Transcript Euclid - Angelfire

Euclid was a Greek mathematician who is best known for his work on
the Elements. He is the most prominent mathematician of antiquity best
known for his treatise on mathematics in The Elements. The long lasting
nature of The Elements must make Euclid the leading mathematics
teacher of all time. However little is known of Euclid's life except that he
taught at Alexandria in Egypt
Lived from 330?-275? B.C.
• Euclid's fame comes from his writings, the best known of
which are his treatise entitled, The Elements. He wrote
many other works including Data, On Division,
Phaenomena, Optics and the lost books Conics and
Prisms.
Euclid's Elements was used as a text for 2,000 years, and even today a modified
version of its first few books forms the basis of high school instruction in plane
geometry
Book
1
2
3
4
5
6
7-10
11
12
13
Contents
Triangles
Rectangles
Circles
Polygons
Proportion
Similarity
Number Theory
Solid Geometry
Pyramids
Platonic Solids
In The Elements Euclid produced five axioms, a.k.a. postulates. The axioms that Euclid
explicitly stated in The Elements were five in number, of which the fifth (parallel
axiom) is the most famous due to it's proof being highly controversial until the 19th
century. Below are the five axioms:
Euclid's Postulates
1. A straight Line Segment can be drawn joining any two points.
2. Any straight Line Segment can be extended indefinitely in a straight Line.
3. Given any straight Line Segment, a Circle can be drawn having the segment as
Radius and one endpoint as center.
4. All Right Angles are congruent.
5. If two lines are drawn which intersect a third in such a way that the sum of the
inner angles on one side is less than two Right Angles, then the two lines inevitably
must intersect each other on that side if extended far enough. This postulate is
equivalent to what is known as the Parallel Postulate.
Proof
Suppose there was a largest prime number; call it N. Then there are only finitely many prime numbers,
because each has to be between 1 and N. Let's call those prime numbers a, b, c, ..., N. Then consider this
number:
M = a * b * c * ... * N + 1
Is this new number M a prime number? We could check for divisibility:
M is not divisible by a, because M / a = b * c * ... * N + 1 / a
M is not divisible by b, because M / b = a * c * ... * N + 1 / b
M is not divisible by c, because M / c = a * b * ... * N + 1 / c
.....
Hence, M is not divisible by a, b, c, ..., N. Since these are all possible prime numbers, M is not divisible by
any prime number, and therefore M is not divisible by any number. That means that M is also a prime
number. But clearly M > N, which is impossible, because N was supposed to be the largest possible prime
number. Therefore, our assumption is wrong, and thus there is no largest prime number.
www-gap.dcs.st-and.ac.uk/~history/ Mathematicians/Euclid.html
http://www-groups.dcs.st-and.ac.uk/~history/PictDisplay/Euclid.html
http://www.shu.edu/html/teaching/math/reals/history/euclid.html
http://www.shu.edu/html/teaching/math/reals/logic/proofs/euclidth.html
http://users.rcn.com/bobmer.javanet/euklid.htm
http://archive.ncsa.uiuc.edu/SDG/Experimental/vatican.exhibit/exhibit/dmathematics/Greek_math.html