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Through the Eyes of the MathematicianThrough the

Eyes of the MathematicianThrough the Eyes of the

MathematicianThrough the Eyes of the

Mathematician

- Cheryl Ooten, Ph.D.
- Mathematics Professor Emerita
- Santa Ana College

- Bernie Russo, Ph.D.
- Mathematics Professor Emeritus
- University of California at Irvine

- Preview
- I. Set the StageSome Models
- Myths
- II. Whats It All About?
- III. Sampling the Branches
- IV. Stories about The Big Ones
- Solved Unsolved

I.Set the StageSome Models Myths

The math world is like a restaurant

The math world is like a restaurant

with a kitchen dining room

Math is like gossip.

Devlin, K.

Math is like gossip.

Its about relationships.

Devlin, K. ?

Models for Managing the Mean Math

Blues

- a) Fight negative math rumors.
- 1. Some people have a math mind and some dont.
- 2. I cant do math.
- 3. Only smart people can do math.
- 4. Only men can do math.
- 5. Math is always hard.
- 6. Mathematicians always do math problems quickly

in

their heads. - 7. If I dont understand a problem immediately, I

never will. - 8. There is only one right way to work a math

problem. - 9. It is bad to count on fingers.
- 10. Negative math-experience memories never go

away.

- b) Use reframes.
- What can a reframe do?
- Affect attitude change feelings.
- Neutralize negativity.
- Change a helpless victim to an in-charge owner.
- Remind us of the YET.

Ref Ooten Moore, Managing the Mean Math Blues

Cognitive Psychotherapy Model

of how people work

THOUGHTS

EMOTIONS

BEHAVIORS

BODY SENSATIONS

Ref Ooten Moore. Managing the Mean Math Blues

EMOTIONS I am frightened by math

THOUGHTS I cant do math.

BEHAVIORS I avoid numbers I dont practice math

BODY SENSATIONS My stomach tenses when I see

numbers

Ref Ooten Moore, Managing the Mean Math Blues,

p 154

EMOTIONS Relief Curiosity about what else I can

learn Joy with skills I have

THOUGHTS I can do some math. I can learn more. I

dont need to get it all right now.

BEHAVIORS Take a deep breath Write a possible

solution. Try something new. Ask questions

BODY SENSATIONS Relax Become calmer Heart rate

slows

Ref Ooten Moore, Managing the Mean Math Blues,

p 154

c) Check your Mindset. (Carol Dweck,

Stanford Psychology Prof)

vs

- Growth Mindset
- Your smartness increases with hard work.

- Fixed Mindset
- You can learn but cant change your basic level

of intelligence

Ref Boaler, Mathematical Mindsets Dweck,

Mindset The New Psychology of Success

Mindset Model vs

- Growth Mindset
- Focus on effort
- Skip judging
- Ask
- What can I learn?
- How can I improve?
- What can I do
- differently?

- Fixed Mindset
- Focus on ability
- Evaluate label Good-bad
- Strong-weak

Choose this!

Ref Boaler, Mathematical Mindsets Dweck,

Mindset The New Psychology of Success

Just because some people can do something with

little or no training, it doesnt mean that

others cant do it (and sometimes do it even

better) with training. Reframe anxiety by

focusing on effort, not ability. Math skills are

learnable!

Ref Dweck

d) Anxiety comes from being required to

stay in an uncomfortable situation over which we

believe we have no control.

References for Managing the Mean Math

Blues Boaler, J. (2016). Mathematical mindsets

Unleashing students potential through

creative math, inspiring messages and

innovative teaching. Jossey-Bass San

Francisco, CA. Boaler, J. (2015). Whats math

got to do with it? Penguin Books New York,

N.Y. Dweck, C. S. (2006). Mindset The new

psychology of success. Ballantine Books,

N.Y. Ooten, C. K. Moore. (2010). Managing the

mean math blues Math study skills for

student success. Pearson Upper Saddle River,

N.J.

Watch for math myths everywhere!

Math Writer John Derbyshire wrote I dont

believe this topic can be explained using math

more elementary than I have used here, so if you

dont understand it after finishing my book,

you can be pretty sure you will never understand

it.

Ref Derbyshire, Prime Obsession, p. viii

?

UC Berkeley Mathematics Professor

Edward Frenkel says One of my teachersused

to say People think they dont understand math,

but its all about how you explain it to them.

If you ask a drunkard what number is larger,

2/3 or 3/5, he wont be able to tell you. But if

you rephrase the question what is better, 2

bottles of vodka for 3 people or 3 bottles of

vodka for 5 people, he will tell you right away

2 bottles for 3 people, of course. My goal is

to explain this stuff to you in terms that you

will understand. Ref

Frenkel, Love and Math, p 6

Our goal is to talk about a few math things we

think are cool that were not taught in school and

hopefully to talk about them in ways that make

sense!

II.Whats It All About? Maths beautiful,

Maths everywhere, and Maths huge!

?

Our goal is to talk about a few math things we

think are cool that were not taught in school and

hopefully to talk about them in ways that make

sense!

II.Whats It All About? Maths beautiful,

Maths everywhere, and Maths huge!

The world of mathematics is a hidden parallel

universe of beauty and elegance, intricately

intertwined with ours. Ref Frenkel,

Love and Math, p 1

These are mathematical models of fractals.

- Example
- Neuroscientist/musician Daniel Levitin says

Music is organized sound. - We say
- Math is the organizing tool.
- Music is intricately twined with math.
- Example
- Dave Brubeck with Paul Desmonds
- Take Five

- Ref Levitin, This Is Your Brain on Music

?

How much math does

one person know?

By the late 1800s, math had passed out of the

era when really great strides could be made by a

single mind working alone. It became a

collegial enterprise in which the work of even

the most brilliant scholars was built upon, and

nourished by, that of living colleagues.

Ref Derbyshire, Prime Obsession, p 165

Four Branches of Mathematics

Arithmetic

Algebra

Geometry

Analysis

Four Branches of Mathematics

1. Arithmetic (Counting)

2. Algebra (Symbolic Manipulation)

3. Geometry (Figures,Drawing)

4. Analysis (Calculus, Limits)

60 Mathematics Research Specialties

(A.M.S.)

Combinatorics Number Theory Algebraic

Geometry K-Theory Topological Groups Real

Functions Potential Theory Fourier

Analysis Abstract Harmonic Analysis Integral

Equations Functional Analysis (Bernie) Operator

Theory Calculus of Variations Optimization Convex/

Discrete Geometry

Differential Geometry General Topology Algebraic

Topology Manifolds Cell Complexes Global

Analysis Probability Theory Statistics Numerical

Analysis Computer Science Fluid Mechanics Quantum

Theory Game Theory Economics Operations

Research Systems Theory Mathematical Education

(Cheryl) AND MORE

Each specialty is related

to one or more branches.

- 1.Arithmetic
- Combinatorics
- Number Theory
- Statistics

2.Algebra Algebraic Geometry K-Theory Group

Theory

4.Analysis Fourier Analysis Differential

Equations Functional Analysis

- 3.Geometry
- Convex/Discrete Geometry
- Differential Geometry
- General Topology

Deeper results are possible when different

research specialties are connected.

- Mathematical research is either
- Pure (theory for its own sake)
- Applied (e.g. credit cards security)

III.Sampling the Branches

- School math is
- centuries old
- tiny part of the whole field of math.
- Lets see some cool things

that arent always shown in school. - ?

1.Arithmetic (Counting or Number Theory)

Number Theory prime numbers have given us

cryptography , thus, the ability to securely use

credit cards online.

Prime numbers are a rich source of ideas.

The FUNdamental Theorem of Arithmetic

Every positive integer greater than 1 is either a

prime or it can be factored as the unique product

of prime numbers.

Review What is a prime?

The numbers 2, 3, 4, 5, are either Composite

or Prime

Composites 4, 6, 8, 9, 10, 12, 14, 15, 16, 18,

20, 21, 22, 24, 25, 26, 27, 28, 30

Primes 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31,

Primes cannot be factored in an interesting way.

Review What is a prime?

The numbers 2, 3, 4, 5, are either Composite

or Prime

Composites 4, 6, 8, 9, 10, 12, 14, 15, 16, 18,

20, 21, 22, 24, 25, 26, 27, 28, 30

Primes 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31,

Primes cannot be factored in an interesting way.

Even numbers after 2 are composite.

The FUNdamental Theorem of Arithmetic

- Every positive integer greater than 1 is either a

prime or it can be factored as the unique product

of prime numbers. - e.g. 2 and 3 are prime
- 4 22
- 5 is prime
- 6 is 23
- 7 is prime ?

The FUNdamental Theorem of Arithmetic

Every positive integer greater than 1 is either a

prime or can be factored as the unique product of

prime numbers.

Lets prove it

Good Numbers primes or can be factored as the

unique product of prime numbers Bad Numbers

all the others

The FUNdamental Theorem of Arithmetic

- Consider the smallest bad number.
- It cant be prime (its bad, not good.)
- That means its composite and can be factored

into the product of smaller numbers.

BAD

Smaller number times Smaller number

The FUNdamental Theorem of Arithmetic

- These smaller numbers must either be prime or

able to be factored as primes. - Whoops! The BAD number isnt bad after all.

BAD

Smaller number times Smaller number

Since the smallest bad number couldnt be bad, we

continue up to the next smallest bad number but

the same thing happens. And on and on. That means

that there are no bad numbers and our theorem

is true.

ITS TRUE Every positive integer greater than 1

is a prime or can be factored as the unique

product of prime numbers.

The FUNdamental Theorem of Arithmetic

ITS TRUE Every positive integer greater than 1

is either a prime or can be factored as the

unique product of prime numbers.

The FUNdamental Theorem of Arithmetic

- This proof is done by induction which is like

setting up dominoes so each domino could push the

next over then starting them to fall. - ?

- Arithmetic has the peculiar characteristic that

it is rather easy to state problems in it that

are ferociously difficult to prove. - Example
- In 1742, Christian Goldbach (age 52) made a

conjecture. - It is not yet proved or disproved
- But it is the subject of a novel called Uncle

Petros and Goldbachs Conjecture.

Ref Derbyshire, Prime Obsession, p 90

- Arithmetic has the peculiar characteristic that

it is rather easy to state problems in it that

are ferociously difficult to prove. - Example
- In 1742, Goldbach made a conjecture.
- It is not proved or disproved yet.
- But its the subject of a novel called Uncle

Petros and Goldbachs Conjecture.

Ref Derbyshire, Prime Obsession, p 90

Goldbachs Conjecture Every even number greater

than 2 is the sum of two primes.

422 633 835 1037

1257 14311 16511

18711 20? 22?

Are there any more???

There are many other interesting hypotheses about

primes. 1-3 are known to be true

- There are infinitely many primes.

(Euclid 300 BC) - The higher you go, the sparser the primes.
- Successive primes can be any distance apart. They

can also be very close. (e.g. 1113 or 3741) - There are infinitely many twin primes. (i.e. 2

apart like 1113 or 2931) - Riemann Hypothesis (More later)

- 2. Algebra (Symbolic Manipulation)

In algebra, we like to look for general symbolic

formulas. Example The Pythagorean Theorem

Example The Pythagorean Theorem

Example The Pythagorean Theorem

15

The Pythagorean Theorem

may be pictured this way But why is

c²a²b²?

Once we find a formula, we need to convince

ourselves that it is true. Here is one of many

(112) proofs of The Pythagorean Theorem.

(Only one is needed! Were in

the kitchen now.) Step 1 Consider a right

triangle.

Proof of The Pythagorean Theorem Step 2 Label

its sides.

Proof of The Pythagorean Theorem Step 3 Make 4

triangles the same size.

a

c

b

Proof of The Pythagorean Theorem Step 4 Place

the 4 triangles together in two different ways to

make two squares that are the same size (ab).

b

a

a

b

a

b

b

a

Proof of The Pythagorean Theorem Step 5 Notice

that the extra white space forms 3 smaller

squares.

Proof of The Pythagorean Theorem Step 6

Notice The white square on the left has area

c². The two white squares on the right have

areas b² a².

Proof of The Pythagorean Theorem Step 7 Since

the entire left square is the same size as the

entire right square, removing the four triangles

gives us c²a²b² ?

The Pythagorean Theorem The square of the

hypotenuse Of a right triangle Is equal to the

sum of the squares Of the two adjacent sides.

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Are there cases of right triangles whose sides

are whole numbers? C 19.209

15

Yes, there are cases when the a, b, c in

c²a²b² are whole numbers. For example 5, 4,

3 or 13, 12, 5 These are called Pythagorean

Triples.

We could say that for Pythagorean Triples, this

theorem separates squares into the sum

of two squares. c² a² b² 5² 3² 4² 25 9

16 For now, lets just think about whole numbers

for a, b, c.

In 1637, Pierre de Fermat wrote On the

other hand, it is impossible to separate a cube

into two cubes, , or generally any power except

a square into two powers with the same exponent.

I have discovered a truly marvelous proof of

this, which, however, the margin is not large

enough to contain.

(Ref Aczel, p. 9)

?

(Fermat was a tease.)

356 years later, after thousands of

mathematicians tried to prove it, Andrew Wiles

did, in 1993. (So much for the myth that

mathematicians do problems quickly in their

heads.)

It is clear to us now that entire fields of math

had to be developed by hardworking mathematicians

to prove Fermats Last Theorem. So much

for Fermats truly marvelous proof.

Sophie Germain

3. Geometry (Figures Drawing)

3. Geometry (Figures Drawing) Its more than

the Euclidean geometry that we learned in high

school.

Geometry Makes Me Happy when it meets art.

Jen Starks kaleidoscope brings to mind geodes,

topographic maps, fractal geometry.

Ref Geometry Makes Me Happy, p 33

Fractals

geometry self-similarity

Fractals can come from nature or the

imagination

Ref http//fractalfoundation.org/OFC/OFC-12-2.htm

l

Fractals decribe the real world better than

Euclidean geometry. Engineers can perform

biomimicry.

Ref http//fractalfoundation.org/OFC/OFC-12-2.htm

l

Fractal patterns (from nature) are used to cool

silicon chips in computers.

Ref http//fractalfoundation.org/OFC/OFC-12-2.htm

l

The Sierpinski Triangle Fractal

Ref Burger Starbird

The Sierpinski triangle was used to design

antennas for cell phones and wifi.

Ref http//fractalfoundation.org/OFC/OFC-12-2.htm

l

Fractals provide ways to mix fluids carefully

when just stirring doesnt work.

Ref http//fractalfoundation.org/OFC/OFC-12-2.htm

l

- 4. Analysis (Limits)
- ?

4. Analysis (Limits)

4. Analysis (Limits)

4. Analysis (Limits)

Using limits, the study of Calculus

has two basic themes Differentiation

Integration

Isaac Newton Gottfried Leibnitz

These two lovely gentlemen came up with calculus

independently about 200 years ago. It took 150

years to prove that

their ideas were correct.

Calculus Idea 1 Differentiation

Differentiation is about finding weird slopes.

Example Slope is steepness of a roof

or the grade of a road or the incline

of a treadmill

Calculus Idea 1 Differentiation

Differentiation is about finding weird slopes.

Example Slope is steepness of a roof

What is the slope of a flat roof? What is the

slope of a curved roof like a dome?

Calculus Idea 1 Differentiation

Differentiation is about finding weird slopes.

In calculus, we try to find the slope of a line

at each point on a curve

Example Slope is steepness of a roof

Calculus Idea 2 Integration

Integration is about finding weird areas.

Not-weird area A red carpet measuring

3 feet by 18 feet has area 54 square feet

or 6 square yards.

Calculus Idea 2 Integration

Integration is about finding weird areas like the

area in gray in this picture.

We know how to find the area of rectangles. So,

we approximate the area with a few rectangles.

Then we increase the number of rectangles.

We keep increasing the rectangles.

We keep increasing the number of rectangles and

use the idea of limit to find the exact area.

AND, miracles of miracles, it turns out that

differentiation integration are opposite

processes of each other in a similar way

that adding subtracting are.

IV.Stories about The Big OnesSolved

Unsolved ?

Solved Problems

1. Arithmetic (Counting)

2. Algebra (Manipulating)

a

d

b

c

4-Color Theorem (1852-1976)

The computer was necessary for

part of the proof.

3. Geometry (Figures/Drawing)

Poincarés Conjecture

(1904-2006) Grigori Perelman proved this in 2006

but turned down 1 million in prize money, left

mathematics, and moved back to Russia with

family. (He also turned down the Fields

Medal.) Refhttps//laplacian.wordpress.com/

2010/03/13/the-poincare-conjecture/

4. Analysis

?

P.S. Another Algebra Theorem

Classification of Symmetries (1870-1985)

- Symmetry
- Is pervasive in nature
- e.g. starfish, diamonds, bee hives,
- Is about connections between different parts of

the same object.

For a mathematician, a symmetry is something

active, not passive. For the mathematician, the

pattern searcher, understanding symmetry is one

of the principal themes in the quest to chart the

mathematical world. Ex Lets look at symmetries

of a round table top aRRed square table top

Ref Du

Sautoy, Symmetry

For a mathematician, a symmetry is something

active, not passive. For the mathematician, the

pattern searcher, understanding symmetry is one

of the principal themes in the quest to chart the

mathematical world. Ex Lets look at symmetries

of a round table top a square table top

Ref Frenkel

Consider all possible transformations of the two

tables which preserve their shape

position. Those transformation are called

symmetries. A round table has many symmetries. A

square table only has four. When we combine these

symmetries, we get a mathematical entity called a

group.

Ex Look at all possible transformations of the

two tables which preserve their shape

position. Those transformation are called

symmetries. A round table has many symmetries. A

square table only has four. When we combine these

symmetries, we get a mathematical entity called a

group.

Other nice examples of symmetries forming a group

can be seen with each of the Five Platonic

Solids.

48

48

24

120

120

- Child Prodigy Evariste Galois (1811-1832)
- recorded a great gift to math in a long letter to

a friend the evening before he was killed in a

duel at age 20. - His concept (the group) proved to be most

significant, with applications in - physics
- chemistry
- engineering
- many fields of math

Ref Frenkel, p 75

- In math, discoveries using Galois idea led to

advances in - Computer algorithms
- Logic
- Geometry
- Number theory

A primary aim in any science is to identify

study basic objects from which all other

objects are constructed. In biologycells In

chemistryatoms In physicsfundamental

particles So too for mathematics In number

theoryprime numbers In group theorythe simple

groups

A primary aim in any science is to identify

study basic objects from which all other

objects are constructed. In biologycells In

chemistryatoms In physicsfundamental

particles So too for mathematics In number

theoryprime numbers In group theorythe simple

groups

Ref Devlin. Mathematics, the New Golden Age

A primary aim in any science is to identify

study basic objects from which all other

objects are constructed. In biologycells In

chemistryatoms In physicsfundamental

particles So too for fields in mathematics In

number theoryprime numbers In group theorythe

simple group

Galois contribution

- Unsolved Problems
- ?

Unsolved Problems

In May, 2000, in Paris, Clay Mathematical

Foundation announced 7,000,000 in prizes. One

million dollars apiece for solution of 7

mathematical problems called The

Millenium Problems.

i. The Riemann Hypothesis (MORE LATER) ii.

YangMills Theory the Mass Gap

HypothesisDescribe mathematically why electrons

have mass iii. The P vs. NP ProblemHow

efficiently can computers solve problems iv. The

Navier-Stokes EquationsMathematically describe

wave/fluid motion v. The Poincaré ConjectureFind

the mathematical difference between an apple a

donut vi. The Birch Swinnerton-Dyer

ConjectureKnowing when an equation cant be

solved vii. Hodge ConjectureClassifying abstract

objects

Ref Devlin, K. The Millennium Problems ?

100 years ago, Hilbert stated 23 problems. All

have been solved except for Riemanns Hypothesis.

Ref Derbyshire

The Great White Whale of the 20th Century has

been (and still is) The Riemann Hypothesis.

Ref Derbyshire, Prime Obsession pp 197-198

?

The Riemann Hypothesis was stated by Bernhard

Riemann in 1859

Ref Derbyshire, Prime Obsession pp 197-198

It is slippery to state and slippery to understand

Ref Derbyshire, Prime Obsession pp 197-198

Ref Derbyshire, Prime Obsession pp 197-198

20th century mathematicians have been obsessed by

this challenging problem. Mathematicians have

worked on it in different ways, according to

their mathematical inclinations. There have

been computational, algebraic, physical, and

analytic threads.

Ref Derbyshire, Prime Obsession pp 197-198

The Golden Key

adding

multiplying

Notice the whole numbers.

Notice the prime numbers.

The link between analysis (Riemanns zeta

function) arithmetic (prime numbers)

Lets end on a nice note.

Ref clip art

www.youtube.com/watch?voomGHjJN-RE

Other References

Aczel, A. D. (1996). Fermats last theorem

Unlocking the secret of an ancient mathematical

problem. Burger, E.D. Starbird, M. (2005). The

heart of mathematics An invitation to effective

thinking. Derbyshire, J. (2004). Prime obsession

Bernhard Riemann and the greatest unsolved

problem in mathematics. Devlin, K. (1999).

Mathematics The new golden age. Devlin, K.

(2002). The math gene How mathematical thinking

evolved and why numbers are like gossip. Devlin,

K. (2002). The millennium problems The seven

greatest unsolved mathematical puzzles of our

time. Du Sautoy, Marcus. (2008). Symmetry A

journey into the patterns of nature. Estrada, S.

(publ). (2013). Geometry makes me happy. Fractal

Foundation http//fractalfoundation.org/OFC/OFC-1

2-2.html Frenkel. E. (2013). Love

math. Levitin, D. J. (2007). This is your brain

on music The science of a human

obsession. Ronan, M. (2006). Symmetry and the

monster One of the greatest quests of

mathematics. Russo, B. Freshman Seminars UCI.

Prime Obsession, Millennium Problems, Love

and Math. http//www.math.uci.edu/brusso/freshw0

5.html http//www.math.uci.edu/brusso/fr

eshs15.html http//www.math.uci.edu/brus

so/freshs14.html

Other References continued

- Russo, B. High School Presentation. The Prime

Number Theorem and the - Riemann Hypothesis.
- http//www.math.uci.edu/brusso/slidRHKevin0

60811.pdf - Singh, S. (1998). Fermats enigma The epic quest

to solve the worlds - greatest mathematical problem.
- Sabbagh, K. (2002) The Riemann hypothesis The

greatest unsolved problem - in mathematics.
- Smith, S. (1996). Agnesi to Zeno Over 100

vignettes from the history of - math.
- Szpiro, G. G. (2008). Poincarés prize The

hundred-year quest to solve one - of maths greatest puzzles.

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The FUNdamental Theorem of Arithmetic

ITS TRUE Every positive integer greater than 1

is either a prime or can be factored as the

unique product of prime numbers.

Said another way Every number is either prime or

divisible exactly by a prime.

Theorem Every number is interesting.

Theorem Every number is interesting. Proof Cons

ider the smallest uninteresting number. Isnt

that interesting? QED

Geometry Makes Me Happy when it meets

architecture. This is an underground car park in

Sydney, Australia.

Ref Geometry Makes Me Happy, p 175

Geometry Makes Me Happy when it meets industrial

design.

Textile designer Elisa Strozyk gave textile

properties to wood to make this throw.

Ref Geometry Make Me Happy, p 149

- Koch Triangle
- Fractals from the imagination

Ref Burger Starbird

The FUNdamental Theorem of Arithmetic

Every positive integer greater than 1 is a prime

or can be factored as the unique product of prime

numbers. e.g. 24 2223

Eenie, meenie, miney, moe Catch a

number by its toe If

composite, make it pay With

prime numbers all the way. My

teacher told me to always look for primes.

The largest known twin primes currently Known

are 37568016956852666669 1 or

37568016956852666669 1 They each have

200,700 decimal digits!

Lets play a number game with primes.

primes product 1 prime? 2,3

23 1 7 yes 2,3,5

235 1 31 yes 2,3,5,7

2357 1 211 yes 2,3,5,7,11

23571112311 ? 2,3,,p

235p1 Q So, Q is either prime or

composite. If prime, we have found a prime

larger than p. If composite, the FUN Theorem of

Arithmetic guarantees a prime divisor. We could

show that we have found a prime larger than

2,3,,p.

Insert the other number game that shows that

Large strings of composites. See Notes.