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MATH 2306
History of Mathematics
Instructor:
Dr. Alexandre Karassev
COURSE OUTLINE
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The Theorem of Pythagoras
(Ch. 1)
Greek Geometry (Ch. 2)
Greek Number Theory (Ch. 3)
Infinity in Greek Mathematics
(Ch. 4)
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Greek Mathematics
(≈ 300 BCE – 250 CE)
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Number Theory in Asia (Ch. 5) } China and India (≈ 300-1200 CE)
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Polynomial Equations (Ch. 6)
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Calculus (Ch. 9)
Infinite Series (Ch. 10)
The Number Theory Revival
(Ch. 11)
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Complex Numbers in Algebra
(Ch. 14)
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Europe (17th – 18th century CE)
Chapter 1
The Theorem of Pythagoras
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Arithmetic and Geometry
Pythagorean Triples
Rational Points on the Circle
Right-angled Triangles
Irrational Numbers
The Definition of Distance
Biographical Notes: Pythagoras
1.1 Arithmetic and Geometry
The Theorem of Pythagoras
If c is the hypotenuse of a right-angled triangle
and a, b are two other sides then
a2+b2=c2
“Let no one unversed in geometry enter here”
was written over the door of Plato’s Academy (≈ 387 BCE)
a2 a
c
b
b2
Remarks
• Converse statement: if a,b and c satisfy
a2+b2=c2 then there exists a right-angled
triangle with corresponding sides.
• One can consider a2+b2=c2 as an equation
• It has some simple solutions: (3,4,5),
(5,12,13) etc.
• Practical use - construction of right angles
• Deep relationship between arithmetic and
geometry
• Discovery of irrational numbers
1.2 Pythagorean Triples
• Definition Integer triples (a,b,c) satisfying
a2+b2=c2 are called Pythagorean triples
• Examples: (3,4,5), (5,12,13), (8,15,17) etc.
• Pythagoras: around 500 BCE
• Babylonia 1800 BCE: clay tablet
“Plimpton 322” lists integer pairs (a,c) such that
there is an integer b satisfying a2+b2=c2
• China (200 BCE -220 CE), India (500-200 BCE)
• Greeks: between Euclid (300 BCE) and
Diophantus (250 CE)
• Diophantine equation (after Diophantus,
300 CE) - polynomial equation with
integer coefficients to which integer
solutions are sought
• It was shown that there is no algorithm
for deciding which polynomial equations
have integer solutions.
General Formula
• Theorem Any Pythagorean triple can be obtained
as follows:
a = (p2-q2)r, b = 2qpr, c = (p2+q2)r
where p, q and r are arbitrary integers.
• Special case:
a = p2-q2, b = 2qp, c = p2+q2
• Proof of general formula:
Euclid’s “Elements” Book X (around 300 BCE)
1.3 Rational Points on the Circle
• Pythagorean triple (a,b,c)
• Triangle with rational sides
x = a/c, y = b/c and hypotenuse c = 1
• x2 + y2 = 1 → P (x,y) is a rational point on the
unit circle.
Y
P
1
O x
y
X
Construction of rational points on the circle
• Base point (trivial solution) Q(x,y) = (-1,0)
• Line through Q with rational slope t
y = t(x+1)
intersects the circle at a second rational point R
• As t varies we obtain allY rational points on the
circle which have the form
R
x = (1-t2) / (1+t2), y = 2t / (1+t2)
where t = p/q Q
X
-1
O
1
of Pythagoras’ Triangles
Theorem
1.4 Proof
Right-angled
a2 a
c
b
b2
1.5 Irrational Numbers
• For Pythagoreans “a number” meant integer
• The ratio between two such numbers is a rational number
• According to the Pythagoras theorem, the diagonal of the
unit square is not a rational number
• Discovery of incommensurable lengths (not measurable
as integer multiple of the same unit)
• Irrational numbers
(diagonal ) 2  12  12  2
1

diagonal  2
1
Consequences of this discovery
• According to the legend, first Pythagorean to
make the discovery public was drowned at
sea
• Split between Greek theories of number and
space
• Greek geometers developed techniques
allowing to avoid the use of irrational
numbers (theory of proportions and the
method of exhaustion)
1.6 The Definition of Distance
• Coordinates of a point on the plane: pair of
numbers (x,y)
• Development of analytic geometry (17th CE)
• Notion of distance
P
R
y
O
P
Y
Y
x
X
?
X
O
Y
Pythagoras’ theorem:
P (x2 , y2 )
x2-x1
R (x1 , y1 )
O
y2-y1
PR 2  ( x2  x1 ) 2  ( y2  y1 ) 2

X
PR  ( x2  x1 ) 2  ( y2  y1 ) 2
Alternative approach:
Definition
A point is an ordered pair (x,y)
Distance between two points R (x1 , y1 )
and P (x2 , y2 ) is defined by formula
( x2  x1 ) 2  ( y2  y1 ) 2
1.7 Biographical Notes: Pythagoras
• Born on island Samos
• Learned mathematics from Thales
(624 - 547 BCE) (Miletus)
• Croton (around 540 BCE)
• Founded a school (Pythagoreans)
– “All is number”
– strict code of conduct (secrecy,
vegetarianism, taboo on eating beans
etc.)
– explanation of musical harmony in
terms of whole-number ratios
Pythagoras (580 BCE – 497 BCE)