Transcript Math 251 Section 001

Math 251 Towson University
About the Course
 What is geometry?
History of Geometry – Early Civs
 One of the earliest branches of mathematics
 Ancient Egyptians, Babylonians, and Indians used some
form of geometry as early as 3000 BC (5000 years ago!)
 How do you think they might have used geometry?
History of Geometry
 Ancient cultures used geometry for:
 Measuring land and distances
 Measuring angles for building
structures and planning cities
 Drawing circles for wheels and artistic designs
 Use of geometric shapes for altar designs
 All three civilizations discovered the Pythagorean
Theorem at least 1000 years before Pythagoras himself
 Why is it called the Pythagorean Theorem then???
Greek Geometry
 Greek mathematicians, starting with Thales (“Thay-lees”) of
Miletus, proposed that geometric statements should be
proved by deductive logic rather than trial and error.
 What is the difference between proving a statement by a
deductive proof rather than a series of examples? Why might
someone prefer a deductive proof?
 No matter how many examples you provide, you can never be
sure that an example exists that disproves your statement
 Even more important, proofs often tell us “why” a statement is
true
Greek Geometry – Pythagoras
 Thales’ student, Pythagoras, continued and expanded on the
method of deductive proofs. Pythagoras and his disciples
used these methods to prove many geometric theorems.
 The most famous -- the Pythagorean Theorem:
 The sum of the squares of the two sides of a right triangle equals
the square of its hypotenuse
c
a
b
a2 + b2 = c2
Greek Geometry – Pythagoras
 Pythagoras and his disciples also discovered a number of
other geometric theorems and mathematical ideas:
 Area of a circle
 Square numbers and square roots
 Irrational numbers
 Pythagoras believed that everything was related to
mathematics and that numbers were the ultimate reality and,
through mathematics, everything could be predicted and
measured.
 They developed a curriculum for students which divided
mathematics into four subjects: Arithmetic, Geometry,
Astronomy, and Music
Greek Geometry – The Liberal Arts
Quadrivium
• Arithmetic
• Geometry
• Music
• Astronomy
Trivium
• Grammar
• Rhetoric
• Logic
How is this the same as / different from the liberal arts at a
university today?
Greek Geometry – Euclid
 While Euclid did not discover many new theorems, he
contributed greatly to the advancement of geometry by
collecting known theorems and presenting them in a single,
logically coherent book – possibly the first textbook?
 Euclid’s goal was to start with a few axioms and use these to
prove other geometric statements, thus creating a logical
system.
 What is an axiom??? Why do we even need them?
 An axiom is “a statement that is assumed to be true without
presenting any reasoning”
 Euclid’s goal was for his axioms to be self-evident
 These then serve as a starting point for proving other
statements.
Euclid’s Axioms
 First Axiom: For any two points, there is a unique line that
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

can be drawn passing through them.
Second Axiom: Any line segment can be extended as far as
desired.
Third Axiom: For any two points, a circle can be drawn with
one point as its center and the other point lying on the circle.
Fourth Axiom: All right angles are congruent to one another.
Fifth Axiom: For every line, and for every point that does not
lie on that line, there is a unique line (only one!) through the
point and parallel to the line.
Euclid’s Axioms
 First Axiom: For any two points, there is a unique line that
can be drawn passing through them.
A
B
A
B
Euclid’s Axioms
 Second Axiom: Any line segment can be extended as far as
desired.
A
B
A
B
Euclid’s Axioms
 Third Axiom: For any two points, a circle can be drawn with
one point as its center and the other point lying on the circle.
A
B
A
B
Euclid’s Axioms
 Fourth Axiom: All right angles are congruent to one
another.
F
C
A
B
D
 Angle CAB is congruent to Angle FDE
E
Euclid’s Axioms
 Fifth Axiom: For every line, and for every point that does not
lie on that line, there is a unique line (only one!) through the
point and parallel to the line.
C
A
B
C
A
B
Euclid’s Axioms
 Does the fifth axiom seem different from the first four?
 Euclid himself put off using this axiom for as long as possible,
proving his first 28 propositions without using it.
 For over 2000 years, mathematicians attempted to deal with
this axiom by proving it based on the first four axioms, or
replacing it with a more self-evident one.
 In the 1800s, mathematicians discovered new systems of
geometry that could be created by using a different fifth
axiom (“Non-Euclidean Geometry”). We will talk more
about this later in the course.