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By: Tao Steven Zheng (郑涛)

Large Numbers

Big-number-table-explain--scaled

Figure 1. Large numbers in modern Chinese

During the Spring-Autumn and Warring States period (771 – 221 BC), the large numbers above “ten thousand” (万wan) were expressed as powers of ten. After the Han Dynasty, the large numbers above “ten thousand” were expressed as powers of ten thousand. According to the first chapter of the Sunzi Suanjing:

凡大数之法,万万曰亿,万万亿曰兆,万万兆曰京,万万京曰陔,万万陔曰秭,万万秭曰壤,万万壤曰沟,万万沟曰涧,万万涧日正,万万正曰载。 The rule of large numbers states: wan-wan is called yi (亿), wan-wan-yi is called zhao (兆), wan-wan-zhao is called jing (京), wan-wan-jing is called hai (陔), wan-wan-hai is called zi (秭), wan-wan-zi is called rang (壤), wan-wan-rang is called gou (沟), wan-wan-gou is jian (涧), wan-wan-jian is called zheng (正), wan-wan-zheng is called zai (载).


Number Description in the Sunzi Suanjing Value in the Sunzi Suanjing
万wan 104 104
亿 yi 104104 108
兆 zhao 104104108 1016
京 jing 1041041016 1024
垓 gai 1041041024 1032
秭 zi 1041041032 1040
壤 rang 1041041040 1048
沟 gou 1041041048 1056
涧 jian 1041041056 1064
正 zheng 1041041064 1072

Problem Study 1: Nine Dykes

The following problem is from the Sunzi Suanjing (Chapter 3, Problem 34).

今有出门望见九堤。堤有九木,木有九枝,枝有九巢,巢有九禽,禽有九雏,雏有九毛,毛有九色。问:各几何?


答曰:木八十一。枝七百二十九。巢六千五百六十一。禽五万九千四十九。雏五十三万一千四百四十一。毛四百七十八万二千九百六十九。色四千三百四万六千七百二十一。

                      

术曰:置九堤,以九乘之,得木之数。又以九乘之,得枝之数。又以九乘之,得巢之数。又以九乘之,得禽之数。又以九乘之,得雏之数。又以九乘之,得毛之敷。又以九乘之,得色之数。

Suppose that after leaving a town gate, one sees 9 dykes. There are 9 trees on each dike, 9 branches on each tree, 9 nests on each branch, 9 birds in each nest, 9 fledglings per bird, 9 feathers on each fledgling, and 9 (different) colours on each feather. Question: How many are there of each?


Answer: 81 trees, 729 branches, 6,561 nests, 59,049 birds, 531,441 young birds, 4,782,969 feathers, 43,046,721 colours.


Method: Set 9 dikes, multiply by 9 to obtain the number of trees. Next multiply by 9 to obtain the number of branches. Next multiply by 9 to obtain the number of nests. Next multiply by 9 to obtain the number of birds. Next multiply by 9 to obtain the number of young birds. Next multiply by 9 to obtain the number of feathers. Next multiply by 9 to obtain the number of colours.


Minute Numbers

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