As a slow learner of Japanese, I often have to look up *kanji* (Chinese characters) in my dictionary. I was intrigued to notice that for some kanji, one of their meanings is a *very* large number:

Kanji | Reading | Number |
---|---|---|

万 | man | 10^{4} |

億 | oku | 10^{8} |

兆 | chō | 10^{12} |

京 | kei | 10^{16} |

垓 | gai | 10^{20} |

𥝱 | jo | 10^{24} |

穣 | jō | 10^{28} |

溝 | kō | 10^{32} |

澗 | kan | 10^{36} |

正 | sei | 10^{40} |

載 | sai | 10^{44} |

極 | goku | 10^{48} |

Table 1. *Kanji for large numbers with readings and number values*

## Kanji and the supercomputer

For example, the character 京, which is the “kyō” in Kyoto (京都) means “metropolis”, but also 10^{16}. For that meaning it is pronounced “kei”, and this is the origin of the name of the famous Riken supercomputer in Kobe, the “K computer”: it is a computer capable of 10 petaflops, so 10^{16} or *kei* flops. Incidentally, it is probably the only supercomputer that has its own railway station, 京コンピュータ前駅.

## Counting systems

The story of how these kanji came to represent such large numbers is quite interesting. It started with the emergence of a number of different counting systems.

Japan has adopted kanji from China – the word *kanji* (漢字) means Chinese characters, the 漢 character meaning “Han”. The traditional numeral systems of China, Korea, and Japan are all decimal-based but grouped into ten thousands, also known as myriads (*man*, 万), rather than thousands. Very long ago, the characters beyond *man* simply indicated degrees of large, so *oku* (億) was larger than *man* (万), *chō* (兆) was larger than *oku*, *kei* (京) was larger than *chō*, etc. Actual counting of numbers beyond one myriad was ten myriads, hundred myriads, thousand myriads.

Then over time the characters *oku*, *chō*, *kei* etc. started to be used for this, in a counting system called *kasū* (下数,かすう, “lower arithmetic”). So in this system, *oku* was ten times *man*, *chō* was ten times *oku*, etc. This was very long ago, more than two thousand years, and clearly it was not entirely satisfactory because already from the Han dynasty onwards (i.e. 200 BCE), a separate counting system called *jōsū*（上数,じょうすう,”upper arithmetic”）also became used. In this system, the character after *man* got the value of *man man* (万万), so *oku* = *man man* = 10^{8}. The next value *chō* became *oku oku* (億億) = 10^{16}, *kei* = *chō chō* (兆兆) =10^{32} and so on.
Later, yet another system, called *chūsū* (中数,ちゅうすう,”middle arithmetic”) came into use. In this system, the values for consecutive kanji where 10^{4} apart. To be precise, in China there were two variants of *chūsū*, *manshin* (万進) and (万万進) *manmanshin*, the latter had intervals of 10^{8}.

The table below summarises the various systems:

Kanji | Reading | kasū (下数) |
chūsū/manshin (中数／万進) |
chūsū/manmanshin (中数／万万進) |
jōsū (上数) |
---|---|---|---|---|---|

万 | man |
10^{4} |
10^{4} |
10^{4} |
10^{4} |

億 | oku |
10^{5} |
10^{4×2}=10^{8} |
10^{8×1}=10^{8} |
10^{4×2}=10^{8} |

兆 | chō |
10^{6} |
10^{4×3}=10^{12} |
10^{8×2}=10^{16} |
10^{4×22}=10^{16} |

京 | kei |
10^{7} |
10^{4×4}=10^{16} |
10^{8×3}=10^{24} |
10^{4×23}=10^{32} |

垓 | gai |
10^{8} |
10^{4×5}=10^{20} |
10^{8×4}=10^{32} |
10^{4×24}=10^{64} |

秭 | jo |
10^{9} |
10^{4×6}=10^{24} |
10^{8×5}=10^{40} |
10^{4×25}=10^{128} |

穣 | jō |
10^{10} |
10^{4×7}=10^{28} |
10^{8×6}=10^{48} |
10^{4×26}=10^{256} |

溝 | kō |
10^{11} |
10^{4×8}=10^{32} |
10^{8×7}=10^{56} |
10^{4×27}=10^{512} |

澗 | kan |
10^{12} |
10^{4×9}=10^{36} |
10^{8×8}=10^{64} |
10^{4×28}=10^{1024} |

正 | sei |
10^{13} |
10^{4×10}=10^{40} |
10^{8×9}=10^{72} |
10^{4×29}=10^{2048} |

載 | sai |
10^{14} |
10^{4×11}=10^{44} |
10^{8×10}=10^{80} |
10^{4×210}=10^{4096} |

極 | goku |
10^{15} |
10^{4×12}= 10^{48} |
10^{8×11}=10^{88} |
10^{4×211}=10^{8192} |

Table 2. *Summary of the different counting systems used historically*

## An Edo-era mathematics book

In Edo-era Japan, the 1634 edition of an influential mathematics book, *jinkōki* (塵劫記), standardised on the *chūsū/manshin* (中数/万進) system as shown in the table above. The work covers a wide variety of mathematical problems but its main purpose was to explain techniques for calculating using the *soroban* (算盤), the Japanese abacus.

## The daimyo, the samurai and the soroban

The oldest *soroban* in existence is the *Shibei Shigekatsu hairyo soroban*, from 1591. It was made from plum tree wood and silver and given to the samurai Shibei Shigekatsu by the famous daimyō Toyotomi Hideyoshi (The word *hairyo* (拝領) means “receiving from a superior”). It shows that this kind of calculating device was highly valued. This soroban has 20 number rods, so it can represent any number up to 10^{20}-1. So in any case the introduction of the *soroban* made it easier to conceive very large numbers.

## Buddhist angels and mustard seeds

The story so far does not quite explain where those different systems came from or *why* these choices were made, and indeed earlier editions of the *jinkōki* used different combinations of *kasū* and *chūsū*. In any case, it turns out that the origin of these very large numbers lies in ancient Buddhist texts.

Buddhism has the notion of a *kalpa* or aeon, a very long unit of time. There are in the old texts two definitions for a kalpa, provided by the Buddha by analogy. I found the following explanation on a Japanese web site for the book “巨大数入門” (“An introduction to very large numbers”):

『仏教では、とても長い時間を表すときに劫（こう）という時間の単位が使われます。「未来永劫」「億劫」「五劫のすり切れ」などの言葉は、ここから来ています。この「劫」には2通りの意味があり、1つ目は「磐石劫」で、四十里四方の大石を、いわゆる天人の羽衣で百年に一度払い、その大きな石が摩滅して無くなってもなお「一劫」の時間は終わらないと譬えています。 2つ目は「芥子劫」で、方四十里の城に小さな芥子粒を満たして百年に一度、一粒ずつ取り去り、その芥子がすべて無くなってもなお尽きないほどの長い時間が一劫であるとのことです。』

Loosely translated, this means that there are two definitions for a kalpa:

- Imagine a huge cube of rock of 40
*ri*on the side. Every hundred years, an angel brushes the rock with its wings. A kalpa won’t have ended by the time it takes to entirely wear away the rock. - Imagine a huge building of 40
*ri*on the side. Every hundred years, a small mustard seed is put into it. The kalpa won’t have ended by the time the building is full.

To estimate how long a kalpa actually lasts, someone made the following assumptions:

- Suppose the angel’s wings remove one layer of atoms over an area of one square meter. Given that 1
*ri*is 500*m*and the diameter of an atom is 0.2*nm*, it would take 4×10^{24}years to wear away the rock. - For the second definition, assuming a mustard seed is 0.5
*mm*, it would take 6.4×10^{24}years to fill the entire volume.

It’s rather striking that both estimates produce the same order of magnitude, 10^{24} years. For comparison, the estimated age of the universe is “only” 1.38×10^{10} years.

## Grains of sand in the Ganges

Regardless, it means that Buddhist scholar monks had a need to express very large numbers. In fact, Buddhism has concepts such as the number of grains of sand in the Ganges (*gōgasha*,恒河沙), which are even larger than *goku* (極, 10^{48}). The largest quantified term I could find in my dictionary is *muryōtaisū* (無量大数,”immeasurably large number”), with a value of 10^{68} using *chūsū*, but often defined as 10^{88}, due to a variant system used for values above *goku* used in the 1631 edition of the *jinkōki*.