sme class notes

8/9/2019 SME Class Notes

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SME 430 Class Notes

Week 2: Numbers and Operations

In your opinion, what are important characteristics for a number

system to have?

Base value 10 instinctual and biological - ?

o Perhaps just to have a base value a number that makes

sense. This limits the number of symbols. Can perform

calculations more easily

Important to have a symbol for zero. This makes it much more

clearer than a space would provide.

Large numbers can be written consistently and concisely. (Dont

need to write down infinite symbols.

Using symbols that are simpler (not intricate)

Using a limited number of symbols.

Symbols should be read in only one direction

The reading of a number should not involve computations.

In your group, come up with your own definition of a positional number

system, then discuss which of the number systems we discussed last weekfit your groups definition of positional?

1. In a positional number system, rearranging the symbols will change

the value./The position of the symbols is important to determining

the value of the number.

i. Positional Roman, Babylonian, Mayan, Chinese/Japanese,

ii. Non-positional Egyptian, Greek

2. Any number system with place value that has some sort of basenumber system

i. Positional Chinese/Japanese, Mayan, Babylonian

ii. Non-positional Greek, Egyptian, Roman


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3. A positional number system has a constant multiplier

i. Positional Babylonian, Chinese/Japanese

ii. Non-positional Mayan, Egyptian, Roman, Greek

To classify things, you have to make sure everyone has the same

definition. Definition was relative depending on different definitions,

different classifications could be obtained.

What are the characteristics of our base 10 number system?

10 symbols (0-9)

Efficient positional structure

Can compute directly with the numbers

Have symbols for non-integers

Base-10

Place value exists

Can use the symbol 0 to hold place value

Large numbers can be represented with few symbols

Every number has a name Has properties (distributive, associative, commutative)

Symbols are always read from left to right

Have standard (and easy to use) algorithms for different operations

Multiple representations for numbers (1/4, .25, 1:4)

Symbolic representation for special numbers (, i, e, ln, c, repeated

digit 0.33333)

Do we need 4 different operations? Can two be enough?

What mathematics can & cant we do if we only use addition and

subtraction in a number system?


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Do multiplication and division provide access to any additional

mathematics?

The 2 Camp

Two may be enough (addition or subtraction) & (multiplication ordivision).

Only addition (multiplication is addition)

If were tricky about it, we can treat some operations like others.

The 4 Camp

Hard to manage with less than 4.

* seems difficult (if not impossible) with only addition

Might not be able to represent all numbers if we dont have all the

operations

Further mathematics might be difficult/impossible

If we start with 0-9 and +/-, then we can get negatives, we can get all

the integers, Natural, Integers. Were missing rational numbers, irrationals,

imaginary numbers.

If we start with 0-9, and +/-/*/, we can get natural numbers, integers,

rational numbers & irrational numbers (real numbers). imaginary?

Do you think that you can use the standard algorithm for addition or

multiplication we are using today in other number systems. Why or why not?

If yes, how?

Can do

o Egyptian could do addition if you go horizontally and may

need regrouping.

o Greek could do addition

o Roman could do addition if you also kept track of subtractions.

Cant do


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o Systems without a zero placeholder, we cant perform the

standard algorithm for multiplication

Week 3: Zeros and Fractions

How has zero evolved in mathematics and as a number?

Why was there a need for zero?

o Before there was a symbol for zero, there was just an empty

space. This led to confusion, so a dot was used (sometimes

the same as the punctuation for the end of a sentence). Zero

began as being a symbol for a placeholder (first recognized as

the absence of a quantity). Later recognized as a quantity.

Our symbol for zero evolved from a tiny circle used as a placeholder.

How was zero named in different cultures?

o Set properties/identities of zero including the additive

property (any number plus zero is the original number) and

the multiplicative identity (any number times zero is zero).

o Confusion existed when dividing by zero? (Is the answer zero,

one, something else?)

What new mathematics became possible with each of these

changes in conceptions of zero?

o Algebra was developed (setting an equation equal to zero).

Originally, variables were used on both sides of an equation.

Now all variables could be put on one side of an equation.

Quadratic equations could be used to find roots of a quadratic

equation (which is where the graph crosses the x-axis, if there

are two real roots/one real root touches the axis/no real roots

does not cross the axis). These are sometimes called thezeros of the equation.

How have fractions evolved in mathematics?

Why was there a need for fractions?


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o Fractions began as representations of parts of a whole. There

was a need for greater precision in measurement (feet -> half

feet -> quarter feet, etc.). Sometimes fractions were given

unique names instead of parts of a larger whole (cups, pints,

quarts, gallons, etc.) How were fractions described and used in different cultures?

o Chinese avoided using unit fractions (7/3 was instead written

as 2 and 1/3).

What is the difference between our current use of fractions and the

unit fraction approach?

o Unit fractions consisted of a 1 in the numerator. Some

fractions were represented as the sum of unit fractions.Chinese were perhaps the first to go from unit fractions to

more conventional notations of fractions (multiples of a small

unit -> instead of +1/4 you can now say ).

o Fractions could be written as decimals. This allowed easier

computations (which led to better understandings of square

roots and pi). Percents (Per=per, cent=100) came from

fractions. Base 60 used in time and navigation (from

Babylonians).

What are some of the logical difficulties that arise when you attempt to

define 0/0 to be 1 or 0?

Conflict anything divided by zero is undefined/anything divided by

itself is 1/zero divided by anything is zero.

Issue what does this problem represent?

0/0=0 is the same as 0 times what equals zero (of which there are

an infinite number of answers). Zero times what equals 1 is not

possible. Quotient Remainder Theorem


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Imagine a teacher shows her students a shortcut for dividing single digits by

9. She says that all you have to do is to write the numerator as a repeating

decimal (for example, 1/9=.1 repeating, that is 0.1111111..., and 4/9=.4

repeating, that is 0.4444444...). A student raises their hand and asks if that

means that 9/9=.9 repeating. A different students says that this is impossiblebecause 9/9 has to be equal 1. What would say to these two students?

3/9=1/3=.333333

0.333.+0.333+0.333=0.999

1/3+1/3+1/3=3/3=1 These are the same.

X=.99999

10X=9.9999.

subtract these two things

10X-X=9.9999-0.99999

9X=9

x=1

Week 4: Negative and Imaginary Numbers

Discussion on Negatives

What shifts in peoples conceptions of numbers were required to

accept negative numbers?

Previously, had counted number (thought of numbers) as objects

and measurements. There needed to be a purpose for negative

numbers (reasons they had to exist). Had a new way of referring to

negatives (as debts). People had to conceptualize something less

than nothing.


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There was confusion about where to put the negatives as compared

to their positive counterparts. People had seen that negatives were

coming up as solutions to equations, but previously they had

ignored them (false root/fictitious solutions).

One descriptions of negative numbers are numbers that are less thannothing. How would you explain or verify that negative numbers are less

than nothing?

A problem may exist with using 0 and nothing interchangeably.

What other descriptions for negative numbers can you think of?

_________________ 0

| |

| |

| |

\______________/

Absolute value -> Distance from zero.

Real world examples

o Owing money.

o Hot air balloon with negative numbers as sandbags.

Have the number line displayed with both positive and negatives.

Operations with negative numbers have defined rules (negative plus

negative is negative, negative times negative is positive, etc.). How

can you justify those rules?

o Proof of Negative times Negative?

Negative times Positive = Negative -> Negative *Negative is theopposite of Negative times Positive, which is Positive.

o Bad things happen to good people is a bad thing, but a bad

thing happening to bad people is a good thing. Good things

happening to good people is good. Good things happening to

bad people is bad. (Friends of friends/friends of enemies)


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o Logic based Not going to the store -> not going, but not not

going to the store is going.

o Using the two negative bars to form a plus sign.

-n * -m = -(-n * m) = -(-nm)= nm


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Discussion on Imaginary Numbers

What shifts in peoples conceptions of numbers were required to

accept imaginary numbers?

o People had to understand that imaginary numbers were NOTuseless. These were not impossible solutions. They had to

create the conception of imaginary numbers. This didnt exist

before and actually had to create the conceptions. Had to

have an understanding of negative numbers, as well as an

understanding of square roots.

Where are the roots of f(x)=x3+5x2+2x-8, and g(x)=x3-3x2+3x-

9?

o

Roots of f(x) are -4, -2, 1o Root of g(x) is 3

Order from least to greatest -2, -1+i, 3-i, 4i, 7-3i. Explain your

methods.

o Possibly square each number then sum them.

o Possibly substitute i for another known number.

Week 5: Euclids Elements and Pythagorean Theorem

What is the structure/organization of Euclids Elements?

Split into 13 different books among 4 topics.

o 1,2,3,6 Plane Geometry

o 11, 12, 13 Solid Geometry

o 5, 10 Magnitudes and Ratios

o 7,8,9 Whole Numbers

Within each book statement -> diagram -> proof -> Q.E.D

o Split into postulate, definitions, common notions

How did Elements contribute to mathematics?


10/23

Taught a different way of thinking (step by step using logic), and

going through proofs. The step by step process of the Elements

were a source of inspiration for two-column proofs.

Defined the entire field of plane geometry. Unified and made

consistent the theorems of geometry, number theory, etc.


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What are some reasons that the Pythagorean Theorem is so useful and

widely known among all different cultures?

Theorem has many practical uses architecture, land use.

Theorem is natural. This theorem is also easily observable.Independently discovered in many cultures. Various justifications

exist for this theorem.

Of the different justifications of the Pythagorean Theorem in the text,

which stood out to you as being the clearest, simplest, most elegant, and

most convincing? Why?

Display 1 Tilted square inscribed in a square. Easy to visualize.

Had seen proof before. Use non-complex skills to arrive at formula.

Didnt have to manipulate picture.

Week 6: Pi & Greek Geometry

What is the mathematical definition/description of pi?

=Ratio of circumference of a circle to its diameter

How have people represented pi throughout history?

Fractions or Mixed numbers - between 3+10/71 and 3+10/70, using

the symbolic symbol pi, decimal approximations, 22/7.

Why has there been so attention paid to pi throughout history?

Since pi is irrational, well never know all the digits of the decimal.

There also is no patterns found in the digits. Challenge to find more

digits. Ratio is used so often (so popular) because of its relation to a

circle. Also, possible discovers await if we can discover about the

nature of irrational numbers within the digits.

Week 7: Platonic Solids

Coordinate Geometry


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What is the mathematical definition/description of analytic

geometry?

o Representation of shapes by equations.

o This provided a numerical address to shapes. Combinedalgebra and geometry, links between the two.

o Related to the coordinate plane, need a concept of distance

and how to measure distance on a plane. Plots and

trajectories.

How has a coordinate system been used throughout history?

o Originally used to divide land into districts.

o Greece, 350 B.C., Apollonius plotted points that were a fixeddistance away from a given point to form a circle.

o A grid has been used to make maps and survey land.

o Pierre de Fermat in 1630, plot relationships between unknown

points.

What new mathematics became available because of the use of

analytic geometry (and the perpendicular y-axis)?

o Helps with negative numbers (provided a boundary betweenpositive and negatives).

o Can represent more than one variable at a time now. Can use

this to find areas and lengths.

o Can represent functions, this also led to discoveries in

Calculus.


13/23

o

Platonic Solids

What are the names/properties of the Platonic Solids?

o Tetrahedron (4 sides)

o Dodecahedron (12 sides)

o Icosahedron (20 sides)

o Hexahedron (Cube) (6 sides)

o Octahedron (8 sides)

o All the faces are the same regular shapes. Every vertex has

the same number of faces meeting (number of edges as well).

o The angles at each vertex must add to less than 360 degrees

(or else itll be flat.)

o Symmetry around each vertex.

o At least three faces must meet at each vertex.

What associations did the Pythagoreans make with these solids?

o

4 elements Tetrahedron (4) -> Fire (looks like a triangle/four

necessities for fire)

Hexahedron (6) -> Earth (belief earth was flat/set on a

surface, wont roll/six major land forms on Earth)

Octahedron (8) -> Air

Icosahedrons (20) -> Water (most spherical, most likely

to roll out of your hand) Dodecahedron (12) -> Universe

What is the distinction between Platonic solids and Archimedean

Solids?


14/23

o Archimedean solids have the same properties as Platonic

solids, but the faces dont all have to be the same one shape.

Each type of shape has to be congruent to all of the same

shapes in the figure.

Week 8: Non-Euclidean Geometries

Non-Euclidean Geometry

How is non-Euclidean Geometry different than Euclidean Geometry?

Give an example.

o Euclidean geometry is on the surface of a plane (non-

Euclidean is not). An example was Riemann which was on thesurface of a sphere.

o A sphere has no parallel lines, so the parallel postulate

doesnt hold. Triangle angle measures add to over 180

degrees. Pi is different

Which of Euclids 5 postulates is not true in spherical geometry.

Explain.

o

Postulate 2 doesnt. Lines are finite (can be measured).o Postulate 5 doesnt. There are no parallel lines.

Explain why the statement, Euclidean Geometry is better than non-

Euclidean geometries. is an unfair statement to make.

o While we may know Euclidean and be more familiar with it,

other subject areas rely on and use different kinds of

geometries. (For example, spherical geometry is good for

astronomy.)

Projective Geometry

What motivated the development of projective geometry?


15/23

o Based on a need from philosophers, artists, and scientists.

Wanted to represent three dimensional scenes on paper (in

two dimensions). Wanted to look at the relationship between

distance on the paper versus distance in real life.

What is the principle of duality? Explain with an example.

o Point-Line duality- For a statement that includes the words

line and point, interchanging the words still produces a

true statement. Points are collinear if they all lie on the same

line and Lines are concurrent if they all intersect at the

same point.

Perspective in Art

How did the use of perspective change the perception and purposeof art in society.

o Non-perspective art (such as Egyptian) was used to tell a

story (or portray a message) for record keeping. Perspective

allowed art to be more enjoyable and more abstract.

Perspective drawing allowed better blueprints for planning.

o In art, perspective fed a realism movement. Going from

cartoonish icons to more real world descriptions. Perspective

allowed showing depth in space.

Week 10: Linear Equations

What is proof?

o Proof (provides/is) evidence that something has to be true (or

not true). Provides a reason that something is true (or not

true).

o I proved the Pythagorean Theorem means

This means I took a given, and used supporting facts in

a step by step (sequential) manner which can be

followed to supports a conclusion, and arrived at a

conclusion.


16/23

How is proof used in mathematics? In math education? In real life?

How are these the same and how are they different?

o Mathematics tend to think its something new, uses

theorems and theories

o Math Education tend to think its something already done,

use manipulatives (not just words), proof can also mean

convincing someone, but not necessarily a formal proof.

Visual signals can be convincing. Understanding a proof can

lead to greater understanding of what was proved.

o Real life the scope (one instance vs. all instances), trying to

get evidence to be convinced. Visual signals can be

convincing.

o All of these areas tend to be about convincing someone that

something is true.

Why do we use proof in these different contexts?

What is the difference between a deductive proof and an inductive

proof?

Discussion of Reading (Writing Algebra) What is your own definition/description of algebra? Is there a formal

definition of algebra? (If so, what?)

o Algebra (colloquial) equations and expressions that include

symbols and variables (that represent quantities). Can use

algebra to manipulate equations to isolate a variable. Solving

for an unknown. Manipulation of polynomials that can also

represent graphs.

o Algebra (formal) A part of mathematics in which letters andother general symbols, are used to represent numbers and

quantities in formulae and equations.


17/23

o Algebra problems (formal) regardless of how it is written, it

is a questions about numerical operations and relations in

which an unknown quantity must be deduced from known

ones.

What are the characteristics of algebra that distinguish it from otherbranches of mathematics?

o Equations are already set equal to something, as opposed to

being a variable.

o Algebra uses arithmetic to find an answer, but arithmetic

doesnt use algebra to find an answer.

o Algebra is a generalization of arithmetic.

o Geometry involves shapes/pictures.

o Algebra may be used in geometry solutions.

How is a symbolic style different from a rhetorical style in algebra?

Give an example of each, and state one advantage of each

approach.

X^2+5x+6=0


18/23

o How is the mathematics used in these two solutions the

same? Different? Are there any similarities between this

method other methods we use today.

o What are the different approaches we use to solve these

problems today?

o Will the false position method work for all first degree

algebraic problems? How do you know?

o

Week 11: Polynomials and Patterns

Sketch 10 Quadratic Equations

What is meant by quadratic equations?

o Dictionary.com equations containing a single variable to the

second degree. (Its general form is ax^2+bx+c=0, where a

is not equal to zero.).

o Wikipedia a polynomial of degree 2.

o Comes from the Latin word quadractum which means

square.

What are different methods that have been used to solve quadratic

equations in the past?

o Solving an equation finding where the graph of the equation

passes through the x-axis. This is also finding the values of x,

such that when you plug them back into the equation, the

equation comes out to zero.

o Different approaches graphic, algebraic (factoring, quadratic

formula, completing the square, difference of squares),

geometric

What are some of the quadratic equations and solutions (current

method & al-Khwarizmis method) you came up with?

X^2 + 8X = 20


19/23

X^2+8X+16=20+16

(X+4)^2=36

X+4=6

X=2

Week 12: Calculus

Imagine you are trying to find different ways of calculating the area of

the shaded region of the parabola. What are different methods you could

come up with?

Integrals taking the integral of the first point minus the integral of

the second point.

Area of the parabola, then subtracting the area underneath it. This

would be using the rectangle approximation.

o

Breaking up into multiple figure. (Would be approximating).o Playdough roll it out to a uniform thickness.

Imagine you are trying to find different ways of determining the speed

of a falling object. What different methods can you come up with?

Use a speed gun (radar gun)

x

x 8

x

x 48

4


20/23

Measure falling distance and falling time. (9.8m/square second)

Compare impact markings to KNOWN impact marking.

Vertex formula

What is origin of the word calculus?

Originally from the word stone or pebble.

What does calculus mean in mathematics?

Rate of change.

What is the distinction between calculus and other mathematical

subjects?

Calculus takes into consideration change. Focuses on limits,derivates, integrals, functions, and infinite series.

What were the origins of calculus?

What problems did calculus help solve?

Helps us solve things that go on to infinity and dont have an ending

point. Things that go to an infinitely small infinity. Also addresses

rates of change and fast things are moving.

Week 13: Calculus (Part 2)

How did calculus help solve some of Zenos paradoxes?

Calculus proved Zeno wrong

What is the sum S of S=1 - 1 + 1 - 1 + 1 - 1...

S = (1-1) + (1-1) + (1-1) += 0

S = 1 (1+1) (1+1) - = 1

S = -1 + (1-1) + (1-1)= -1

So the solution = no sum. It is a divergent series


21/23

What is Archimedes Method of Equilibrium/Method of Exhaustion?

Method of Exhaustion: regular polygons inside of a shape inscribed

inside of a circle

Method of Equilibrium: like the playdough experiment take acertain amount of playdough with a certain thickness, find the area

of a circle, then transcribe the playdough into a square or other

polygon with an easy area formula and make it the same thickness

How may these have contributed to the creation and structure of

calculus.

Working with infinity

Finding the area under curves

Limits

Breaking it up into parts/integration by parts

What are some of the contributions that Newton and Leibniz made to

the creation of calculus?

Newton: Laws of Motion

Newton: Principia, fluxions

Leibniz: Symbols for integration

Leibniz: Rules for integration and anti-derivatives

Week 14: Probability and Statistics

Discussion of Probability

Design your own dice game (similar to the one listed on page 165).Describe the rules of the game, and describe how you would

calculate the mathematical expectation of winning.

How could the expected value of a game be related to the cost to

play the game?


22/23

What advantages became available by looking at the expected

outcomes of events instead of just describing equally likely

outcomes?

Discussion of Statistics

What is the relationship between data and statistics?

How are statistics and probability related?

How would you design an experiment to test the fastest route by

car from Brody Hall to Hubbard Hall? How would you control for

error? Does this prove that one route is faster than the other?

What different designs did you use?

What different variables did you come up with?

What can you conclude?

Week 15: Big Ideas

What does logic mean to you?

Logic is using data or statistics to reach a reasonable conclusion.

(Well thought out).

Logic deals with using step by step processing (as opposed to

skipping over mathematical details)

Using knowns and assumptions to get to new ideas.

How is mathematics and logic related?

Give an example of the equivalence not(P and Q) not (P) or not Q)

You cant do (A & B) is the same as not doing A or not doing B

P=I am a boy

Q=I like baseball

I am not (a boy who likes baseball) same as not being a boy or not

liking baseball.

P=Go to store


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Q=Got to movies

Not going to (store and movies) same as not going to store or not

going to movies.

Give an example of the equivalence not(P or Q) not (P) and not(Q)

P=She is old

Q=She is grumpy

She is not (old or grumpy) is the same as she is not old and she is

not grump.

If we wanted to see if every student in our class had a desk to sit in,

how could we find out besides counting the number of students and counting

the number of desks?

Have everyone go sit in a desk (see if theres any of either left

over).

Assigning labels to desks and to students.

o (One-to-one correspondence)

How did society react to the idea of infinity?

Church believed that infinity was a challenge and heresy.

Do you think the size of the set of odd numbers is the same as the size

of the set of even numbers? Support your claim with some of Cantors ideas.

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