the euclidean algorithm...al-kitab al-mukhtas¯ .ar f¯ı h¯ıs ab al-¯ gabr wa’l-muq˘ abala¯...
TRANSCRIPT
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The Euclidean Algorithm
Paul Tokorcheck
Department of MathematicsIowa State University
September 26, 2014
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The Elements
China
India
Islam
Europe
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A map of Alexandria, Egypt, as it appeared shortly afterEuclid and during the expansion of the Roman Empire.
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Euclid’s Elements, Book VII, Propositions 1 and 2.This copy of The Elements is from Constantinople, writtenin 888CE.
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Detail. Here you can see some familiar drawings.
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The Silk Road between Rome and the Han Dynasty(206 BCE - 220 CE)
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The Suanshu Shu.
This copy was foundwithin a tomb inZhangjiashan, Hubeiprovince, China. Thetomb was sealed in 186BCE and opened byarcheologists in 1983.
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Zhou Bi Suan Jing
(The Arithmetical Classicof the Gnomon and theCircular Paths of Heaven)
Jiuzhang Suanshu
(The Nine Chapters on theMathematical Art)
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Liu Hui, on a modernChinese postagestamp. He lived inthe 3rd Century CE,and wrote hiscommentaries on theZhou Bi Suan Jingand theJiuzhang Suanshuaround 263 CE.
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The mathematicians of China knew how to find the GCDof two numbers, which they called the Deng Shu. Here istheir method:
17163→ 108
63→ 45
63→ 45
18→ 27
18→ 9
18→ 9
9
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Asia in the early first millenium.
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Aryabhata was born near Bihar, India in 476 CE, andwrote his Aryabhat.ıya in 499 CE.
Bhaskara I later transcribed it and wrote his owncommentary in 629 CE.
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Kut.t.aka:
“[A quantity when divided] by twelve has a remainderwhich is five, and furthermore, it is seen by me [having] aremainder which is seven, when divided by thirty-one.What should one such quantity be?” (Bhaskara I in hiscommentary, 629 CE.)
N = 12y + 5 = 31x + 7
12y − 31x = 2
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Kut.t.aka:
“[A quantity when divided] by twelve has a remainderwhich is five, and furthermore, it is seen by me [having] aremainder which is seven, when divided by thirty-one.What should one such quantity be?” (Bhaskara I in hiscommentary, 629 CE.)
N
= 12y + 5 = 31x + 7
12y − 31x = 2
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Kut.t.aka:
“[A quantity when divided] by twelve has a remainderwhich is five, and furthermore, it is seen by me [having] aremainder which is seven, when divided by thirty-one.What should one such quantity be?” (Bhaskara I in hiscommentary, 629 CE.)
N = 12y + 5
= 31x + 7
12y − 31x = 2
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Kut.t.aka:
“[A quantity when divided] by twelve has a remainderwhich is five, and furthermore, it is seen by me [having] aremainder which is seven, when divided by thirty-one.What should one such quantity be?” (Bhaskara I in hiscommentary, 629 CE.)
N = 12y + 5 = 31x + 7
12y − 31x = 2
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Kut.t.aka:
“[A quantity when divided] by twelve has a remainderwhich is five, and furthermore, it is seen by me [having] aremainder which is seven, when divided by thirty-one.What should one such quantity be?” (Bhaskara I in hiscommentary, 629 CE.)
N = 12y + 5 = 31x + 7
12y − 31x = 2
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The Pulverizer: we attempt to find an integer solution to
12y − 31x = 2
by repeated division of polynomials.
y =31x + 2
12= 2x + w
x =12w − 2
7= 1w + v
w =7v + 2
5= 1v + u
v =5u − 2
2.
Choosing u = 2 and v = 4 gives the solutioncorrespoding to N = 317.
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“Whoever, when a given remainder of the Sun inrevolutions and so on is on a Monday or a Thursday or aWednesday, tells the zodiacal sign [and so on of the Sun’slongitude], he knows the pulverizer.”
From Brahmasphut.asiddhanta by Brahmagupta, 628 CE.
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Aryabhata - 1975 Bhaskara I - 1979
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Islamic expansion before the Abbasid Caliphate.
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Abu ‘Abdallah Muh.ammad ibnMusa al-Khwarizmı, on aSoviet-era postage stamp.c.780 - c.850 CE
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al-Khwarizmı was author ofAl-Kitab al-mukhtas. ar fıhısab al-gabr wa’l-muqabala.
(The Compendious Book onCalculation by Completionand Balancing.)
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Hulagu Khan, grandson of Genghis, with his wife.
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The House of Wisdom was destroyed during the Siege ofBaghdad, in 1258 CE.
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Europe at the start of the 13th Century.
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A translation ofal-Khwarizmı’s workinto Latin.
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Saint Thomas Aquinas.
c.1225 - 1274 CE.
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Leonardo da Vinci.
1452 - 1519 CE.
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Galileo Galilei.
1564 - 1642 CE.
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Leonhard Euler.
1707 - 1783 CE.
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Etienne Bezout.
1730 - 1783 CE.