Download Elementary matematical ideas, theorems from the Ancient Greece

Elementary matematical ideas, theorems from the Ancient Greece
In this time the greatest mathematicians were Thales of Miletos, Pythagoras of Samos,
Aristoteles of Strageria and Euclides of Alexandria in chronological sequence. They also were
philosophers except for Euclides.
Thales is the first well-known greek matematician lived from BC 624 to BC 546. He founded
the school of Miletos and taught there. He has reached famous results in geometry. He said
that the sum of angles in the triangle equals 180 degrees, declared the Thales’s theorem for
triangles with right angle and the theorem of paralell lines in the circle. Foreign people
(italians, frenches, spanishes, russians) call the first one the large Thales’s theorem and the
second one the little Thales’s theorem.
Pythagoras, who lived from BC 582 to BC 496 was a presocratist philosopher. He travelled to
to Egypt in order to learn that culture. After wards he moved to Kroton, Italy, where he
founded the phytagorian school. In math, he declared the Phytagoras’s theorem for triangles
with right angle, saying that the square of the longest sideline equals the sum of square of the
second sideline and square of the third sideline.
Aristoteles has lived from BC 384 until BC 322. He was born in Strageria and he was a
student of Platon. Finally he was the master of Alexander the Great. Aristoteles was a wellknown philosopher, however he cared for logic too. Also he showed the irrationality of
squareroot of two.
Aristoteles and squareroot of two
a
2
b
2
a
2 2
b
2
a  2b 2  a  2k
2b 2  4k 2
It’s impossible, contradiction.□
Euclides lived in the third century BC. He studied at Platon’s Academy, later he founded a
school in Alexandria. He was a simple matematician, and published his mathematical ideas in
his book, the Elements, moreover founded the classic geometry. Furthermore he has analysed
and solved some problems in number theory. For example he verificated the infinitude of
prime numbers firstly and found out the Euclides’s algorithm. The Euclides’s algorithm helps
us to find the gcd, greatest common divisor of two numbers.
Euclides and the prime numbers
pn-s are prime numbers, so N = p1p2p3…pn, but N + 1 is a prime again.□
Euclides’s algorithm
A  Bq1  r1
B  r1 q 2  r2
r1  r2 q3  r3
r2  r3 q 4  r4

rn  2  rn 1 q n  rn
rn 1  rn q n 1  0
rn is the gcd of A and B.□