Euclidean Algorithm By Ryan Winders A Little on
Euclidean Algorithm By: Ryan Winders
A Little on Euclid l Lived from 323 – 285 BC l He taught in Alexandria, Egypt
Interesting Facts l 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. l It is believed that Euclid of Alexandria never existed!
The Elements l 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) l This led to Euclidean Geometry l
The Elements l Divided into 13 books l 1 -6 deal with Plane Geometry l 7 -10 with Number Theory l Book 7 contains the Euclidean Algorithm l 11 -13 with 3 -D Geometry l He also wrote 9 other books – 5 of which are currently lost.
Euclidean Algorithm l The Euclidean Algorithm is a method of finding the GCD of two numbers. l It makes two observations 1. 2. If b/a, then GCD (a, b) = b If a = bt + r, for integers t and r, then GCD (a, b) = GCD (b, r)
Euclidean Algorithm l Let a = 2322, b=654 l 2322 = 654*3 + 360 654 = 360*1 + 294 360 = 294*1 + 66 294 = 66*4 + 30 66 = 30*2 + 6 30 = 6*5 l l l Therefore, gcd(2322, 654) = 6. l gcd(2322, 654) = gcd(654, 360) l l gcd(654, 360) = gcd(360, 294) = gcd(294, 66) = gcd(66, 30) = gcd(30, 6) l gcd(30, 6) = 6 l l
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