The Chinese multiplication table

Multiplication table

In mathematics, a multiplication table is a mathematical table used to define a multiplication operation for an algebraic system....

is the first requisite for using the Rod calculus

Rod calculus

Rod calculus or rod calculation is the method of mathematical computation with counting rods in China from the Warring States to Ming dynasty before the counting rods were replaced by the more convenient and faster abacus.-Hardware:...

 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

Abacus

The abacus, also called a counting frame, is a calculating tool used primarily in parts of Asia for performing arithmetic processes. Today, abaci are often constructed as a bamboo frame with beads sliding on wires, but originally they were beans or stones moved in grooves in sand or on tablets of...

; 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 song or 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 two. In the opinion of Wang Guowei

Wang Guowei

Wang Guowei , courtesy name Jing'an or Baiyu , was a Chinese scholar, writer and poet...

, 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.

The table consists of eighty-one sentences with five Chinese characters per sentence; this is easy for children 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

Abacus

The abacus, also called a counting frame, is a calculating tool used primarily in parts of Asia for performing arithmetic processes. Today, abaci are often constructed as a bamboo frame with beads sliding on wires, but originally they were beans or stones moved in grooves in sand or on tablets of...

 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 table in Chinese literature

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

• Zhou Bi Suan Jing: "nine nine eighty one"
• Guan Zi has sentences of the form "three eights beget twenty four, three sevens beget twenty-one"
• The Nine Chapters on the Mathematical Art
  The Nine Chapters on the Mathematical Art
  The Nine Chapters on the Mathematical Art is a Chinese mathematics book, composed by several generations of scholars from the 10th–2nd century BCE, its latest stage being from the 1st century CE...
  
  : "Fu Xi invented the art of nine-nine".
• In Huainanzi
  Huainanzi
  The Huáinánzǐ is a 2nd century BCE Chinese philosophical classic from the Han dynasty that blends Daoist, Confucianist, and Legalist concepts, including theories such as Yin-Yang and the Five Phases. It was written under the patronage of Liu An, Prince of Huainan, a legendarily prodigious author...
  
  , 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 in Dun 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 eigthy one". This suggests that the table has begun with the smallest term since the Song dynasty.
• Song dynasty
  Song Dynasty
  The Song Dynasty was a ruling dynasty in China between 960 and 1279; it succeeded the Five Dynasties and Ten Kingdoms Period, and was followed by the Yuan Dynasty. It was the first government in world history to issue banknotes or paper money, and the first Chinese government to establish a...
  
   mathematician Yang Hui
  Yang Hui
  Yang Hui , courtesy name Qianguang , was a Chinese mathematician from Qiantang , Zhejiang province during the late Song Dynasty . Yang worked on magic squares, magic circles and the binomial theorem, and is best known for his contribution of presenting 'Yang Hui's Triangle'...
  
  '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 dynasty
  Yuan Dynasty
  The Yuan Dynasty , or Great Yuan Empire was a ruling dynasty founded by the Mongol leader Kublai Khan, who ruled most of present-day China, all of modern Mongolia and its surrounding areas, lasting officially from 1271 to 1368. It is considered both as a division of the Mongol Empire and as an...
  
   mathematician Zhu Shijie
  Zhu Shijie
  Zhu Shijie , courtesy name Hanqing , pseudonym Songting , was one of the greatest Chinese mathematicians lived during the Yuan Dynasty....
  
  's Suanxue qimeng (Elemenary 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

• At the end of 19th century, archeologists unearthed pieces of written bamboo script from the Han dynasty in Xin 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 the Warring 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 the Warring States period
• The nine-nine table was transmitted to Japan, and appeared in a Japanese primary mathematics book in the 10th century, beginning with 9×9.