LESSON 1

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LESSON 1 NATURE AND ESSENCE OF GEOMETRY
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Geometry

• "measuring the earth
• is the branch of math that has to do with
  spatial relationships.

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Fact No one has been able take a tape measure
around the earth.

• The circumference of the planet at the equator
  is 24,901.473 miles

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HOW DO WE KNOW THAT?

• The first known case of calculating the distance
  around the earth was done by Eratosthenes around
  240 BCE.

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Eratosthenes

• was a Greek mathematician, poet, athlete,
  geographer and astronomer.

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Eratosthenes

• His contemporaries nicknamed him "beta" (Greek
  for "number two") because he supposedly proved
  himself to be the second in the ancient
  Mediterranean region in many fields.

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Eratosthenes

• He is noted for devising a system of latitude and
  longitude, and for being the first known person
  to have calculated the circumference of the Earth.

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Eratosthenes

• He also created a map of the world based on the
  available geographical knowledge of the era.
  Eratosthenes was also the founder of scientific
  chronology he endeavored to fix the dates of the
  chief literary and political events from the
  conquest of Troy.

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A BIT OF HISTORY

• Among the mathematical sciences, Geometry is the
  earliest and historically most influential. The
  Babylonians and Egyptians knew many geometric
  facts more than a thousand years before Christ.

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• Their geometrical knowledge was PRACTICAL in
  nature as exemplified by their temples and
  pyramids that needed very accurate plans and
  models

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Great Sphinx More than 4000 years old, the
Great Sphinx of Giza is the most famous emblem of
ancient Egypt.
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A BIT OF HISTORY

• Geometry first became associated with land
  measurement in Egypt. The Egyptians were obliged
  to invent it in order to restore the landmarks
  that were destroyed by the periodic inundation of
  Nile River.

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Herodotus, the Greek historian, gave the
following account of the origin of Geometry.

• King Sesostres divided the land among all
  Egyptians so as to give each a four-sided piece
  of equal size and draw from each his revenue by
  imposing a tax to be levied annually.

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• Whenever the river tore part of the land due to
  inundation, the citizen concerned had notify the
  king. The king then sent his overseers to measure
  how much the land was reduced so that the citizen
  would pay the tax imposed proportionally to the
  land lost.

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The first important document on mathematics was
written about 1700 B.C. by the Egyptian priest
AHMES

• His work called Directions for Knowing All Dark
  Things, consisted of a collection of problems in
  Geometry and Arithmetic. In the problems
  concerning the area of a circular field, Ahmes
  used the value of pi ( ?) equal to 3.1604. at
  that time the value of pi was taken to equal 3.

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Thales of Miletus (about 640-546 B.C) who was
engaged both in commerce and public affairs
visited Egypt and brought his acquired knowledge
of Geometry to Greece.
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• Thales
• Name Thales of Miletos (Ta??? ? ????s???)
• Birth ca. 624625 BC
• Death ca. 547546 BC
• School/tradition Ionian Philosophy, Milesian
  school, Naturalism
• Main interests Ethics, Metaphysics, Mathematics,
  Astronomy
• Notable ideas Water is the physis, Thales'
  theorem
• Influenced Pythagoras, Anaximander, Anaximenes

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• Pythagoras was one of his pupils who through his
  advice went to Egypt and gained extensive
  experience. Having become more famous than his
  teacher, Thales, Pythagoras laid the foundation
  of Geometry.

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Our textbook is based on Euclidean (or
elementary) geometry. "Euclidean" (or
"elementary") refers to a book written over 2,000
years ago called "The Elements" by a man named
Euclid.
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• Euclid Born fl. 300 BCResidence Alexandria,
  Egypt
• Nationality Greek
• Fields Mathematics
• Known for Euclid's Elements

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In the book, Euclid started with some basic
concepts. He built upon those concepts to create
more and more concepts. His structure and method
influence the way that geometry is taught today.
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WHY DO WE NEED TO STUDY GEOMETRY?

• WE HAVE TO STUDY GEOMETRY TO
• ENHANCE OUR ANALYTICAL SKILLS TO ENABLE US TO
  EXPRESS OUR THOUGHTS ACCURATELY AND TRAIN US TO
  REASON LOGICALLY.

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WHY DO WE NEED TO STUDY GEOMETRY?

• WE HAVE TO STUDY GEOMETRY TO
• PROVIDE US WITH MANY IMPORTANT FACTS OF PRACTICAL
  VALUE.

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WHY DO WE NEED TO STUDY GEOMETRY?

• WE HAVE TO STUDY GEOMETRY TO
• PREPARE US FOR THE STUDY OF HIGHER MATHEMATICS.

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WHY DO WE NEED TO STUDY GEOMETRY?

• WE HAVE TO STUDY GEOMETRY TO
• UNDERSTAND AND APPRECIATE OUR NATURAL AND
  MAN-MADE ENVIRONMENT.

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WHY DO WE NEED TO STUDY GEOMETRY?

• REMEMBER
• A KNOWLEDGE OF BASIC GEOMETRY IS USEFUL TO
  EVERYDAY LIFE, PARTICULARLY IN MEASURING AND
  DESIGNING SUCH ITEMS AS

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ROOM CARPETING
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PAINTING OF A HOUSE
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CONSTRUCTION OF A PICTURE FRAME
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Types of Reasoning1. INDUCTIVE REASONNG-
reaching a conclusion based on previous
observation.- CONCLUSIONS CAN BE TRUE BUT NOT
NECESSARILY TRUE.
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EXAMPLES

• 1² 1 1 1
• 2² 4 2 4
• 3² 9 3 9
• (-1)² 1 -1 1

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BASED ON THE EXAMPLES, WE NOTICED A PATTERN FROM
WHICH WE DRAW A CONCLUSIONS.

• THROUGH INDUCTIVE REASONING, IT MAY BE CONCLUDED
  THAT WHENEVER A NUMBER IS SQUARED, THE RESULT IS
  A NUMBER WHICH IS GREATER THAN OR EQUAL TO THE
  ORIGINAL NUMBER.

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A BEGINNING OBSERVER OF AMERICAN BASEBALL MAY
INCLUDE, AFTER WATCHING SEVERAL GAMES, THAT THE
IS OVER AFTER 9 INNINGS. HE WILL ONLY REALIZE
THAT THIS OBSERVATION IS FALSE AFTER OBSERVING A
GAME WHICH IS TIED AFTER 9 INNINGS.
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INDUCTIVE REASONING IS USEFUL BUT NOT
CERTAIN.THERE WILL ALWAYS BE A CHANCE THAT THERE
IS AN OBSERVATION THAT WILL SHOW THE REASONING TO
BE FALSE. ONLY ONE OBSERVATION IS NEEDED TO PROVE
THE CONCLUSION TO BE FALSE.