Through the Eyes of the Mathematician Through the Eyes of the Mathematician Through 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 StageSome Models
& Myths II. Whats It All About? III. Sampling the Branches IV. Stories about The Big 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. 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.
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 BEHAVIORS EMOTIONS
BODY SENSATIONS Ref: Ooten & Moore. Managing the Mean Math Blues THOUGHTS EMOTIONS
I cant do math. I am frightened by math BEHAVIORS BODY SENSATIONS
I avoid numbers I dont practice math My stomach tenses when I see numbers Ref: Ooten & Moore, Managing the Mean Math Blues, p 154
THOUGHTS EMOTIONS I can do some math. Relief I can learn more.
I dont need to get it all right now. Curiosity about what else I can learn Joy with skills I have BEHAVIORS
Take a deep breath Write a possible solution. Try something new. BODY SENSATIONS Relax
Become calmer Heart rate slows Ask questions Ref: Ooten & Moore, Managing the Mean Math Blues, p 154 c) Check your Mindset.
(Carol Dweck, Stanford Psychology Prof) Fixed Mindset You can learn but cant change vs 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
Growth Mindset Focus on effort Skip judging Ask: What can I learn? How can I improve? Choose this! What can I do differently? Ref: Boaler, Mathematical
Mindsets; Fixed Mindset vs Focus on ability Evaluate & label Good-bad Strong-weak 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 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 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. 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 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 Analysi s Four Branches of Mathematics:
24, 25, 26, interesting way. 27, 28, Even 30 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 smallerBAD numbers.
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 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 Prime Obsession, p Uncle PetrosRef: andDerbyshire, Goldbachs
Conjecture. 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. 4=2+2 6=3+3 10=3+7 14=3+11 12=5+7
16=5+11 20=? 22=? 18=7+11 8=3+5 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 Example: The Pythagorean Theorem Example: The Pythagorean Theorem
1 5 The Pythagorean Theorem may be pictured this way: But why is c=a+b? 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 a size. 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 (a+b). a b a a
b b a b 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 c. The two white squares on the right have areas b & a. 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 c=a+b 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. 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 c=a+b 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. c = a+ b 5 = 3+ 4 25 = 9 + 16 For now, lets just think
about whole numbers for a, 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.
Sophie Germa in So much for Fermats truly marvelous proof. 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 Fractals geometry: self-similarity Fractals can come from nature or the imagination Ref: http://fractalfoundation.org/OFC/OFC-12-2.h tml
Fractals decribe the real world better than Euclidean geometry. Engineers can perform biomimicry. Ref: http://fractalfoundation.org/OFC/OFC-12-2.h tml
Fractal patterns (from nature) are used to cool silicon chips in computers. Ref: http://fractalfoundation.org/OFC/OFC-12-2.ht ml 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.ht ml 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 & 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. Example: Slope is steepness of a roof In calculus, we try to find the slope of a
line at each point on a curve: 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 b d
c 4-Color Theorem (1852-1976) The computer was necessary for part of the proof. 3. Geometry (Figures/Drawing) Poincars 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/ 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 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 Other nice examples of symmetries forming a group can be seen with each of the Five Platonic Solids.
48 48 24 12 0 120
Child Prodigy Evariste Galois (18111832) 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 Ref: Frenkel, p 75 many fields of math 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 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. 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 Equations Mathematically describe wave/fluid motion v. The Poincar ConjectureFind the mathematical difference between an apple & a donut vi. The Birch & Swinnerton-Dyer Conjecture Knowing when an equation cant be solved
vii. Hodge ConjectureClassifying Ref: Devlin, K. The Millennium abstract 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 addin g multiplyi ng
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?v=oomGHjJN-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-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.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). Poincars prize: The hundred-year quest to solve one of maths greatest puzzles. 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: Consider 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
imaginati on 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. 24miney, = 2223 Eenie, meenie, 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 37568016956852^666669 + 1 or 37568016956852^666669 1 They each have 200,700 decimal digits! Lets play a number game with primes. primes
yes 2,3,5,7,11 235711+1=2311 ? 2,3,,p 235p+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, Insert the other number game that shows that Large strings of composites. See Notes.
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