Math 3305 – Chapter 1 Notes
Beem F’14
1.1
Euclid’s Postulates and Axioms
SMSG Axioms for Euclidean Geometry
Spherical Geometry and Hyperbolic Geometry, a mention
1.2
Truth Tables Summary
The contrapositive
1.3
A brief review of units analysis
Find the equation of a line: TWO different ways
Distance formula in 1, 2, and 3 dimensions
Square roots – a worksheet on approximating them
1.4
The definition of a regular polyhedron and all kinds of other objects. Lots of these!
Spherical Geometry – Great Circles as lines
Topological equivalence: cube::circle, doughnut::coffee cup
Euler characteristic
1.5
Surface area of a sphere modeled for you in real time
1
Sections 1.1 and 1.2
Euclid’s Postulates and Axioms:
Postulates:
1.
2.
3.
4.
5.
Two distinct points lie on exactly one line.
One may extend a line segment indefinitely in each direction.
Given an two distinct points, one may construct a circle with one point as center and the
segment joining the second point as a radius.
Any two right angles are equal in measure to each other.
If a straight line falling on two straight lines makes the interior angles on the same side
less than two right angles, then the two straight lines, if extended indefinitely, meet on
that side on which the angles are less than the two right angles.
Euclid defined point and line and several other objects in long, circular sentences.
Common Notions
1.
Things that are equal to the same thing are also equal to each other.
2.
If equals are added to equals, the sums are equal.
3.
If equals are subtracted from equals, the remainders are equal.
4.
Things that coincide with each other are equal to each other.
5.
The whole is greater than the part.
2
The SMSG Axioms for Euclidean Geometry
A new format called an Axiomatic System with four parts:
Undefined terms
Axioms
Theorems
Definitions
Undefined Terms:
point, line, and plane
We take as our beginning point the undefined terms:
point, line, and plane.
Most people visualize a point as a tiny, tiny dot. Lines are thought of as long, seamless
concatenations of points and planes are composed of finely interwoven lines: smooth, endless
and flat.
Think of undefined terms as the basic sounds in a language – the sounds that make up our
language for the most part have no meaning in themselves but are combined to make words.
The grammar of our language and a good dictionary are what make the meaning of the sounds.
This part of language corresponds to the axioms and definitions that you will find next in the
module.
From there the facts, flights of fancy, and content-laden sentences are built – these are the
theorems and definition in an axiomatic system.
The conventions of the Cartesian plane are well suited to assisting in visualizing Euclidean
geometry. However there are some differences between a geometric approach to points on a line
and an algebraic one, as we will see in the explanation of Axiom 3.
3
Axioms:
The axioms are grouped in 5 sections. The first eight axioms deal with points, lines, planes, and
distance. Then comes convexity and separation issues – these two axioms deal with facts about
the relationships among our undefined objects on a set theoretic basis. Axioms 11 through 14
introduce angles: measuring and constructing them as well as some fundamental facts about
linear pairs. With Axiom 15, we begin to look at congruent triangles – note that this is so
fundamental a notion that it requires its own axiom. Axiom 16 introduces parallel lines. We then
look at area for polygons and congruent triangles (axioms 17 – 20) and we finish up with two
axioms about solid figures.
Common notions for this system:
Logic
Arithmetic
Numberlines
A bit of history:
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SMSG Postulates for Euclidean Geometry:
A1.
Given any two distinct points there is exactly one line that contains them.
Exactly the same as with Euclid.
A2.
The Distance Postulate:
To every pair of distinct points there corresponds a unique positive number. This number
is called the distance between the two points.
A3.
The Ruler Postulate:
The points of a line can be placed in a correspondence with the real numbers such that
A.
To every point of the line there corresponds exactly one real number.
B.
To every real number there corresponds exactly one point of the line,
and
C.
The distance between two distinct points is the absolute value of the
difference of
the corresponding real numbers.
A4.
The Ruler Placement Postulate:
Given two points P and Q of a line, the coordinate system can be chosen in such a way that
the coordinate of P is zero and the coordinate of Q is positive.
A5.
A.
B.
Every plane contains at least three non-collinear points.
Space contains at least four non-coplanar points.
A6.
If two points line in a plane, then the line containing these points lies in the same plane.
A7.
Any three points lie in at least one plane, and any three non-collinear points lie in exactly
one plane.
5
A8.
If two planes intersect, then that intersection is a line.
A9.
The Plane Separation Postulate:
Given a line and a plane containing it, the points of the plane that do not lie on the line form
two sets such that
A.
B.
each of the sets is convex, and
if P is in one set and Q is in the other, then segment PQ intersects the
line.
Convexity:
A10.
The Space Separation Postulate:
The points of space that do not line in a given plane form two sets such that
A.
B.
A11.
each of the sets is convex, and
if P is in one set and Q is in the other, then the segment PQ intersects
the plane.
The Angle Measurement Postulate:
To every angle there corresponds a real number between 0 and 180.
6
A12.
The Angle Construction Postulate:
Let AB be a ray on the edge of the half-plane H. For every r between 0 and 180 there is
exactly one ray AP with P in H such that m  PAB = r.
A13.
The Angle Addition Postulate:
If D is a point in the interior of  BAC, then
m  BAC = m  BAD + m  DAC.
A14.
The Supplement Postulate:
If two angles form a linear pair, then they are supplementary
A15.
The SAS Postulate:
Given an one-to-one correspondence between two triangles (or between a triangle and
itself). If two sides and the included angle of the first triangle are congruent to the
corresponding parts of the second triangle, then the correspondence is a congruence.
A16
The Parallel Postulate:
Through a given external point there is at most one line parallel to a given line.
Since Euclid’s days we have discovered that there are several major categories of geometries.
There are geometries that have a finite number of points and no parallel lines, or more than one
line parallel to the given line. We will look at a couple of these in Chapter 6.
There are the Big Three: Euclidean, Spherical and Hyperbolic. Each has it’s own axiom system
and each is distinctly different about parallel lines.
Euclidean: exactly one
Spherical: no parallel lines
Hyperbolic: more than one line parallel to the given line through the external point.
7
A17.
To every polygonal region there corresponds a unique positive number called its area.
A18.
If two triangles are congruent, then the triangular regions have the same area.
A19.
Suppose that the region R is the union of two regions R1 and R2. If R1 and R2 intersect at
most in a finite number of segments and points, then the area of R is the sum of the areas of
R1 and R2.
8
A20.
The area of a rectangle is the product of the length of its base and the length of its altitude.
A21.
The volume of a rectangular parallelpiped is equal to the product of the length of its
altitude and the area of its base.
A22.
Cavalieri’s Principal:
Given two solids and a plane. If for every plane that intersects the solids and is parallel to
the given plane, the two intersections determine regions that have the same area, then the
two solids have the same volume.
9
Truth Tables – Summary
Discuss “iff”
10
Fill in: Original Statement, Converse, Contrapositive, Inverse above the correct columns
See page 7 in the text.
Write out a table for the Contrapositive:
11
Together let’s make the whole truth table for
(P  Q)  ~P
What about P  ~Q
when P is true and Q is true. What is the truth value of the statement?
Now do Logic Work from the Activities Handout
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Section 1.3
We’ll do this together and then you work in groups on the next page!
Given the following points find the equation of the line in standard form and y-intercept form.
(2, 4) and (4, 7). The standard form for a line is Ax + By + C = 0. What are the unknowns in
this problem?
Do the work by solving:
Look closely at how we handle C!
2A + 4B + C = 0
4A + 7B + C = 0
Then do the work by solving for the slope and using the Slope - Point Formula:
( y  y0 )  m( x  x0 )
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In groups:
(5, 3) and (8, 9)
The new way:
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In groups:
(0, 1/3) and (1, ¼)
The new way:
15
Write a list of steps here for The New Book Way:
Now do the Solving for A and B in the Activities handout
16
Distance Formula by dimension
One dimension is the number line:
The distance from Point A with coordinate a to Point B with coordinate b: a  b
Try this with a = −10 and b = 7
Sketch it:
Try this with a = 16 and b = − 1
Sketch it:
In two dimensions you are in the Cartesian Plane. Point A will have coordinates ( x1 , y1 ) and
Point B will have coordinates ( x2 , y2 ) . The distance from Point A to Point B is
 x1  x2    y1  y2 
2
2
Do you want those subscripts reversed?
Try it with (1, 3) and (4, 7) and then try it with the points reversed….
Sketch it!
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Now in three dimensions, you are in space with 3 axes: x, y, and z.
The distance formula is
 x1  x2    y1  y2 
2
2
 ( z1  z2 )2
Let’s find (1, 1, 1) and (1, 2, 3). Now, let’s find the distance between them.
18
Square roots – approximating them easily
The graph of f ( x)  x
23
Do the
not in the book!
7
with your group:
19
Write out a list of steps for finding the square root of a number that is not a perfect square here:
Now do the Square root of 17 from the Activities handout
20
1.4
Solids
Polygon: p.19
Convexity
Polyhedron, p. 20
Prism
Right
Oblique
Pyramid
Hexahedron
Octahedron
Icosahedron
21
On page 23 is a definition of a regular polyhedron:
A polyhedron is regular if, given any first vertex and face at that vertex along with any second
vertex and face at that second vertex, there is a rotation, taking the polyhedron onto itself and
taking the first face at the first vertex to the second face at the second vertex.
Let’s try this with a cube:
vertex 2
face 2
face 1
vertex 1
22
Platonic Solids
5 of these!
Tetrahedron
Cube
Octahedron
Dodecahedron
Icosahedron
Spheres – Spherical Geometry!
GREAT CIRCLES – on your orange with your rubber bands put on some
great circles.
Parallel lines
Circles as oppose to lines
23
Euler Number – p. 25
F+V–E=2
for objects equivalent to a sphere…
Check on one sugar cube.
Why equivalent?
What’s different?
Sugar cubes: one hole
F
V
E
24
Now look at Two Holes on page 3 of the Activities booklet.
It is due as class starts the next time
1.5
The formula for the area of a circle is  r 2 .
The formula for the surface area of a sphere is 4 r 2 .
Could it really be true that one is just 4 times the other? Let’s check it out.
Use the string to get the diameter of your orange. Sketch 4 circles of this diameter on this paper.
Peel the orange and fit the peels into the 4 circles. What do you think?
25
Formula for a sphere continued
26
Spherical Geometry introductory discussion
Biangles:
Great Circles and no parallel lines
Has its own axiomatic system!
More to come
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