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Dear friends, both known and unknown to me, fellow Russians, and people of all countries and continents, in a few minutes a mighty spaceship will carry me into the far-away expanses of space. What can I say to you in these last minutes before the start? At this instant, the whole of my life seems to be condensed into one wonderful moment. Everything I have experienced and done till now has been in preparation for this moment. You must realize that it is hard to express my feeling now that the test for which we have been training long and passionately is at hand. I don’t have to tell you what I felt when it was suggested that I should make this flight, the first in history. Was it joy? No, it was something more than that. Pride? No, it was not just pride. I felt great happiness. To be the first to enter the cosmos, to engage single handed in an unprecedented duel with nature – could anyone dream of anything greater than that? But immediately after that I thought of the tremendous responsibility I bore: to be the first to do what generations of people had dreamed of; to be the first to pave the way into space for mankind. This responsibility is not toward one person, not toward a few dozen, not toward a group. It is a responsibility toward all mankind – toward its present and its future. Am I happy as I set off on this space flight? Of course I’m happy. After all, in all times and epochs the greatest happiness for man has been to take part in new discoveries. It is a matter of minutes now before the start. I say to you, ‘Until we meet again,’ dear friends, just as people say to each other when setting out on a long journey. I would like very much to embrace you all, people known and unknown to me, close friends and strangers alike. See you soon! – Yuri Alekseevich Gagarin, first person in space, April 12, 1961

 

Today, of course, is the Ides of March, when Anonymous is slated to protest Scientology again, two days after founder L. Ron Hubbard’s birthday.  But let’s look back to the future to Pi Day, the 14^th of March, which is Einstein’s birthday, and, according to Sean at Cosmic Variance, Talk Like a Physicist Day (not to be confused with any other Talk Like a PD).

To celebrate you could read Lucas Kovav’s paper, Electron Band Structure In Germanium, My Ass, which concludes:

Going into physics was the biggest mistake of my life.

Or you might want to see scientists explain their research results on video.  In that case check out SciVee.tv.  In keeping with our back to the future theme, below is Dr. David Frisch and James Smith’s demonstration, atop Mount Washington, of time dilation, first predicted by Einstein.

For more physics phun try the free Phun Software package.

My blogoverse buddy (B_vB^TM) Jonathan asked me to contribute to CoM-18, which he is hosting and I am happy to do so. But first a belated shoutout to my B_vB, and best reviewer, Dave Marain, who interviews Professor Lynn Steen, a principal architect of the original NCTM Standards and a highly respected voice in reform mathematics.

Don’t expect any research topics here. This is about solving a practical problem in automotive gas economy which involves a pricing anomoly, a Greek mathmatician who may have tutored Alexander the Great, and an 18th century Scottish math professor who almost loses his job by taking an unauthorized 2-year sabbatical to tour the Continent. Can you imagine not losing your job today?

So what is the problem? I have 2 cars, an older model which uses regular gas, and a slightly newer model which only uses the more expensive high-test gas, but also gets more miles per gallon (mpg). The question is which is more economical to  drive? The pricing anomaly I have observed is that no matter the price per gallon (ppg) of gasoline, and the price has fluctuated up and down quite a bit, the difference in price between regular and high-test is almost always a constant, viz., 25 cents. So which car  is more economical to drive actually depends on the price of gasoline! Observe.

The price per mile, ppm = price per gallon / miles per gallon.  The percent difference in ppm between regular and high-test,

%diff = (ppm_regular / ppm_high-test) – 1 = (p / m) – 1,

where p = ppg_regular / ppg_high-test and

m = mpg_regular / mpg_high-test.

If we take

d = ppg_high-test – ppg_regular = $0.25 and

m = 25 / 27 and

then plot ppg_high-test against %diff we get:

So if the price of high-test is below about $3.35 the gas guzzler is more economical fuel-wise, and versa vice. But what is this  curve? A cubic polynomial trend line fits it almost perfectly. So is it a polynomial? Zooming out gives us a clue.

It is a rectangular hyperbola (first studied by Menaechmus, a student of Plato, about 350 BC) flipped on the X axis and asymptotic to X = 0 and Y = 0.08 = (1 / m) -1. You can easily see this by rewriting the equation for %diff as

%diff = A / ppg_high-test + B where

A = -d / m and

B = (1 / m) – 1

So this leaves the question of why a cubic polynomial fits the data so well. And here is where our Scottish math professor, Colin MacLaurin, comes in. In 1742 he wrote A Treatise on Fluxions (pdf), the first systematic application  of Newton’s calculus, in which he shows, among other trigonometric marvels, how the equation for a hyperbola could be closely approximated by truncating an infinite polynomial series.

“Your small minds are musclebound with suspicion … because the only exercise you ever get is jumping to conclusions.”

That’s Danny Kaye as Walter Mitty in a 1947 movie to be remade casting Mike Myers. [Kaye, author James Thurber and many others were accused of being Communists in the propaganda pamphlet Red Stars #3 published in 1960 (!) by the Cinema Educational Guild.]

In the 1830s a Professor of Mathematics to the College of Nantes France published the following works:

• The Calculation of simple and compound interest rates, simplified with a mental addition of two numbers,
• Arithmetic, commercial, industrial and administrative, simplifed to addition by the Reformed system of calculation,
• A Theorem of Mr. Sturm, and its numerical applications,
• A novel Shorthand, improvement to what is written in the margins (re: Fermat’s Last Theorem?), and
• a 21-page book, De Quelques Propriétés des Nombres et des Fractions Décimales Périodiques proving a theorem eventually named after him.

Midy’s Theorem is a very pretty result.  It was neglected by mathematicians until 2003 when Yale undergraduate Brian Ginsberg published an extension in a student paper.  Since then there have been numerous papers with generalizations. So what is this theorem?

Rational numbers, which are defined as fractions of two integers p and q, viz., p/q, when represented as a decimal expansion come in two forms: finite decimals, e.g., 1/5 = 0.2; and repeating decimals, e.g., 1/6 = 0.1666…, which is usually shown using the vinculum, viz., 1/6 = 0.16.  Since 1/6 is the product of a finite decimal 0.5 and a repeating decimal 0.3, as David Wells shows in his wonderful book, 1/6 has a nonperiodic part whose length is the same as that of 1/2, and a periodic part with period equal to the period of 1/3.  By multiplying by a power of 10 we can convert any p/q to  p’/q, where p’/q is a pure repeating decimal.

The nice thing is that the period only ever depends on q.  Anticipating by 120 years mathematicians’ keen interest in modular forms, these periods were first related to the multiplicative order of 10 (mod q), in other words, the smallest exponent, e, such that 10^e (mod q) = 1, by J. W. L. Glaisher in 1878! [Papa Glaisher’s balloon exploits] Thus:

• 10 mod 7 = 3
• 100 mod 7 = 2
• 1000 mod 7 = 6
• 10000 mod 7 = 4
• 100000 mod 7 = 5
• 1000000 mod 7 = 1

So the multiplicative order of 10 (mod 7) is 6, and 1/7 as a repeated decimal has period 6.

Now consider rationals that have representations as pure repeating decimals where the period is even, such as:  

• 1/7 = 0.142857
• 1/77 = 0.012987
• 1/121 = 0.0082644628099173553719
• 1/803 = 0.00124533

Breaking these periodic strings in half and adding yields 999, 999, 99999999999, and 4545, respectively.  Those strings of 9‘s can’t be a coincidence, and Midy’s Theorem tells us exactly when to expect them, namely:

If p/q is a fraction written in lowest terms and a pure repeating decimal of period 2k, and q is not divisible by 10, then the ‘nines property’ holds iff:

1. q is a prime or
2. q is a prime  power; or
3. the gcd(q, 10^k – 1) = 1, where gcd is the greatest common denominator.

Thus, 7 is a prime, 121 is a prime power (11²), and gcd(77, 999) = 1, whereas gcd(803, 9999) = 11.  In 2003 Ginsberg extended Midy’s result and things progressed quickly.

Ginsberg chopped in three repeating decimals where the period is of length 3k, and showed that the nines property holds if p is 1 and q is prime.  In 2005 Gupta and Sury (pdf) solve the problem of 1/q where q is prime in complete generality.  Also in 2005 Abdul-Baki (pdf) has some imaginative, true things to say about Midy.

In early 2006 Gil and Weiner (pdf) extend Midy’s Theorem, for 1/q where q is prime, to other bases. Later in 2006 Lewittes (pdf) extends Midy in general to other bases, and extends Ginsberg’s extension, e.g., it holds for 1/21 = 0.047619, 04 + 76 + 19 = 99.  Also in 2006 Martin (Integers vol. 7) generalizes Midy’s result for chopping pure repeating decimals of the form 1/q into arbitrary fixed size commensurable pieces. 

Chopping into arbitrary fixed size commensurable pieces in arbitrary bases does not seem to obey any obvious pattern.  There is more work ahead on extending this no-longer-so-secret theorem.

If only one person knows the truth, it is still the truth. – Mahatma Gandhi

Thanks, Bassam!

    

Khodayar Akhavi points out that education funding is nowhere near as generous after the 911 crisis as it was after the Sputnik crisis (which may have ended with Gene Cernan’s last footprint on the Moon). Nevertheless, PhD production in the U.S. seems to have picked up after both crises.

The College Board just released figures for the 2007 SATs.  I looked at the scores for Math in States where at least 40% of the student population took the exams, and compared the rankings for 1997 (left) to 2007 (right).  The rankings are color coded in quintiles, with legend Red, Yellow, Green, Blue, and Violet (Red the worst).  Making double jumps up in the rankings are NC, VA, and VT; double jumps down are MD and ME.  Maine should get a bye, since this year it forced 100% of its eligible students to take the exam.  IN, TX, and NV should get special mention as the only non-coastal States to have at least 40% SAT participation. 

Update: While SC as a State improved, Miss SC Teen has issues with geography & numericity.

Welcome to the 15^th Carnival of Mathematics.  While the quantity is low during these doldrum months, I think the quality of this episode is excellent.  The contents also speak to the question du Jore of whether the CoM should be bifurcated into separate teaching and research Carnival?  We actually have a research contribution that is explainable in elementary terms and has everyday application – if you eat McNuggets everyday (not recommended)! 

But before that, because I surely will be asked (again), let me point out I-am-only-related-by-name-to-the-late-famous-Professor-John-G-Kemeny.  My favorite anecdote about him is when he was appointed President of Dartmouth he requested to continue teaching.  The Board thought that was beneath the position.  So he asked what they would say if he requested time to play golf.  He got to teach.

My father-in-law, Bill Curnin, is a retired professor, and pointed out a Brief History of Mathematics, which he has used in the classroom since 1948.  While there I noted the 1954 poem A Song … Against Mathematicians which begins:

Of all the lunatic professions which are practised on this earth
Mathematics is the craziest, and has been from its birth.

I thought to enlarge upon this in poet laureate style:

I must read and study math again, where tensor algebras Lie,
And all I need is a Kronecker delta, an epsilon and pi;

I immediately realized I wasn’t up to the poetry nor math – so you’ll have to bear with my limericks.

Chapter 1: 3599..

Taking pairs from the number nine closet,
And appending to 35, I’ll posit,
When factoring’s done,
Some 6*10^N plus one,
Is a divsor. The results’ all composite!

In Teaching Factoring – Should we?, Jonathan (aka jd2718) kicks off a series of 4 articles on rationales for what we should teach our children.  In the second article published today, I ban FOIL, he explains how his school handles polynomials to make factoring easier.  The next article will be on the trinomial factoring technique, breaking the middle.  Jonathan says, “It’s not uncommon, but many people have never seen it.”  The last article will cover other sorts of factoring and anecdotes.  For reference, there are internet discussions on factoring, and/or you can just do communal factoring on your PC.

Chapter 2: A tale of five gentlemen

Henricus Hubertus van Aubel (1830-1906), was a Dutch mathematician who proved a pretty theorem on quadrilaterals.  A. Gutierrez has a javascript conventional proof, accompanied by Chopin, online.  Wesley Cowans of FOXMATHS! did one better. His proof of van Aubel’s theorem uses complex numbers instead of standard geometric techniques.

 This problem has an interesting history.  It is related to a theorem attributed to Napoleon, often called the most rediscovered theorem in mathematics.  

A generalization of both theorems is attributed to three men, Karel Petr, Bernhard Newmann, and Jesse Douglas (proof and interactive gizmo by Alex Bogomolny).  Petr (1868-1950) was a high-powered Czech math professor before, during, and after two World Wars – with many students-turned-refugees.  Newmann has a history with Napoleon’s Theorem explained in an Australian interview entitled Napoleon, my Father and I. Douglas, co-winner of the first Fields Medal, working independently (pdf) at the same time as Newmann (1939-1941) had an incredibly prolific year in 1939.  He is also a fellow Bronxite.  Of course, as Stephen Gray points out in his 2002 article Generalizing the Petr-Douglas-Neumann Theorem on n-gons (pdf), Petr has precedence, publishing in 1908!  

Chapter 3: Pay me now or and pay me later 

John Armstrong, the Unapologetic Mathematician, tackles a solution to an actual, real-world  data analysis problem, CEO Compensation and its relationship to corporate profits. What relationship? John has been in an Ivory Tower too long. Read Dilbert. (just kidding – actually a thoughtful article)

 

 

 

Chapter 4: Study now or and study later

Dave Marain at MathNotations sent in How Recursion is Tested on the SATs and much more… Designed for middle- and high schoolers, this detailed investigation for the classroom explores the ideas of recursive-defined sequences using an SAT-type problem as a springboard.

That’s my son Alan studying at the blackboard. He should be ready for SATs in about 15 years.  I wonder if they will have changed much, viz., no portable AI-bots in the exam room?

 

Chapter 5: 26³

Julie Rehmeyer reports some news about advances in solving Rubik’s Cube in Cracking the Cube in MathTrek. Northeastern professor Gene Cooperman and grad student Dan Kunkle have discovered a way, using high-powered computation, to reduce the maximum number of moves to solve the puzzle by 1, to 26. However, there is room left for improvement, as it is believed that 20 moves may be the minimum. If that’s too simple, here is an online 4D Magic Cube to play.

Julie also writes an interesting article on mathematically modeling word acquisition in Calculating the Word Spurt.

 

Chapter 6: 43 McNuggets to don’t go

A student of Shalit’s named Xu,
Discovered some math that is new,
There are Frobenius things,
Taken from co-finite strings,
Constructed so they exponentially grew.

Here is the fresh research article, The Noncommutative Frobenius Problem is Solved!, from professor Jeffrey Shalit at Recursivity.  It is well explained, and the tie-in to McNuggets is delicious.

Chapter 7: For Smarties Only, NOT!

Secret Blogging Seminar is a group blog by 8 recent and future Berkeley mathematics Ph.D.’s. Sound intimidating?  Not always.  Take, for example, Scott Carnahan’s piece p-adic fields for beginners.  You could learn something.  Then there is Noah Snyder’s The Minkowski Bound. Not frivolous!

The following rating system I devised, analogous to the movie rating system (G, PG, PG-13, R, NC-17), should help you identify appropriate articles:-)

• γ = Graduate Level
• Ï€ Γ = Post-Graduate
• Ï€ Γ – XIII = 13-year old Post-Graduate
• ρ = Rudimentary
• ν Χ – XVII= Nearly Comatose 17-year old

Epilogue: Off to the movies

That’s where I’m going.  But you should stay and exercize your brain here (thanks Alvaro Fernandez & Praveen).  Oh, I wanted to congratulate Sherry Gong, from Exeter, N.H., who earned a gold medal and tied for first place at the 2007 China Mathematical Olympiad for Girls, which was held in Wuhan, China, from August 11-16.

 

That’s right, on 24-August I will be publishing the 15^th CoM.  Right again, Alon mispelt the name of the blog, substiuting ‘blog’ for ‘bog’ in a mispelt bog, which is just irony (pun intended).

Anyway, send your entries to the Carnival through here or direct an email to johnkemeny at yahoo dot com.  It would be helpful to include ‘Carnival’ or ‘CoM’ in the subject line.

 

As reported by the BA, the Space Shuttle (STS-117) and the International Space Station (ISS) are flying tandem, and should be visible from Westford tonight 9:17 through 9:24 pm (details at heavens-above, btw., Distan=distance in km).  Unfortunately, the cloud cover is predicted overcast here.  But if it is clear where you are, look up! You won’t even need binoculars.