Chinese multiplication table

TheChinesemultiplication tableis the first requisite for using theRod calculusfor 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 theSpring and Autumn period,and survived through the age of theabacus;pupils in elementary school today still must memorise it.

TheTsinghua Bamboo Slips,containing the world's earliestdecimalmultiplication table,dated 305 BC during theWarring Statesperiod

The Chinese multiplication table consists of eighty-one terms. It was often called thenine-nine table,or simplynine-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 two. In the opinion ofWang 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 theBook of Change.See alsoNumbers in Chinese culture § Nine.

It is also known asnine-nine song(or poem),as the table consists of eighty-one lines with four or five Chinese characters per lines; this thus created a constantmetreand render the multiplication table as a poem. For example, 9 × 9 = 81 would be rendered as "Chín chín tám mươi mốt", or "nine nine eighty one", with the world for "begets" "Đến" 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 theabacusreplaced the counting rods in the Ming dynasty, many authorson the abacus advocated the use of the full table instead of the shorter one. They claimed that memorising it without needing ag 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

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

Cửu cửu phép nhân khẩu quyết biểu (The Nine-nine multiplication table)

	1 một yī	2 nhị èr	3 tam sān	4 bốn sì	5 năm wǔ	6 sáu liù	7 bảy qī	8 tám bā	9 chín jiǔ
1 một yī	Nhất nhất đến một
2 nhị èr	Một vài đến nhị	Nhị nhị đến bốn
3 tam sān	Một tam đến tam	Nhị tam đến sáu	Tam tam đến chín
4 bốn sì	Một bốn đến bốn	Nhị bốn đến tám	Ba bốn mươi nhị	Bốn 46
5 năm wǔ	Một năm đến năm	Hai lăm một mười	Ba năm mười lăm	Bốn năm hai mươi	Năm năm 25
6 sáu liù	Một sáu đến sáu	Hai sáu mười hai	Tam 68	Bốn sáu 24	Năm sáu 30	Sáu sáu 36
7 bảy qī	Một bảy đến bảy	Hai bảy mười bốn	Ba bảy hai mốt	Bốn bảy 28	Năm bảy 35	Sáu bảy 42	Bảy bảy bốn mươi chín
8 tám bā	Một tám đến tám	Nhị bát mười sáu	38 24	Bốn tám ba mười hai	Năm tám 40	Sáu tám 48	Bảy tám 56	Bát bát 64
9 chín jiǔ	Một chín đến chín	Nhị 98	Tam chín 27	Bốn chín ba mươi sáu	Năm chín 45	Sáu cửu ngũ mười bốn	Bảy 96 mười ba	Tám chín 72	Chín chín tám mươi mốt

Nine-nine multiplication table

	1 One yī	2 Two èr	3 Three sān	4 Four sì	5 Five wǔ	6 Six liù	7 Seven qī	8 Eight bā	9 Nine jiǔ
1 One yī	One times one equals one
2 Two èr	One times two equals two	Two times two equals four
3 Three sān	One times three equals three	Two times three equals six	Three times three equals nine
4 Four sì	One times four equals four	Two times four equals eight	Three times four equals twelve	Four times four equals sixteen
5 Five wǔ	One times five equals five	Two times five equals ten	Three times five equals fifteen	Four times five equals twenty	Five times five equals twenty-five
6 Six liù	One times six equals six	Two times six equals twelve	Three times six equals eighteen	Four times six equals twenty-four	Five times six equals thirty	Six times six equals thirty-six
7 Seven qī	One times seven equals seven	Two times seven equals fourteen	Three times seven equals twenty-one	Four times seven equals twenty-eight	Five times seven equals thirty-five	Six times seven equals forty-two	Seven times seven equals forty-nine
8 Eight bā	One times eight equals eight	Two times eight equals sixteen	Three times eight equals twenty-four	Four times eight equals thirty-two	Five times eight equals forty	Six times eight equals forty-eight	Seven times eight equals fifty-six	Eight times eight equals sixty-four
9 Nine jiǔ	One times nine equals nine	Two times nine equals eighteen	Three times nine equals twenty-seven	Four times nine equals thirty-six	Five times nine equals forty-five	Six times nine equals fifty-four	Seven times nine equals sixty-three	Eight times nine equals seventy-two	Nine times nine equals eighty-one

The Nine-nine table in Chinese literature

Nine nine song inMing dynastyCheng DaweiSuanfa tongzongVolume II

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

• Zhoubi Suanjing:"nine nine eighty one"
• Guan Zihas sentences of the form "three eights beget twenty four, three sevens beget twenty-one"
• The Nine Chapters on the Mathematical Art:"Fu Xiinvented the art of nine-nine ".
• InHuainanzi,there were eight sentences: "nine nines beget eighty one", "eight nines beget seventy two", all the way to "two nines beget eighteen".
• A nine-nine table manuscript was discovered inDun Huang.
• Xia Houyang's Computational Canons:"To learn the art of multiplication and division, one must understand nine-nine".
• The Song dynasty author Hong Zhai's Notebooks said: "three threes as nine, three fours as twelve, two eights as sixteen, four fours as sixteen, three nines as twenty seven, four nines as thirty six, six sixes as thirty six, five eights as forty, five nines as forty five, seven nines as sixty three, eight nines as seventy two, nine nines as eighty one". This suggests that the table has begun with the smallest term since the Song dynasty.
• Song dynastymathematicianYang Hui's mathematics text book:Suan fa tong bian ben mo,meaning "You must learn nine nine song from one one equals one to nine nine eighty one, in small to large order"
• Yuan dynastymathematicianZhu Shijie'sSuanxue qimeng(Elementary mathematics): "one one equals one, two by two equals four, one by three equals three, two by three equals six, three by three equals nine, one by four equals four... nine by nine equals eight one"

Archeological artifacts

Further information:Tsinghua Bamboo Slips

• At the end of the 19th century, archeologists unearthed pieces of written bamboo script from the Han dynasty inXin Jiang.One such Han dynasty bamboo script, from Liusha, is a remnant of the nine-nine table. It starts with nine: nine nine eighty one, eight nine seventy two, seven nine sixty three, eight eight sixty four, seven eight fifty six, six eight forty eight,... two two gets four, altogether 1100 Chinese words.
• In 2002, Chinese archeologists unearthed a written wood script from a two-thousand-year-old site from theWarring States,on which was written: "four eight thirty two, five eight forty, six eight forty eight." This is the earliest artifact of the nine-nine table that has been unearthed, indicating that the nine-nine table, as well as a positional decimal system, had appeared by theWarring Statesperiod.
• Tsinghua Bamboo Slips Calculation Table, is an ancient calculator artifact from the Warring States period in 305 BC. It is included in the "Tsinghua University Collection of Warring States Bamboo Slips (Part IV)," predating the previously discovered Liye Qin Bamboo Slips and Zhangjiashan Han Bamboo Slips nine-nine tables by a century.
• In 2023, a bamboo slip from the 4th century BC, containing a multiplication formula, was found in a Jingzhou tomb in Hubei Province, China. The formula was deciphered using infrared scanning, revealing calculations such as "five times seven is thirty plus five, four times seven is twenty plus eight, three times seven is twenty plus one." As of December 2023, this represents the earliest discovery of a nine-nine table artifact.
• The nine-nine table was transmitted to Japan, and appeared in a Japanese primary mathematics book in the 10th century, beginning with 9×9.