Are non-base-10 numbers ever written in kanji?


This morning I was thinking about the joke "[There are only 10 types of people
in the world: those who understand binary, and those who
don't"](http://en.wikipedia.org/wiki/Mathematical_joke#Jokes_with_numeral_bases).
Considering if this would translate well to Japanese, I figured it would have
to use `十` to represent the 10. But `十` represents the actual **value** of 10,
not just the digits '1' and '0' juxtaposed. However, if you wrote it as just
`一〇` (or possibly `一零`), it (almost) completely ruins the joke.

So does Japanese mathematics even write numeric notation using kanji? If so,
how are radices greater than 10 handled? Because traditionally, the symbols
used are letters, starting with A=10, B=11, and so on. Presumably Japan had
mathematicians long before encountering (Latin?) letters, so how did they
represent the >10 values?

* * *

Realize this may be too localized and not generally helpful. Close if
necessary.




A distinction is usually made between **positional numeral systems** and
**non-positional**.

Let's use Arabic numerals as an example of a positional numeral system. In
this kind of system, if we write **100** , each digit represents a coefficient
in an exponential series. Let's use _b_ to represent the base:

> **1** _b_ 2 \+ **0** _b_ 1 \+ **0** _b_ 0 = **100**

Okay, so what about 漢数字? The numbers 一, 十, and 百 each consist of one numeral
and each represent different powers of ten, but they do so _without respect to
position_. So in this use of 漢数字, we say they're _non-positional_. In this
system, it makes the most sense to say 十 represents the _value_ ten. It
doesn't represent an exponential series like the one above because there's no
correspondence between position and degree.

Notice how in the example of Arabic numerals above, I didn't specify a base.
It might be decimal, or it might be binary. You can't say the same for 一, 十,
and 百. With the numbers represented by these numerals, **there is no
meaningful definition of _b_ , so it doesn't make sense to talk about what
base the numbers are written in**.

However, there is a positional use of 漢数字, as well. 漢数字 supplanted
[算木](http://ja.wikipedia.org/wiki/%E7%AE%97%E6%9C%A8) as a positional numeral
system in Japan after the 16th Century, and in modern Japan both positional
and non-positional uses exist. For example, [user1205935 wrote in a
comment](https://japanese.stackexchange.com/users/1628/user1205935) that a
restaurant price may be written `一三八〇円`. This number is clearly written using
positional numerals:

> **一** _b_ 3 \+ **三** _b_ 2 \+ **八** _b_ 1 \+ **〇** _b_ 0 = **一三八〇**

We've demonstrated that a base exists! We haven't demonstrated _what_ base it
is, and that's the crux of your question. In this case, it's obvious the base
is 10. But is it always?

I can't prove it, but I've searched and looked through history books, and I
can't find any evidence that it's ever _not_ 10\. It seems like there's a very
strong preference for Arabic numerals when writing numbers in any base but
ten. In fact, I found people joking about what the numeral sequence would be
if it _did_ continue, implying it doesn't. (Someone suggested 十土王圭, which I
include here for humor value.) So, in the absence of evidence to the contrary,
I conclude that the answer is:

**No, kanji is not ever used to write values in bases other than decimal.**

Two historical notes:

  1. The historical Chinese numeral 十 was once written as a vertical line, and this old form was combined with 一, 二, 三 and the old form of 四 to form single numerals representing the values 11-14. See [here](http://en.wikipedia.org/wiki/File:Shang_numerals.jpg) and [here](http://en.wikipedia.org/wiki/File:%E8%A5%BF%E5%91%A8%E9%87%91%E6%96%87.jpg) on Wikipedia. However, I believe these forms had passed out of use before kanji was adopted in Japan.

  2. According to [Wikipedia](http://ja.wikipedia.org/wiki/%E6%BC%A2%E6%95%B0%E5%AD%97#.E4.BD.8D.E5.8F.96.E3.82.8A.E8.A8.98.E6.95.B0.E6.B3.95), in the old non-positional system, the juxtaposition 二八 represented the value 16 (二×八） rather than 28. I believe this sort of use is still around to some extent in, for example, [四六時中](http://dic.yahoo.co.jp/dsearch?enc=UTF-8&p=%E5%9B%9B%E5%85%AD&dtype=0&dname=0ss&stype=0&pagenum=1&index=109881600000), which represents 四×六 through juxtaposition.




A distinction is usually made between **positional numeral systems** and
**non-positional**.

Let's use Arabic numerals as an example of a positional numeral system. In
this kind of system, if we write **100** , each digit represents a coefficient
in an exponential series. Let's use _b_ to represent the base:

> **1** _b_ 2 \+ **0** _b_ 1 \+ **0** _b_ 0 = **100**

Okay, so what about 漢数字? The numbers 一, 十, and 百 each consist of one numeral
and each represent different powers of ten, but they do so _without respect to
position_. So in this use of 漢数字, we say they're _non-positional_. In this
system, it makes the most sense to say 十 represents the _value_ ten. It
doesn't represent an exponential series like the one above because there's no
correspondence between position and degree.

Notice how in the example of Arabic numerals above, I didn't specify a base.
It might be decimal, or it might be binary. You can't say the same for 一, 十,
and 百. With the numbers represented by these numerals, **there is no
meaningful definition of _b_ , so it doesn't make sense to talk about what
base the numbers are written in**.

However, there is a positional use of 漢数字, as well. 漢数字 supplanted
[算木](http://ja.wikipedia.org/wiki/%E7%AE%97%E6%9C%A8) as a positional numeral
system in Japan after the 16th Century, and in modern Japan both positional
and non-positional uses exist. For example, [user1205935 wrote in a
comment](https://japanese.stackexchange.com/users/1628/user1205935) that a
restaurant price may be written `一三八〇円`. This number is clearly written using
positional numerals:

> **一** _b_ 3 \+ **三** _b_ 2 \+ **八** _b_ 1 \+ **〇** _b_ 0 = **一三八〇**

We've demonstrated that a base exists! We haven't demonstrated _what_ base it
is, and that's the crux of your question. In this case, it's obvious the base
is 10. But is it always?

I can't prove it, but I've searched and looked through history books, and I
can't find any evidence that it's ever _not_ 10\. It seems like there's a very
strong preference for Arabic numerals when writing numbers in any base but
ten. In fact, I found people joking about what the numeral sequence would be
if it _did_ continue, implying it doesn't. (Someone suggested 十土王圭, which I
include here for humor value.) So, in the absence of evidence to the contrary,
I conclude that the answer is:

**No, kanji is not ever used to write values in bases other than decimal.**

Two historical notes:

  1. The historical Chinese numeral 十 was once written as a vertical line, and this old form was combined with 一, 二, 三 and the old form of 四 to form single numerals representing the values 11-14. See [here](http://en.wikipedia.org/wiki/File:Shang_numerals.jpg) and [here](http://en.wikipedia.org/wiki/File:%E8%A5%BF%E5%91%A8%E9%87%91%E6%96%87.jpg) on Wikipedia. However, I believe these forms had passed out of use before kanji was adopted in Japan.

  2. According to [Wikipedia](http://ja.wikipedia.org/wiki/%E6%BC%A2%E6%95%B0%E5%AD%97#.E4.BD.8D.E5.8F.96.E3.82.8A.E8.A8.98.E6.95.B0.E6.B3.95), in the old non-positional system, the juxtaposition 二八 represented the value 16 (二×八） rather than 28. I believe this sort of use is still around to some extent in, for example, [四六時中](http://dic.yahoo.co.jp/dsearch?enc=UTF-8&p=%E5%9B%9B%E5%85%AD&dtype=0&dname=0ss&stype=0&pagenum=1&index=109881600000), which represents 四×六 through juxtaposition.

