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Euclidean Algorithm

Euclidean Algorithm. By: Ryan Winders. A Little on Euclid. Lived from 323 – 285 BC He taught in Alexandria, Egypt. Interesting Facts.

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Euclidean Algorithm

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  1. Euclidean Algorithm By: Ryan Winders

  2. A Little on Euclid • Lived from 323 – 285 BC • He taught in Alexandria, Egypt

  3. Interesting Facts • Euclid worked on a team of mathematicians that all contributed in writing books. The group continued writing these books under Euclid’s name even after his death. • It is believed that Euclid of Alexandria never existed!

  4. The Elements • Euclid’s most famous book • Begins with 5 postulates. The first three deal with construction, the fourth with right angles, and the fifth states that only one line can be drawn through a point parallel to a given line. (Parallel Postulate) • This led to Euclidean Geometry

  5. The Elements • Divided into 13 books • 1-6 deal with Plane Geometry • 7-10 with Number Theory • Book 7 contains the Euclidean Algorithm • 11-13 with 3-D Geometry • He also wrote 9 other books – 5 of which are currently lost.

  6. Euclidean Algorithm • The Euclidean Algorithm is a method of finding the GCD of two numbers. • It makes two observations • If b/a, then GCD (a,b) = b • If a = bt + r, for integers t and r, then GCD (a,b) = GCD (b,r)

  7. Let a = 2322, b=654 2322 = 654*3 + 360   654 = 360*1 + 294   360 = 294*1 + 66  294 = 66*4 + 30   66 = 30*2 + 6  30 = 6*5   Therefore, gcd(2322,654) = 6. gcd(2322, 654) = gcd(654, 360) gcd(654, 360) = gcd(360, 294) gcd(360, 294) = gcd(294, 66) gcd(294, 66) = gcd(66, 30) gcd(66, 30) = gcd(30, 6) gcd(30, 6) = 6 Euclidean Algorithm

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