The Euclidean Algorithm

Paul Tokorcheck

Department of MathematicsIowa State University

September 26, 2014

The Elements

China

India

Islam

Europe

A map of Alexandria, Egypt, as it appeared shortly afterEuclid and during the expansion of the Roman Empire.

Euclid’s Elements, Book VII, Propositions 1 and 2.This copy of The Elements is from Constantinople, writtenin 888CE.

Detail. Here you can see some familiar drawings.

The Silk Road between Rome and the Han Dynasty(206 BCE - 220 CE)

The Suanshu Shu.

This copy was foundwithin a tomb inZhangjiashan, Hubeiprovince, China. Thetomb was sealed in 186BCE and opened byarcheologists in 1983.

Zhou Bi Suan Jing

(The Arithmetical Classicof the Gnomon and theCircular Paths of Heaven)

Jiuzhang Suanshu

(The Nine Chapters on theMathematical Art)

Liu Hui, on a modernChinese postagestamp. He lived inthe 3rd Century CE,and wrote hiscommentaries on theZhou Bi Suan Jingand theJiuzhang Suanshuaround 263 CE.

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

Asia in the early first millenium.

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.

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

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

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

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

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

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.

“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.

Aryabhata - 1975 Bhaskara I - 1979

Islamic expansion before the Abbasid Caliphate.

Abu ‘Abdallah Muh.ammad ibnMusa al-Khwarizmı, on aSoviet-era postage stamp.c.780 - c.850 CE

al-Khwarizmı was author ofAl-Kitab al-mukhtas. ar fıhısab al-gabr wa’l-muqabala.

(The Compendious Book onCalculation by Completionand Balancing.)

Hulagu Khan, grandson of Genghis, with his wife.

The House of Wisdom was destroyed during the Siege ofBaghdad, in 1258 CE.

Europe at the start of the 13th Century.

A translation ofal-Khwarizmı’s workinto Latin.

Saint Thomas Aquinas.

c.1225 - 1274 CE.

Leonardo da Vinci.

1452 - 1519 CE.

Galileo Galilei.

1564 - 1642 CE.

Leonhard Euler.

1707 - 1783 CE.

Etienne Bezout.

1730 - 1783 CE.