【九因歌】 Multiplication


<九因歌>

一一如一
一二如二
一三如三
一四如四
一五如五
一六如六
一七如七
一八如八
一九如九

 

二一如二
二二如四
二三如六
二四如八
二五得一十
二六一十二
二七一十四
二八一十六
二九一十八

 

三一如三
三二如六
三三該九
三四一十二
三五一十五
三六一十八
三七二十一
三八二十四
三九二十七

 

四一如四
四二如八
四三一十二
四四一十六
四五中二十
四六二十四
四七二十八
四八三十二
四九三十六

 

五一如五
五二得一十
五三一十五
五四中二十
五五二十五
五六中三十
五七三十五
五八中四十
五九四十五

 

六一如六
六二一十二
六三一十八
六四二十四
六五中三十
六六三十六
六七四十二
六八四十八
六九五十四

 

七一如七
七二一十四
七三二十一
七四二十八
七五三十五
七六四十二
七七四十九
七八五十六
七九六十三

 

八一如八
八二一十六
八三二十四
八四三十二
八五中四十
八六四十八
八七五十六
八八六十四
八九七十二

 

九一如九
九二一十八
九三二十七
九四三十六
九五四十五
九六五十四
九七六十三
九八七十二
九九八十一

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【九因歌】 
http://blog.yahoo.com/_4LV7HFWVLSU7U6EQ4CW6ZADTQM/articles/107082

相信大家也認識九因歌,或許在小二時已琅琅上口,但你們又是否知道九因歌的歷史呢?

九因歌就是九九乘法表,因從「九九八十一」起頭而得名。九九乘法表誕生在什麼時候,已經無從稽考。漢代數學家劉徽所著的《九章算術注》中曾提及「九九」起源於中國神話人物伏羲之時,由此可見「九九」的起源非常早。

在春秋時代,九九乘法表已經是比較普及的知識,而精通數學的人都很會借助九九乘法表幫助心算。《韓詩外傳》等典籍便記載了以下一個與九因歌有關的故事。

春秋時代,齊王齊桓公十分重視有才幹的人,他深知人才對於一個國家、一個國君來說是十分重要的。於是他決心廣納賢才,命人在宮廷外面燃起火炬,照得宮廷內外一片通明,一方面造成聲勢,另一方面也便於不分日夜接待前來晉見的八方英才。然而,火炬燃了整整一年,人們經過不是議論紛紛,就是前來看看熱鬧,始終沒有一人進宮求見。大臣們只是面面相覷,也不知是什麼原因。

有一天,竟然來了一個鄉下人,在宮門外請求進去見齊桓公。

守門的士兵問這個鄉下人:「你有什麼才幹求見大王?」

鄉下人回答說:「我能熟練地背誦九因歌,我希望大王接見我。」

士兵如實稟告了齊桓公。齊桓公覺得十分好笑,心想背誦九因歌算什麼才能?於是讓士兵回覆鄉下人說:「念九因歌的才能太淺陋了,怎麼可以接受國君的召見呢?回去吧!」

鄉下人不卑不亢地說:「聽人們說,這裡的火炬燃燒了整整一年,卻一直沒有人前來求見,我想,這是因為大王雄才大略名揚天下,各地賢才雖敬重大王,希望為大王出力,但深恐自己的才幹遠不及大王而不被接納,因此不敢前來求見。今天我以念九因歌的才能來求見大王,我這點本事的確算不了什麼,可是如果大王能對我以禮相待,天下人知道了大王真心求才、禮賢下士的一片誠意,何愁那些有真才實學的能人不來呢?泰山就是因為不排斥一石一土,才有它的高大;江海也因為不拒絕涓涓細流,廣納百川,才有它的深邃。古代那些聖明的君王,也要經常去向農夫樵夫請教,集思廣益,才會使自己更加英明啊!」

齊桓公聽了鄉下人的這一番話,被深深打動,認為鄉下人說得太有道理,於是馬上以隆重的禮節接見了他。這件事很快傳開了,不到一個月,各地賢才紛紛前來,絡繹不絕。齊桓公大為高興。一個統治者若真心求賢,就必須有誠意、禮賢下士,以寬廣的胸懷接納人才。

出處:http://www.newasiabooks.com/subject/maths/learn_maths/learn_maths_3a_0512_a.htm

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Chinese multiplication table

http://en.wikipedia.org/wiki/Chinese_multiplication_table

From Wikipedia, the free encyclopedia

 

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 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, 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 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: “Fu Xi invented the art of nine-nine”.
  • In Huainanzi, 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 mathematician Yang 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 dynasty mathematician Zhu Shijie‘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”

 

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