Chinese multiplication table

The Tsinghua Bamboo Slips, containing the world's earliest decimal multiplication table, dated 305 BC during the Warring States period

The Chinese multiplication table is the first requisite for using the Rod calculus for carrying out multiplication, division, the extraction of square roots, and the solving of equations based on place value decimal notation. It was known in China as early as the Spring and Autumn period, and survived through the age of the abacus; pupils in elementary school today still must memorise it. The Chinese multiplication table consists of eighty-one terms. It was often called the nine-nine table, or simply nine-nine, because in ancient times, the nine nine table started with 9×9: nine nines beget eighty-one, eight nines beget seventy-two... seven nines beget sixty three, etc. two ones beget one. In the opinion of Wang Guowei, a noted scholar, the nine-nine table probably started with nine because of the "worship of nine" in ancient China; the emperor was considered the "nine five supremacy" in the Book of Change. See also Numbers in Chinese culture#Nine.

It is also known as nine-nine song (or poem), as the table consists of eighty-one lines with four or five Chinese characters per lines; this thus created a constant metre and render the multiplication table as a poem. For example, 9x9=81 would be rendered as “九九八十一”, or "nine nine eighty one", with the world for "begets" "得" implied. This makes it easy to learn by heart. A shorter version of the table consists of only forty-five sentences, as terms such as "nine eights beget seventy-two" are identical to "eight nines beget seventy-two" so there is no need to learn them twice. When the abacus replaced the counting rods in the Ming dynasty, many authors on the abacus advocated the use of the full table instead of the shorter one. They claimed that memorising it without needing a moment of thinking makes abacus calculation much faster.

The existence of the Chinese multiplication table is evidence of an early positional decimal system: otherwise a much larger multiplication table would be needed with terms beyond 9×9.

The Nine-nine song text in Chinese

一二得二 二二得四
一三得三 二三得六 三三得九
一四得四 二四得八 三四十二  四四十六
一五得五 二五一十 三五十五  四五二十  五五二十五
一六得六 二六十二 三六十八  四六二十四 五六三十  六六三十六
一七得七 二七十四 三七二十一 四七二十八 五七三十五 六七四十二 七七四十九
一八得八 二八十六 三八二十四 四八三十二 五八四十  六八四十八 七八五十六 八八六十四
一九得九 二九十八 三九二十七 四九三十六 五九四十五 六九五十四 七九六十三 八九七十二 九九八十一

It can be read in either row-major or column-major order.

The Nine-nine table in Chinese literature

Nine nine song in Ming dynasty Cheng Dawei Suanfa tongzong Volume II

Many Chinese classics make reference to the nine-nine table:

Archeological artifacts

Further information: Tsinghua Bamboo Slips
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