Mat360 lecture 1 euclid s geometry
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MAT360 Lecture 1 Euclid’s geometry. The origins of geometry. Office hours. Th 10-11pm, Tu 1-3m. All in 4-103 Math Tower. Correction in class My questions. Homework due Tuesday Sept 18th. Chapter 1: Exercises 1,2,3,4, Mayor Exercise 1.

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MAT360 Lecture 1 Euclid’s geometry

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Mat360 lecture 1 euclid s geometry

MAT360 Lecture 1Euclid’s geometry

  • The origins of geometry


Office hours

Office hours

  • Th 10-11pm, Tu 1-3m. All in 4-103 Math Tower.

  • Correction in class

  • My questions


Homework due tuesday sept 18th

Homework due Tuesday Sept 18th

  • Chapter 1: Exercises 1,2,3,4, Mayor Exercise 1.

  • To think (but not submit): Exercises 8 and 11, and how can “straight” be defined.

  • Homework 0: Due Sept 12th.


A jump in the way of thinking geometry

A “jump” in the way of thinking geometry

  • Before Greeks: experimental

  • After Greeks: Statements should be established by deductive methods.

  • Thales (600 BC)

  • Pythagoras (500 BC)

  • Hippocrates (400 BC)

  • Plato (400 BC)

  • Euclid (300 BC)


The axiomatic method

The axiomatic method

  • A list of undefined terms.

  • A list of accepted statements (called axioms or postulates)

  • A list of rules which tell when one statement follows logically from other.

  • Definition of new words and symbols in term of the already defined or “accepted” ones.


Question

Question

  • What are the advantages of the axiomatic method?

  • What are the advantages of the empirical method?


Undefined terms

Undefined terms

  • point,

  • line,

  • lie on,

  • between,

  • congruent.


More about the undefined terms

More about the undefined terms

  • By line we will mean straight line (when we talk in “everyday” language”)


How can straight be defined

How can straight be defined?

  • Straight is that of which the middle is in front of both extremities. (Plato)

  • A straight line is a line that lies symmetrically with the points on itself. (Euclid)

  • “Carpenter’s meaning of straight”


Mistake the slides of the course are here remove the dot of the previous address

Mistake! The slides of the course are here (remove the dot of the previous address).

  • http://www.math.sunysb.edu/~moira/mat360fall07/slides/

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Administrative remarks

Administrative remarks.

  • Grader: Pedro Solorzano Office Hours.

    • Tu 4-6pm in 2-119 Math Tower

    • Th 4-6pm in MLC.

    • Contact Pedro for problems with grading, but take into account that he is not allowed to accept overdue homework.

  • From Class on: The class will take place in CHECK YOUR EMAIL and/or Blackboard.

  • Check if you receive an email from me next Thursday. If not, I contact me.

  • Discussion Board in Blackboard.


Euclid s first postulate

Euclid’s first postulate

  • For every point P and every point Q not equal to P there exists a unique line l that passes through P and Q.

  • Notation: This line will be denoted by


According with the definitions we made what is wrong in the previous postulate

According with the definitions we made, what is wrong in the previous postulate?

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More ways to express line l passes through point p

More ways to express “line l passes through point P”.


More undefined terms

More undefined terms

  • Set

  • Belonging to a set, being a member of a set.

  • We will also use some “underfined terms” from set theory (for example, “intersect”, “included”, etc) All these terms can be defined with the above terms (set, being member of a set).


Definition

Definition

  • Given two points A and B, then segment AB between A and B is the set whose members are the points A and B and all the points that lie on the line and are between A and B.

    • Notation: This segment will be denoted by AB


Second euclid s postulate

Second Euclid’s postulate

  • For every segment AB and for every segment CD there exists a unique point E such that B is between A and E and the segment CD is congruent to the segment BE.

  • Another formulation

  • Let it be granted that a segment may be produced to any length in a straight line.


Definition1

Definition

  • Give two points O and A, the set of all points P such that the segment OP is congruent to the segment OA is called a circle. The point O is the center of the circle. Each of the segments OP is called a radius of the circle.


Euclid s postulate iii

Euclid’s postulate III

  • For every point O and every point A not equal to O there exists a circle with center O and radius OA.


Definition2

Definition


Question1

Question

  • What terms are defined in the previous slide?


Definition of opposite rays

Definition of opposite rays.


Definition of angle

Definition of angle


Notation

Notation

We use the notation

for the angle with vertex A defined previously.


Questions

Questions

  • Can we use segments instead of rays in the definition of angles?

  • Is the zero angle (as you know it) included in the previous definition?

  • Are there any other angles you can think of that are not included in the above definition?


Definition3

Definition


Definition of right angle

Definition of right angle.


Euclid s postulate iv

Euclid’s Postulate IV

  • All right angles are congruent to each other.


Definition of parallel lines

Definition of parallel lines

  • Two lines are parallel if they do not intersect, i.e., if no point lies in both of them.

  • If l and m are parallel lines we write l || m


Euclidean parallel postulate equivalent formulation

Euclidean Parallel Postulate (equivalent formulation)

  • For every line l and for every point P that does not lie on l there exists a unique line m through P that is parallel to l.


Euclid s postulates modern formulation

Euclid’s postulates (modern formulation)

  • For every point P and every point Q not equal to P there exists a unique line l that passes for P and Q.

  • For every segment AB and for every segment CD there exists a unique point E such that B is between A and E and the segment CD is congruent to the segment BE.

  • For every point O and every point A not equal to O there exists a circle with center O and radius OA

  • All right angles are congruent to each other

  • For every line l and for every point P that does not lie on l there exists a unique line m through P that is parallel to l.


Euclid s postulates another formulation let the following be postulated

Euclid’s postulates (another formulation)Let the following be postulated:

  • Postulate 1. To draw a straight line from any point to any point.

  • Postulate 2. To produce a finite straight line continuously in a straight line.

  • Postulate 3. To describe a circle with any center and radius.

  • Postulate 4. That all right angles equal one another.

  • Postulate 5. That, if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.


Two worlds

Two “worlds”

  • Concrete plane

  • Abstract plane

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Exercise define

Exercise: Define

  • Midpoint M of a segment AB

  • Triangle ABC, formed by tree noncollinear points A, B, C

  • Vertices of a triangle ABC.

  • Define a side opposite to a vertex of a triangle ABC.


Exercise

EXERCISE

  • Warning about defining the altitude of a triangle.

  • Define lines l and m are perpendicular.

  • Given a segment AB. Construct the perpendicular bisector of AB.

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Exercise1

Exercise

  • Prove using the postulates that if P and Q are points in the circle OA, then the segment OP is congruent to the segment OQ.


Common notion

Common notion

  • Things which equal the same thing also equal to each other.


Exercise euclid s proposition 1

Exercise (Euclid’s proposition 1)

  • Given a segment AB. Construct an equilateral triangle with side AB.


Exercise prove the following using the postulates

Exercise. Prove the following using the postulates

  • For every line l, there exists a point lying on l

  • For every line l, there exists a point not lying on l.

  • There exists at least a line.

  • There exists at least a point.

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Second euclid s postulates are they equivalent

Second Euclid’s postulates: Are they equivalent?

  • For every segment AB and for every segment CD there exists a unique point E such that B is between A and E and the segment CD is congruent to the segment BE.

  • Any segment can be extended indefinitely in a line.


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