.

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.

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.

~~~
itry
> human culture will have forked

We have that situation already. The Pirahã have only 2 numers in their
languge. Roughly translatable as "few" and "many".

And scientists use different bases and scientific notation.

------
instakill
Probably not.

