Geometry/Chapter 1 - HS

Section 1.1 - Introduction

Geometry comes from two words: geo which means Earth and metry meaning measure. Therefore, geometry means "measuring the Earth". This branch of mathematics deals with the understanding of point and lines and their combinations. In other words, geometry is a type of maths used to measure things that are impossible to measure with devices. For example, no one has been able take a tape measure around the Earth, yet we are pretty confident that the circumference of the planet at the equator is 40,075.036 kilometers (24,901.473 mi). How do we know that? The first known case of calculating the distance around the Earth was done by Eratosthenes around 240 BCE. What tools do you think current scientists might use to measure the size of planets? The answer is geometry.

However, geometry is more than measuring the size of objects. If you were to ask someone who had taken geometry in high school what it is that s/he remembers, the answer would most likely be proofs. If you were to ask him/her what it is that s/he liked the least, the answer would probably be proofs. A study of Geometry does not have to include proofs. Proofs are not unique to Geometry. Proofs could have been done in algebra or delayed until calculus. The reason that High School Geometry almost always spends a lot of time with proofs is that the first great Geometry textbook, "The Elements," was written exclusively with proofs.

This textbook is based on Euclidean geometry. Euclidean refers to a book written over two thousand years ago called The Elements by a man named Euclid. In this book Euclid provides methods using just a compass, ruler and a protractor to prove geometrical statements. His method influences the way geometry is taught today. Euclid's book was part of the math curriculum until the beginning of the twentieth century.

Section 1.2 - Reasoning

There are two general ways of reaching conclusions: inductive reasoning and deductive reasoning.

Inductive Reasoning

Inductive reasoning is the method we use more often, reaching a conclusion based on previous observations. For example, if I notice that the Sun rises in the east every day, then through inductive reasoning, I could conclude that the Sun will rise from the east tomorrow. In math, we may notice a pattern from which we draw conclusions. Look at the following pattern:

$1^{2}=1$ ,	$1\leq 1$
$2^{2}=4$ ,	$2\leq 4$
$3^{2}=9$ ,	$3\leq 9$
$(-1)^{2}=1$ ,	$-1\leq 1$
$(-2)^{2}=4$ ,	$-2\leq 4$

Through inductive reasoning, we may conclude that whenever a number is squared, the result is a number which is greater than or equal to the original number. Based on the result of squaring whole numbers, this appears to be true. Inductive logic is not certain, though. There are some numbers for which our conclusion does not hold:

\left({\frac {1}{2}}\right)^{2}={\frac {1}{4}}

{\frac {1}{2}}>{\frac {1}{4}}

The same can be applied to problems outside of Math. A foreign observer of American baseball may conclude after watching several games that the game consists of nine innings. He will only realize that this observation is false after observing a game that is tied after nine 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. This theory was first formulated by Kuhn in his study of the laws of induction. The question being one of whether a paradigm has the correct generalizations to proceed to a conclusion that is factual.

Much of the reasoning in geometry is like this, consisting of three simple stages (see example A):

1. Look for commonalities: A pattern.
2. Make a conjecture: An unproven statement that you will prove.
3. Prove/disprove: The conjecture.

Deductive Reasoning

Deductive reasoning is reaching a conclusion by combining known truths to create a new truth. Unlike inductive reasoning, deductive reasoning is certain, provided that the normal rules of logic are used to conclude such truths. In order to use deductive reasoning there must be a starting point, normally called the axioms or postulates of the theory. For example, an axiom in geometry asserts that given two points, there is only one line that contains both points. Observe that while this is an axiom, it can be used to deduce that two different lines that are not parallel will intersect at only one point.

Not only axioms can be used to deduce new truths. Other knowledge deduced from the axioms using the rules of logic can be used to validate new truths. For example, we can conclude that if three points A, B and C are not in the same line, the lines determined by two of them can only meet at A, B and C (since we already know that two lines can only intersect at one point, all that is necessary to prove is that the lines determined by two of the three points are different, and that is immediate since the given points do not belong to any one line).

Vocabulary

conjecture: A statement in need of proof.

Example A: Making a Conjecture

The sum of the first x odd positive integers can be expressed as $n^{2}$

Solution - Inductive:
sum of the first 1 odd positive integers: 1 = 1 = 1²
sum of the first 2 odd positive integers: 1 + 3 = 4 = 2²
sum of the first 3 odd positive integers: 1 + 3 + 5 = 9 = 3²
sum of the first 4 odd positive integers: 1 + 3 + 5 + 7 = 16 = 4²
sum of the first 5 odd positive integers: 1 + 3 + 5 + 7 + 9 = 25 = 5²
sum of the first 6 odd positive integers: 1 + 3 + 5 + 7 + 9 + 11 = 36 = 6²

The sum of the first x odd positive integers is x².

Solution - Deductive

Prove that the sum of the first n odd numbers: $\sum _{x=1}^{n}(2x-1)=n^{2}$ .
	$\sum _{x=1}^{n}(2x-1)$
$=$	$2\sum _{x=1}^{n}x-\sum _{x=1}^{n}1$
$=$	$2{\frac {n(n+1)}{2}}-n$
$=$	$n^{2}+n-n$
$=$	$n^{2}$

Note that the deductive example is an equation that is simplified until the left terms are reduced to equality with the right term. This is the proof and the root of the word equation acts as a constant reminder of the Euclidean Common Notions on Equality:

Things equal to the same thing are also equal to one another.
And if equal things are added to equal things, the wholes are equal.†
And if equal things are subtracted from equal things, the remainders are equal.†
And things coinciding with one another are equal to one another.
And the whole [is] greater than the part.

† As an obvious extension of C.N.s 2 & 3 — if equal things are added or subtracted from the two sides of an inequality, the inequality remains.

The Σ is the summation sign that tells the reader that the first number to be used is at the bottom of the summation sign. The lowercase "n" when used in maths refers to a natural positive number whose value is not stated. The uppercase $\mathbb {N}$ is reserved for showing sets of natural numbers e.g. $\mathbb {N} =\{1,2,3,\ldots \}$ .

The brackets on the right of the summation sign is the expression for odd numbers:

(2x-1)

You can confirm this by inputting the starting value for $x$ as given under the summation sign and then complete the calculation. Keep increasing the value of $x$ by one.

Exercises

Below are simple statements of logic that have a major premise that is then further elucidated by a minor premise with the conclusion either being affirmative or negative. The answer must contain the subject of the premise.

1) All vegetables are good for you. Broccoli is a vegetable. Therefore, broccoli is good for you. This is an example of what type of reasoning?

2) Broccoli is a vegetable. Broccoli is green. Therefore, all vegetables are green. Why is this conclusion invalid?

3) Berries are sweet. Berries are fruits. Therefore, all fruits are sweet. Why is this conclusion invalid?

4) If all of x is positive, and y is part of x, then y is positive. What type of reasoning is this?

Examples

 Group A     
    (D is negative)  
    (B is negative)            
    (E is positive)  
    (C is positive)                  
    (F is positive)

Just because D is negative, and is part of group A, doesn't mean that all of group A is negative.

 Group Z  [ = 5 ]
    (Y is 5)
    (X is 5)
    (W is 5)
    (V is 5)
    (U is 5)

Since all of group Z equals 5, you can say that Y=W, W=U, U=X, X=V, and that V=Y, etc.

Because you know that all of group Z equals the same thing, you can say that because T = 5, it is part of group Z.

Section 1.3 - Undefined terms

In Geometry, there are three undefined terms: points, lines, and planes. Although most terms in geometry are defined based on previously defined terms, it is impossible to define every geometric term this way. The first geometric term cannot be defined based on previously defined terms.

Although we cannot formally define these three terms, we can informally describe them. We also use these terms to help us write definitions of other terms such as segment or ray. There is no axiom that says that lines are drawn straight. What this means is that the definition of line depends on the theory that you are studying. So, in Hyperbolic Geometry a line does not look like a line in Euclidean Geometry, since they are defined differently.

In Euclidean Geometry, a point is thought of as having no breadth, width or height. Now imagine taking a very sharp pencil, and making a dot on a piece of paper. Now imagine looking at it under a magnifying glass, the dot would be big, and we would be able to see it has a height and a breadth. A point is not a dot, because a point would have neither height nor breadth, but we can imagine that in the very middle of the dot is a point. This was used by Hume in his A Treatise Of Human Nature to prove that postulates and axioms were not innate but a human construct of understanding and therefore a posteriori.

With the non-definitions out of the way, let's look at how these things work. A point is usually represented by a dot on a piece of paper. A point is useful because it tells us exactly where something is, and we can then build observations, conjectures, and rules from that information. For example, we can say that two points determine a line. What this means is that once you know where two points are, you know where the line that contains both of the points must be. Notice that if you only know where one point is, there are an infinite number of lines that can contain that one point, and if you know where three points are, there is a pretty good chance that there isn't any single line that would contain all three points.

In Euclidean Geometry, a line is thought of as having length but neither width, nor height. A line is such that any two points on the line describe the shortest distance between those two points. Lines also carry on forever in both directions. Imagine a piece of string, hold the two ends and pull them tight. The string represents the shortest distance between the two ends. Remember though that a line does not have any width or height. Under a magnifying glass, we see the string has a width. A tightly drawn string is not a line, because a line would not have a width, but we can imagine a line in the exact middle of the string.

Now usually when we talk about a line in geometry we mean a straight line as described above, but there are other lines in Euclidean geometry, called curves. Curves are not straight. The circumference of a circle is an example of a curve. (We will get to circles later in the syllabus).

Okay, we have talked about lines, now, so how do they behave? We usually represent a line by drawing it on a piece of paper using a ruler to connect the points and extending it past the points. We can take pieces of a line and call them line segments and we can cross two lines and get both a point (where they intersect) and some angles. We can also choose to ignore half of a line by cutting it off at a point and calling what we have left a ray.

A plane has two dimensions: width and length. Both of these dimensions are infinite, and, because there are only two dimensions, a plane is perfectly flat and infinitely thin, meaning it has no thickness dimension. Because of this, a plane doesn't really have a top or a bottom because whatever is on the top is also on the bottom. If you take two planes and make them intersect, you get a line (more on that later) and if you take three points that are not all in the same line, there is only one plane that can contain all three (more on that later too). Planes are useful, because a plane can hold all of the two dimensional (flat) shapes that geometry uses. We usually think of one side of a piece of paper (or a computer screen) as part of a plane. While this is not exactly correct, like the representations of a point and a line, this is useful.

Exercises

Section 1.4 - Axioms/postulates

A postulate or axiom is a statement which is taken to be self-evident, and cannot be proved. They are the starting point from which any system in mathematics, such as geometry, is built up from. The axioms of geometry state properties of points, lines, and planes that are consistent with our intuitive understanding of them. For example, one axiom states that given two points, there is a unique line that passes through those two points (a property of incidence between points and lines). In Euclidean geometry, there are five axioms:

A straight line segment can be drawn joining any two points.
Any straight line segment can be extended indefinitely in a straight line.
Given any straight line segment, a circle can be drawn having the segment as its radius and one endpoint as its center.
All right angles are congruent.
Given a line and a point off the line, exactly one new line can be drawn through the point that is parallel to the given line.

From these postulates, we can deduce all the theorems of Euclidean geometry.

Section 1.5 - Theorems

Postulates:

1) Between any two points, there exists one and only one line.

2) If two lines intersect, their intersection is a point.

3) Given any three non-collinear points, there is exactly one plane that can be constructed, which will include all of them.

4) If two planes intersect, their intersection is a line.

Exercises

1) Draw a point on a piece of paper. How many lines can you draw through that point?

2) Draw two points on a piece of paper. How many lines can you draw through both points?

3) Draw three points on a piece of paper. How many lines can you draw through all three points? Why? What undefinable object could connect all three points? Is there a way to draw the points so that a line goes through all three?

Chapter Review

Vocabulary

Axiom - a formal logical expression used in a deduction to yield further results
Conjecture - a mathematical statement that has been proposed as a true statement, but which no one has yet been able to prove or disprove
Deductive reasoning - process of reasoning in which the argument supports the conclusion based upon a rule
Hypothesis - a proposed explanation which can be a proposition ("A causes B")
Inductive reasoning - process of reasoning in which the assumption of an argument supports the conclusion, but does not ensure it
Postulate - a mathematics statement which is used but cannot be proven
Theorem - a proposition that has been or is to be proved on the basis of explicit assumptions