through the eyes of the mathematician through the eyes of the mathematician cheryl ooten, ph.d....

Through the Eyes of the Mathematician

Through the Eyes of the MathematicianThrough the Eyes of the Mathematician

Through 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 Stage—Some Models & Myths

II. What’s It All About?

III. Sampling the Branches

IV. Stories about The Big Ones— Solved & Unsolved

I.Set the Stage—Some

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.

Devlin, K. ∆

It’s about relationships.

Models for Managing the Mean Math Blues

a) Fight negative math rumors.

1. Some people have a math mind and some don’t.

2. I can’t 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 don’t 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

EMOTIONS THOUGHTS

BEHAVIORSBODY SENSATIONS

Ref: Ooten & Moore. Managing the Mean Math Blues

Cognitive Psychotherapy Model of how people “work”

EMOTIONS

I am frightened by math

THOUGHTS

I can’t do math.

BEHAVIORS

I avoid numbersI don’t 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 don’t 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

Fixed Mindset•You can learn but can’t change your basic level of intelligence

Growth Mindset•Your smartness increases with hard work.

Ref: Boaler, Mathematical Mindsets;Dweck, Mindset: The New Psychology of Success

Mindset Model

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

Growth Mindset•Focus on effort •Skip judging •Ask: What can I learn?How can I improve?What can I do differently?Ref: Boaler, Mathematical Mindsets;

Dweck, Mindset: The New Psychology of Success

Choose this!

“Just because some people can do something with little or no

training, it doesn’t mean that others can’t 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). What’s 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.

Math Writer John Derbyshire wrote:

“I don’t believe [this topic] can be explained using math more elementary than I have used here, so if you don’t understand [it] after finishing my book, you can be pretty sure you will never understand it.”

Ref: Derbyshire, Prime Obsession, p. viii ∆

Watch for math myths everywhere!

UC Berkeley Mathematics Professor Edward Frenkel says:

“One of my teachers…used to say: ‘People think they don’t understand math, but it’s all about how you explain it to them.

If you ask a drunkard what number is larger, 2/3 or 3/5, he won’t 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

II.What’s It All About?

Math’s beautiful, Math’s everywhere,

and Math’s 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.What’s It All About?

Math’s beautiful, Math’s everywhere,

and Math’s 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!

“[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 Desmond’s

“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.)CombinatoricsNumber TheoryAlgebraic GeometryK-TheoryTopological GroupsReal FunctionsPotential TheoryFourier AnalysisAbstract Harmonic AnalysisIntegral EquationsFunctional Analysis (Bernie)Operator TheoryCalculus of VariationsOptimizationConvex/Discrete Geometry

Differential GeometryGeneral TopologyAlgebraic TopologyManifolds & Cell ComplexesGlobal AnalysisProbability TheoryStatisticsNumerical AnalysisComputer ScienceFluid MechanicsQuantum TheoryGame Theory; EconomicsOperations ResearchSystems TheoryMathematical Education (Cheryl)AND MORE…

Each specialty is related to one or more branches.1.Arithmetic:

CombinatoricsNumber TheoryStatistics···

3.Geometry:Convex/Discrete GeometryDifferential GeometryGeneral Topology···Deeper results are possible when

different research specialties are connected.

2.Algebra:Algebraic GeometryK-TheoryGroup Theory··· 4.Analysis:

Fourier AnalysisDifferential EquationsFunctional Analysis ···

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.

Let’s see some cool things that aren’t 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.

Every positive integer greater than 1 is either a prime or it can

be factored as the unique product of prime numbers.

The FUNdamental Theorem of Arithmetic:

The numbers 2, 3, 4, 5, … are either

“Composite” or “Prime”

Review: What is a 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.

The numbers 2, 3, 4, 5, … are either

“Composite” or “Prime”

Review: What is a 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.

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 = 2·2

5 is prime

6 is 2·3

7 is prime … ∆


Every positive integer greater than 1 is either a prime or can be factored as the unique product of

prime numbers.


Good Numbers = primes or can be factored as the unique product of prime numbers

Bad Numbers = all the others

Let’s prove it:


• Consider the smallest bad number.

• It can’t be prime (it’s “bad,” not

“good.”) • That means it’s composite and

can be factored into the product of smaller numbers.BAD

Smaller number times Smaller number


• These “smaller numbers” must either be prime or able to be factored as primes.

• Whoops! The BAD number isn’t “bad” after all.

BAD

Smaller number times Smaller number

IT’S TRUE: Every positive integer greater than 1 is a prime or can be factored as the unique product of

prime numbers.

Since the smallest bad number couldn’t 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.

IT’S TRUE: Every positive integer greater than 1 is either a prime or can be factored as the unique product of

prime numbers.


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 Goldbach’s 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 it’s the subject of a novel called Uncle Petros and Goldbach’s Conjecture.

Ref: Derbyshire, Prime Obsession, p 90

4=2+2 6=3+3 8=3+5

10=3+7 12=5+7 14=3+11

16=5+11 18=7+11

20=?

22=?

Goldbach’s Conjecture: Every even number greater than 2 is the sum of two primes.

Are there any more???

There are many other interesting hypotheses about

primes. #1-3 are known to be true:

1. There are infinitely many primes. (Euclid 300 BC)

2. The higher you go, the sparser the primes.

3. Successive primes can be any distance apart. They can also be very close. (e.g. 11&13 or 37&41)

4. There are infinitely many twin primes. (i.e. 2 apart like 11&13 or 29&31)

5. Riemann Hypothesis (More later)

2. Algebra (Symbolic

Manipulation)

In algebra, we like to look for general symbolic formulas.

Example: The Pythagorean Theorem


15

The Pythagorean Theorem may be pictured this way:

But why is c²=a²+b²?

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! We’re in the kitchen now.)

Step 1: Consider a right triangle.

Proof of The Pythagorean Theorem

Step 2: Label its sides.


Step 3: Make 4 triangles the same size.

c

b

a


Step 4: Place the 4 triangles together in two different ways to make two

squares that are the same size (a+b).ba

b

a

b

b

a

a


Step 5: Notice that the extra white space forms 3 smaller squares.


Step 6: Notice: The white square on the left has area

c². The two white squares on the right

have areas b² & a².


Step 7: Since the entire left square is the same size as the entire right

square, removing the four triangles gives us c²=a²+b²

∆

The Pythagorean Theorem

The square of the hypotenuseOf a right triangle

Is equal to the sum of the squares

Of the two adjacent sides.

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 c²=a²+b²

are whole numbers.

For example:5, 4, 3

or13, 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+ b²5² = 3²+ 4²25 = 9 + 16

For now, let’s 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)

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.)

(Fermat was a tease.)

It is clear to us now that entire fields of math had to be developed by hardworking

mathematicians to prove Fermat’s Last Theorem.

So much for Fermat’s “truly marvelous proof.”

Sophie Germain

3. Geometry (Figures & Drawing)

3. Geometry (Figures & Drawing)

It’s more than the Euclidean geometry that we learned in high school.

Geometry Makes Me Happy when it meets art.

Jen Stark’s 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.html

Fractals decribe the real world better than Euclidean

geometry.

Engineers can perform

“biomimicry.”


Fractal patterns (from nature) are used to cool silicon chips in

computers.


The Sierpinski Triangle Fractal

Ref: Burger & Starbird

The Sierpinski triangle was used to design antennas for cell phones and

wifi.


Fractals provide ways to mix fluids carefully when just stirring doesn’t work.


4. Analysis (Limits)

∆

4. Analysis (Limits)

Using limits, the study of Calculus

has two basic themes:

Differentiation

&

Integration

These two lovely gentlemen came up with calculus independently about 200

years ago.

It took 150+ years to prove that their ideas were correct.

Isaac Newton Gottfried Leibnitz

Differentiation is about finding weird slopes.

Calculus Idea #1: Differentiation

Example: “Slope” is steepness of a roof

or the grade of a road or the incline of a treadmill


What is the slope of a flat roof?

What is the slope of a curved roof like a dome?






In calculus, we try to find the slope of a line at each point on a curve:

Integration is about finding weird areas.

Calculus Idea #2: Integration

Not-weird area:

A red carpet

measuring

3 feet by 18 feet

has area

54 square feet

or

6 square yards.

Integration is about finding weird areas like the area in gray in this picture.

Calculus Idea #2: Integration

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

Ones—Solved & Unsolved

∆

Solved Problems:

1. Arithmetic (Counting)

a

b cd

2. Algebra (Manipulating)

4-Color Theorem (1852-1976)

The computer was necessary for part of the proof.

Poincaré’s 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.)

Ref:https://laplacian.wordpress.com/ 2010/03/13/the-poincare-conjecture/

3. Geometry (Figures/Drawing)

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: Let’s 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: Let’s 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.

120

24

48

120

48

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 biology—cellsIn chemistry—atomsIn physics—fundamental particles

So too for mathematics:In number theory—prime numbersIn group theory—the “simple” groups



So too for mathematics:In number theory—prime numbersIn group theory—the “simple” groupsRef: Devlin. Mathematics, the New Golden

Age



So too for fields in mathematics:In number theory—prime numbersIn group theory—the “simple” groupGalois’ 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. Yang—Mills Theory & the Mass Gap Hypothesis—Describe mathematically why electrons have mass

iii. The P vs. NP Problem—How efficiently can computers solve problems

iv. The Navier-Stokes Equations—Mathematically describe wave/fluid motion

v. The Poincaré Conjecture—Find the mathematical difference between an apple & a donut

vi. The Birch & Swinnerton-Dyer Conjecture—Knowing when an equation can’t be solved

vii. Hodge Conjecture—Classifying abstract objects

Ref: Devlin, K. The Millennium Problems ∆

100 years ago, Hilbert stated 23 problems.

All have been solved

except for Riemann’s

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 stateand slippery to

understand


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—

Notice the

whole numbers.

Notice the prime numbers.The link between:

analysis (Riemann’s zeta function) & arithmetic (prime numbers)

adding

multiplying

Ref: clip art

Let’s end on a nice

note.

www.youtube.com/watch?v=oomGHjJN-RE

Other References:Aczel, A. D. (1996). Fermat’s 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-12-2.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/freshw05.html

http://www.math.uci.edu/~brusso/freshs15.html

http://www.math.uci.edu/~brusso/freshs14.html

Other References continued:

Russo, B. High School Presentation. “The Prime Number Theorem and the Riemann Hypothesis.” http://www.math.uci.edu/~brusso/slidRHKevin060811.pdfSingh, S. (1998). Fermat’s enigma: The epic quest to solve the world’s 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 math’s greatest puzzles.

IT’S 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:Consider the smallest uninteresting number.Isn’t 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

Ref: Burger & Starbird

Koch Triangle

Fractals from the imaginati

on

Every positive integer greater than 1 is a prime or can be

factored as the unique product of prime numbers.

e.g. 24 = 2·2·2·3


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 3756801695685·2^666669 + 1or 3756801695685·2^666669 – 1

They each have 200,700 decimal digits!

Let’s play a number game with primes.

primes product + 1 prime?2,3 2·3 + 1 = 7 yes2,3,5 2·3·5 + 1 = 31 yes2,3,5,7 2·3·5·7 + 1 = 211 yes2,3,5,7,11 2·3·5·7·11+1=2311 ? …2,3,…,p 2·3·5···p+1 = 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.

through the eyes of the mathematician through the eyes of the mathematician cheryl ooten, ph.d....

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