LESSON 1 

NATURE AND ESSENCE OF GEOMETRY

Geometry "measuring the earth“

is the branch of math that has to do with spatial relationships.

Fact:Fact: No one has been able No one has been able take a tape measure around take a tape measure around

the earth.the earth.

The The circumference circumference of the planet at of the planet at the equator is the equator is 24,901.473 miles24,901.473 miles

HOW DO WE KNOW THAT?

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

Eratosthenes

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

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.

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.

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.

A BIT OF HISTORYAmong 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.

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

Great SphinxMore than 4000 years old, the Great Sphinx of Giza is the most famous emblem of ancient Egypt.

A BIT OF HISTORYA BIT OF HISTORY

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

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.

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”.

The first important document The first important document on mathematics was written on mathematics was written

about 1700 B.C. by the about 1700 B.C. by the Egyptian priest Egyptian priest AHMESAHMES

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

Thales of Miletus Thales of Miletus (about (about 640-546 B.C) 640-546 B.C)

who was engaged both who was engaged both in commerce and public in commerce and public affairs visited Egypt and affairs visited Egypt and

brought his acquired brought his acquired knowledge of Geometry knowledge of Geometry

to Greece.to Greece.

Thales Thales Name Thales of Miletos Name Thales of Miletos

(Θαλής ο Μιλήσιος) (Θαλής ο Μιλήσιος) Birth ca. 624–625 BC Birth ca. 624–625 BC Death ca. 547–546 BC Death ca. 547–546 BC School/tradition Ionian School/tradition Ionian

Philosophy, Milesian Philosophy, Milesian school, Naturalism school, Naturalism

Main interests Ethics, Main interests Ethics, Metaphysics, Metaphysics, Mathematics, Mathematics, Astronomy Astronomy

Notable ideas Water is Notable ideas Water is the physis, Thales' the physis, Thales' theorem theorem

Influenced Pythagoras, Influenced Pythagoras, Anaximander, Anaximander, Anaximenes Anaximenes

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

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.

Euclid

Born fl. 300 BCResidence Alexandria, Egypt

Nationality GreekFields MathematicsKnown for Euclid's Elements

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.

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.

WHY DO WE NEED TO STUDY GEOMETRY?

WE HAVE TO STUDY GEOMETRY TO:

PROVIDE US WITH MANY IMPORTANT FACTS OF PRACTICAL VALUE.

WHY DO WE NEED TO STUDY GEOMETRY?

WE HAVE TO STUDY GEOMETRY TO:

PREPARE US FOR THE STUDY OF HIGHER MATHEMATICS.

WHY DO WE NEED TO STUDY GEOMETRY?

WE HAVE TO STUDY GEOMETRY TO:

UNDERSTAND AND APPRECIATE OUR NATURAL AND MAN-MADE ENVIRONMENT.

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

ROOM CARPETING

PAINTING OF A HOUSE

CONSTRUCTION OF A PICTURE FRAME

Types of Reasoning1. INDUCTIVE REASONNG

- reaching a conclusion based on previous observation.- CONCLUSIONS CAN BE TRUE BUT NOT NECESSARILY TRUE.

EXAMPLES1² = 1 1 ≤ 12² = 4 2 ≤ 43² = 9 3 ≤ 9(-1)² = 1 -1 ≤ 1

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.

A BEGINNING OBSERVER OF A BEGINNING OBSERVER OF AMERICAN BASEBALL MAY AMERICAN BASEBALL MAY INCLUDE, AFTER WATCHING INCLUDE, AFTER WATCHING SEVERAL GAMES, THAT THE SEVERAL GAMES, THAT THE IS OVER AFTER 9 INNINGS. IS OVER AFTER 9 INNINGS.

HE WILL ONLY REALIZE THAT HE WILL ONLY REALIZE THAT THIS OBSERVATION IS FALSE THIS OBSERVATION IS FALSE AFTER OBSERVING A GAME AFTER OBSERVING A GAME

WHICH IS TIED AFTER 9 WHICH IS TIED AFTER 9 INNINGS. INNINGS.

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.