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

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Presentation on theme: "Euclidean Algorithm By: Ryan Winders. A Little on Euclid Lived from 323 – 285 BC Lived from 323 – 285 BC He taught in Alexandria, Egypt He taught in Alexandria,"— Presentation transcript:

1 Euclidean Algorithm By: Ryan Winders

2 A Little on Euclid Lived from 323 – 285 BC Lived from 323 – 285 BC He taught in Alexandria, Egypt 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. 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! It is believed that Euclid of Alexandria never existed!

4 The Elements Euclid’s most famous book 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) 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 This led to Euclidean Geometry

5 The Elements Divided into 13 books Divided into 13 books 1-6 deal with Plane Geometry 1-6 deal with Plane Geometry 7-10 with Number Theory 7-10 with Number Theory Book 7 contains the Euclidean Algorithm Book 7 contains the Euclidean Algorithm 11-13 with 3-D Geometry 11-13 with 3-D Geometry He also wrote 9 other books – 5 of which are currently lost. 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. The Euclidean Algorithm is a method of finding the GCD of two numbers. It makes two observations It makes two observations 1. If b/a, then GCD (a,b) = b 2. If a = bt + r, for integers t and r, then GCD (a,b) = GCD (b,r)

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

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