## Tuesday, 30 December 2008

### Division of Fractions by the Alien Method

The day before Christmas break one of my seminar students brought in the old (1951) video of "The The Day the Earth Stood Still". I worked at my desk as they watched, and about thirty minutes in they called my attention to ask if the math on the blackboard was "real". The Alien in the movie, Klatu(Michael Rennie), in the company of a young boy who lived in the house where he was renting a room, had entered the home of a professor who was supposedly knowledgable about Astro Physics. I did not recognize any physics I knew from the brief shot of what looked like differential equations of no particular relation, but that could be my limited physics more than the actual images.
I returned to work, but in a few minutes in another scene, Klatu is helping Bobby with his homework and the only line you hear is "All you have to remember is first find the common denominator, and then divide." My head pops up... what were they doing? "Common denominators" leads to thoughts of fractions, but almost no one teaches finding common denominators as a prelude to dividing fractions (which is sort of a shame because it makes division of fractions work like multiplication...the way kids think it should.) It works in fact, if you do not find the common denominator first, but sometimes the answer is as confusing as the problem.
When you multiply fractions, as every fifth grader learns, you just multiply top times top and bottom times bottom... 2/3 x 5/7 = 10/21. The fact that division works the same way is often missed, or misunderstood because it so often leads to nothing simpler... 2/3 divided by 5/7 is indeed (2 divided by 5) over (3 divided by 7) but that seems not to give the classic simple fraction we seek. For some fractions, it will work out fine... if 4/27 is divded by 2/3, the answer is (four divided by two ) over (27 divided by 3) = 2/9 and that is the answer you get by the method you memorized (but never understood, most likely) in the fifth grade.
But what if we follow the advice of the alien Klatu. If we convert 2/3 and 5/7 to fractions with a common denominator, we get 14/21 and 15/21, and if we divide top by top and bottom by bottom we get 14/15) over 1, which is just 14/15... job done...
I can imagine including some visuals and suggestive images to help it make sense... It is after all, just a reversal of the multiplication process. If we say "3 dogs times 5 = 15 dogs" then by division we should have the eqivalent expressions that "15 dogs divided by 3 = 5 dogs." and just as naturally "15 dogs divided by 5dogs = 3" . Students who have learned (I've been in England too long, I just had to edit "learnt") that "eighths" and "fifths" are just units like "dogs" and "kittens" should then undertand that 5 eighths divided by three eighths is just as clearly 5/3.

## Monday, 29 December 2008

### Viete on Pythagorean Triples

Reading The Analytic Art by Francois Viete, or at least the T R Witmer translation, and came across an interesting way of combining the legs of any two Pythagorean triples to create two others. Viete calls the two methods synaeresis and diaeresis, which seem to be language terms Viete appropriated. Synaeresis is cramming two vowel sounds together to make one... like the way people in New Orleans say "Nor"leans. I think the official term is diphthong, but check with an English major for confirmation. The actual Greek roots mean “a joining or bringing together" or something similar Diaeresis is stretching one vowel out into two....and you can find your own example...

To illustrate Viete's approach, we can take two simple right triangles, say a 3-4-5 and a 5-12-13 as examples. Viete's method would produce two triangles whose hypotenuses( hypotenii?) were both 5x13 = 65 units. Viete distinguished between the legs calling them base and the perpendicular, so in the 3-4-5 triangle the base is 3 and the perpendicular is 4. It doesn't matter which is called what name, of course except that it reverses the outcomes of the two methods. The Synaeresic method would be to add the products of each base with the perpendicular of the other triangle; 3x12+ 4x5 = 56. This would give one leg of the new triangle. To find the other leg take the difference of the products of the two bases from the two perpendiculars; 4x12 - 3x5 = 33. This completes a triple of 33-56-65.

The second method, simply reverses the signs of conjunction. Subtract the two perpendicular x base products and add the two products of a common part. The crossed terms gives 3x12-4x5 = 16 for one leg, while the products of like parts gives 4x12+3x5=63 for the other, completing a 16-63-65 right triangle.

## Saturday, 27 December 2008

### Just Fold and Crease

If you think Origami is child's play, watch...

From the Ted descrtiption .......

"Lang creates creatures of such complexity that it seems impossible that each is composed of a single sheet of paper, no cuts, no glue."

"Robert Lang is a pioneer of the newest kind of origami -- using math and engineering principles to fold mind-blowingly intricate designs that are beautiful and, sometimes, very useful. "
Origami, as Robert Lang describes it, is simple: "You take a creature, you combine it with a square, and you get an origami figure." But Lang's own description belies the technicality of his art; indeed, his creations inspire awe by sheer force of their intricacy. His repertoire includes a snake with one thousand scales, a two-foot-tall allosaurus skeleton, and a perfect replica of a Black Forest cuckoo clock. Each work is the result of software (which Lang himself pioneered) that manipulates thousands of mathematical calculations in the production of a "folding map" of a single creature.

The marriage of mathematics and origami harkens back to Lang's own childhood. As a first-grader, Lang proved far too clever for elementary mathematics and quickly became bored, prompting his teacher to give him a book on origami. His acuity for mathematics would lead him to become a physicist at the California Institute of Technology, and the owner of nearly fifty patents on lasers and optoelectronics. Now a professional origami master, Lang practices his craft as both artist and engineer, one day folding the smallest of insects and the next the largest of space-bound telescope lenses.

## Wednesday, 24 December 2008

### Merry Christmas to Everyone

Merry Christmas...

Hoping that everyone has a wonderful holiday season. And that you and yours are wam, safe and healthy...

Looking at the dozens of gifts under my tree, and aware that I don't really need anything, I am led to think about all the people in the world who will go hungry, cold, and ill this Christmas.

While such ideas were percolating through my head, Erin Fry sent me a note about some people who are out there making a difference, and so I thought I would tell you about a few ways I think we can help out..

My students contribute to Water for People and it is one of my two favorite charities. What can be a more basic human need than clean water, yet for much of the world it is difficult or impossible to come by.

I just learned about el puente a charity in Costa Rica, it started out sharing a dish of soup with a local who searched through their garbage cans for food, and has become a source of support for poor in the area to feed, help pay the educational support, and provide microloans. "Although public education is free in Costa Rica, there are mandatory costs of attending a public school for things such as shoes, uniforms, books, supplies, and class photographs. The "start-up" expenses total more than $156 per child, with ongoing expenses of nearly$2.25 per week, to include breakfast/lunch and copy charges for exams. The Bridge helps families meet this need. " Sounds like the kind of thing a teacher like me could support, and I do... How about you?

If you are not familiar with micro-loans, it is a great way to get money into the hands of the poor around the world and build an economic base instead of just continued dependence on someone else. The hard part is having someone in all the regions of the world to manage the loans... Presto! Kiva to the rescue.. If you can't afford to donate in this economy, how about a zero interest loan for six months to a year.... build the third world economy and then get your money back... or reinvest it in another Kiva loan... you can even screen your own loans. Kiva is one of my personal favorite charities.

Another great project is the Brenda Boone Hope Center Foundation in Meru, Kenya.
They house, feed, educate and care for up to 100 girls at a time who have experienced or are at risk of sexual, physical, or psychological abuse. The services are focused on the resuce of girls that have experienced rape and provides medical and legal service. They also provide training and certification school for those girls who had to drop out of school. See more below

## Saturday, 20 December 2008

### A Play on Words

It never happened, but it could have...
Student reading a medical Journal to Teacher: What is the meaning of ideopathic?
Teacher, without looking up: "I don't know."
Student: "Ok, I'll look it up."

He walks to the dicitionary and looks it up... moments later the dictionary is slammed shut, and as he passes the teacher..."That was NOT funny."
Teacher:"My personal feeling is that it would be, if you checked the etymology."

If you are curious, or confused, see here and don't forget to check the etymology.

## Thursday, 18 December 2008

### unExpected Value and ilLogic

Dan Gilbert, a Harvard Psychologist, begins this talk with Bernoulli's formula for expected value and goes on to give a nice explanation of some things having to do with conditional probability, happyness and human judgement. Some nice illustrations of the how human's make (often poor) decisions. Enjoy

## Wednesday, 17 December 2008

### Another Approach to Pythagorean Triples

Click on image to enlarge

I wrote a couple of posts a while back on the Barning Tree method of finding Pythagorean triples using matrices, and then a followup. I recently came across another approach to Pythagorean triples that involves a clever relationship between points on the positive y-axis, and points on the unit circle. (ok, maybe I should have known this, but I didn't.. and I think it is really a neat idea)

On the unit circle, x2 + y 2 = 1, if we draw a secant from the point (-1,0) through (0,t) on the y-axis, it turns out that if t is a rational number, then the coordinates of P=(x,y) where the secant intersects the circle, will also be rational. Since the slope of the line is also t, the equation is y=tx+t ... and so and t2= (1-x2)/(1+x)2. That means t = y/(x+1) which leads to x= (1-t^2)/(1+t^2) and y= (2t)/(1+t^2) If we pick some rational number to be t, say t=2/7,
then x= 45/53 and y= 28/53.... Then by similar triangles, there must be a circle with radius 53 and a point on the circle would be (28, 45) and in fact 282+452=532... and any such rational point will produce another Image of unit circle

## Tuesday, 16 December 2008

### Play That Funky Number, Math Guy

My pre-calc kids are neck deep in trying to remember the 92 trig values around the unit circle (you know, sin(150o)=? etc...16 for each of the six trig functions) and I came across this on someone’s blog a few days ago... thought they might enjoy it..... If bad music math parodies make you smile, this could be your Christmas present. Presented by that incomparable mathematical duo, Al G Bra and Cal Q Lus, a little ditty about trigonometry...

If you are a student, let your parents listen. They may actually remember the original, which you can hear here .

## Sunday, 14 December 2008

### The Rational Mean

In my last post I mentioned the mediant of a pair of fractions and its application in what is called the Farey Sequence. I mentioned that the property was considered a "curiosity" in mathematical number theory. Then I was called to task for this usage by Domingo Gómez Morín who has spent a considerable amount of time trying to convince people that the mediant, or rational mean as he prefers to call it, is a valuable mathematical tool. He has some nice work on a web page where he shows that the rational mean is a) a general case of both the harmonic and arithmetic means as well as the geometric (not shown on his page) and golden mean, and b) an excellent tool for finding roots (both real and complex). He has thought through all this for so long that it would be a travesty for me to try to explain his work, so check it out for yourself. And thanks to Mr Morin for the reminder...he had sent me a link before and I have a note to look more deeply into it... now I shall.

## Saturday, 13 December 2008

### A Curious Property of Vulgar Fractions

John Farey was a geologist, not a mathematician, but he is better rememberd today for a single short (four paragraphs) paper he wrote in 1816 than for all his good works in geology. In that year he sent a paper to the Philosophical Magazine (or at least it was published in that year) called On a curious property of vulgar fractions .and described a pattern that appears in sequences of what we would today call common (vulgar is the latin term for common) fractions, like 3/8 etc, which are in simplest terms. They observation has almost NO practical use, not even to prove other things mathematically, and yet, it seems to have all kinds of interesting properties that tend to keep us fascinated with it. If you have never been introduced to it, here is a brief description, and some novel relationships that I think are interesting, with some links to places where they are made clearer than I could do in this brief space.

So First... What is it we are talking about? If you take ALL the fractions that could be written in simplest terms with a denominator less than some number n, say n=5 (since that is the one Farey used in his paper), and put them in order from lowest to highest... you get
The "curious" thing that Farey noticed is that if you ignore the way your fifth grade teacher taught you to add fractions, and do it the way YOU would have added them, "add the tops, add the bottoms", then each number in the sequence is the sum of the terms on each side of it... for example 1/4 and 2/5 are on each side of 1/3, and if you add by this approach you get (1+2)/(4+5) = 3/9 and that simplifies to 1/3 . The number obtained by adding two fractions in this fashion is often called the mediant .

OK, so that is how you make them. Our first question might be, how many of them are there? F(5) obviously has eleven terms (I counted). If we picked a value of N, what would be the number of fractions in the set F(N). A little investigation would show that F(1) = 2 (0 and 1); and F(2) = 3 (0, 1/2, and 1). So how many would be added to the next set... and the next... it turns out that each new set will have all the values of the previous set (of course) and will add one for every value of one through n that is co-prime (has no common divisors) to n. So the set for N=6 will have the eleven terms of F(5) plus 1/6 (one has no common factor with six), and 5/6. (notice that 2/6, 3/6, and 4/6 are already in the sequence in F(5) in simplified form)... thus F(6) has thirteen terms.. and in general we get a recursive formula that say the Order of F(N) = Order of F(n-1) + φ(n).

A second nice curiosity related to Farey Sequences are the Ford Circles. "Ford circles are named after American mathematician Lester R. Ford, Sr., who described them in an article in American Mathematical Monthly in 1938, volume 45, number 9, pages 586-601" (from Wikipedia). In fact, the
wikipedia article is a nice place to see how they work, and I need say no more as it shows the circles for F(5). What is amazing is that each circle is tangent to every other circle for a fraction it will be adjacent to in ANY sequence.... ahhh, go on..say "cool".

I decided to mention this when I came across another curious property of Farey sequences that relate them to lines on the plane and Pick's Theorem. If you treated each fraction a/b as a point (b,a) then none of the lines cross. If you make a triangle with the origin and any two adjacent Farey fractions, since each of the triangles have a determinant of one (meaning the area is 1/2) and therefore, by PIck's theorem, they cannot contain any other lattice points in their interior. A nice explanation of this, including the photo below, is at the Cut-The-Knot web site

## Wednesday, 10 December 2008

### Curious Properties of 17

Came across some notes on curious properties of the number 17 in a blog, called Mathnexus. Then today one of my students announced that she was 17 years old... So I shared them with her and the class......and now with you:

It is the only known prime that is equal to the sum of digits of its cube (173 = 4913, and 4 + 9 + 1 + 3 = 17)

It is the only prime that is the average of two consecutive Fibonacci numbers. ... (Ok, that would be 13 and 21... now the only way Fibonacci numbers can have an integer value is if they are both odd [there are no consecutive even Fibonacci numbers}, and all the even Fibonacci numbers are twice the average of the two previous values... so there can be no Fibonacci number after 34 which is twice a prime) It is interesting that there are lots of odd prime Fibonacci numbers, 2, 3, 5, 13, 89, 233, 1597, 28657, for example , Sloane's A005478, and each of them has a prime index (except three, which is f4).

It is the least integer such that the sum of its digits in every base B = 2, 3, 4, 5, 6, 7, 8 is prime. (In base two ,17 is represented as 10001 and 1+0+0+0+1=2, a prime number; in base three it becomes 122, 1+2+2=5; in base four it is 41, in base five it is 32; in base six it is 25; in base seven it is 23; and in base eight it is 21... but in base nine it becomes 18, and 1+8 = 9 is not prime.) which makes me wonder... is there a prime number for which the sum of the digits in all bases two through ten is also prime?

There are exactly 17 ways to express 17 as the sum of one or more primes.(can you list them all?)

And just one more, there is no odd Fibonacci number that is divisible by 17. (Ok, how special IS that?... are there other (odd) numbers that do not divide evenly into any of the odd Fibonacci numbers?..YES, the smallest is nine. A good strategy for attacking this kind of problem is given in this blog by Tanya Khovanova. Tanya goes on to state that none of the odd Fibonacci numbers are divisible by 19, 23, or 27 also, so maybe this really isn't such an unusual event at all.)

## Monday, 8 December 2008

Everyone knows the stress of testing can be detrimental to students, but it seems that some college professors have observed that it can be even more hazardous to their family members. Recently I came across a (very tongue-in-cheek) article by Mike Adams of the Biology Department at Eastern Connecticut State Univ. about the strength of this effect.

"It has long been theorized that the week prior to an exam is an extremely dangerous time for the relatives of college students. Ever since I began my teaching career, I heard vague comments, incomplete references and unfinished remarks, all alluding to the "Dead Grandmother Problem." Few colleagues would ever be explicit in their description of what they knew, but I quickly discovered that anyone who was involved in teaching at the college level would react to any mention of the concept. In my travels I found that a similar phenomenon is known in other countries. In England it is called the "Graveyard Grannies'' problem, in France the "Chere Grand'mere," while in Bulgaria it is inexplicably known as "The Toadstool Waxing Plan" (I may have had some problems here with the translation. Since the revolution this may have changed anyway.) Although the problem may be international in scope it is here in the USA that it reaches its culmination, so it is only fitting that the first warnings emanate here also."

"
The basic problem can be stated very simply: A student's grandmother is far more likely to die suddenly just before the student takes an exam, than at any other time of year.

"While this idea has long been a matter of conjecture or merely a part of the folklore of college teaching, I can now confirm that the phenomenon is real. For over twenty years I have collected data on this supposed relationship, and have not only confirmed what most faculty had suspected, but also found some additional aspects of this process that are of potential importance to the future of the country. The results presented in this report provide a chilling picture and should waken the profession and the general public to a serious health and sociological problem before it is too late."

The rest of the article goes on to "document" the existance of the effect, and prescribe potential interventions."

You can find the whole article here

## Thursday, 4 December 2008

### More Detail on Birthday Problems

A couple of email questions asked about the birthday problems... One questioned the assumption that births are not uniformly distributed in the months (or days of the month)... which is quite true, and worth backing up with some info, but it actually makes the probability of a match MORE likely at n=23 than it would be if the births were uniformly distributed.

A Math Trek article by Ivars Peterson has a table of monthly probabilities showing the daily frequency of birth each month. September seems to be the most popular month, but the differences are almost negligible in the total probability calculation.

A greater difference is due to the fact that in modern times, far fewer people are born on a weekend. Induced labor saves many doctors from a spoiled yachting weekend. The Fathom Graph below shows the distribution of birthdays for births in the U.S. in 1978. It was used by Professor Geoffrey Berresford in his article: "The uniformity assumption in the birthday problem, Math. Mag. 53 1980, no. 5, 286-288." If you plot a times series of the data you will have a nice example of periodic data. The saturdays and sundays show up well below the others, (yearday.jpg)... The atctual data can be found at the Chance Data Base at Dartmoth.

One more graph, this one from the Skeptical Inquirer on line magazine. It relates to the probabilty of a match or near match (one day apart) with n people. The curve shows the probability of a match on the vertical axis, and the number of people on the horizontal. The dots are for the traditional problem, and the solid line is the "near match" probability.

## Wednesday, 3 December 2008

### North of the Border, A visit to Scotland

Took a trip up to Edinburgh over the holiday (Thanksgiving in America, but it was St. Andrews day in Scotland) and had a wonderful time. Must have walked past the Sheraton five times before I noticed that they had huge models of the Neolithic Stone balls that resemble the Platonic Solids. In the picture above I am standing by one that shows the symmetry of the dodecahedron, and others show all the Platonic solids, as well as some other polyhedral models. The actual balls range from two to eight inches in diameter.

Historians usually date the knowledge of all five Platonic solids no earlier than about 500 BC, but the dates of man's discovery (creation) of the Platonic solids is made more complicated by the existance of these neolithic Scottish balls (the real ones, not the ones at the Sheraton) that have been unearthed dating back to about 2000 BC. I was made aware of these a few years ago by discussions on the Historia Matematica discussion group, and quote here from a posting of Dick Tahta:

"On neolithic carved stone balls: There are nearly 400 of these objects found in various sites in Scotland and now in various museums and private collections. They include various regular and semi-regular solids, alternatively they can be seen as arrangements of knobs - from 3 to 10 and then various numbers up to 160! Some of them are decorated, notably the tetrahedral Towie stone. now in the Edinburgh museum (which stocks an excellent coloured postcard). . I have a baked clay model of this stone, bought from a shop in Avebury, Wiltshire."

"J Frazer is quoted as taking the grooves to be meant for thongs so that the balls could be hurled through the air "uttering oracles in a whistling voice which a wizard was able to interpret". I have been unable to trace this quotation, and the author was unable to help me at the time I inquired. The Ashmolean museum, Oxford, has a number of balls, kept in a drawer - I have handled these and they are certainly as remarkable as Keith Critchlow has pointed out. "These neolithic objects display the regular mathematical symmetries normally associated with the Platonic solids, yet appear to be at least a thousand years before the time of either Pythagoras or Plato." (K Critchlow, Time stands still, London - Gordon Fraser, 1979, p133 - this book has some splendid photos of various stone balls.)"

There have been various attempts to guess at what the balls might have beenmade for. The nineteenth century archeologists who excavated them thought they might be weapons - whether as pike heads, or hurled from slings, or used in games or perhaps for divination. It seems that the balls were never found in personal graves, so it has been suggested they were a sort of ceremonial conch, a prized possession of the tribe. Contemporary archeologists tend to be more cautious. According to Dorothy Marshall, "there is so little hard fact to be extracted from the evidence available about the carved stone balls that postulation as to their evolution and use if very difficult." ( D Marshall, Carved stone balls, Proc Soc Antiq Scotland, 108 (1976-7) 40-72 - this is the most up-to-date and authoritative account. Some previous papers in the same journal are to be found in 11 (1874-6) 29-62 and 48 (1913-4) 407-20.)"

Edinburgh is also home to several campuses of the Napier University, one of which includes the home of the famous inventor of the logarithm, John Napie, Merchiston Tower, at Napier University, just off hwy 702 (Morningside drive) two miles or so from downtown Edinburgh...
. Nearby on Morningside Drive is the Eric Liddell Center, a living memorial to Eric Liddell, the rugby star and first Scottish Olympic gold-medalist (you remember, "Chariots of Fire!") which has been serving the local and wider Edinburgh community for more than twenty five years. (Liddell died in 1945 in China just before the end of the war). In an August 8, 2008 poll in The Scotsman newspaper Eric Liddell was voted as the most popular athlete Scotland has ever produced. Sometimes good guys finish first.

## Tuesday, 2 December 2008

### Sarah Palin is NOT alone!

Click on image to see full picture

It started out as just a joke. I sent a copy of a cartoon from xkcd (see above) to the local Gov/History teacher who is my resident Election junkie, Mike Keegan, that I thought he would like. He Responded with a survey that apparently was failed by most Americans and the people they elected to represent them.

The Story caption reads: "WASHINGTON (AFP) – US elected officials scored abysmally on a test measuring their civic knowledge, with an average grade of just 44 percent, the group that organized the exam said Thursday." It continues to add that "Ordinary citizens did not fare much better, scoring just 49 percent correct on the 33 exam questions compiled by the Intercollegiate Studies Institute (ISI)."

He also sent me a link to the quiz, so if you are brave enough to face the truth about what you know about Government and US History, click here... and good luck

## Monday, 1 December 2008

### More on Birthday Problems...

Four questions about Birthdays left unanswered, so I wanted to get to that. The first:

"What happens to the birthday problem if you switch to a different number of days in the year (say 400 or 1000) or what if we use weeks or months... in short, is there a general formula for the number of people to reach the p=1/2 that there is at least one match?" The problem can be attacked in the same way as shown above for any number of categories, but a nice approximation (I believe this was from Perci Diaconis) is that for n categories, the approximate number of selections to have a probability of 1/2 of a match is 1.2 times the square root of n. So for 1000 days it would be about 38 people. You can reach 95% confindence of a match by adjusting the formula to 2.5 times the square root of n. This means that you can be 95% sure of a match for the conventional 365 day case with only 48 people.

The second problem asks, "what is the number of people needed to make the probability that EVERYONE has a match equal to or greater than 1/2?" This is called the "Strong Birthday Problem." I will leave it to you to search the derivation of the solution, but it seems that with more than 3064 people, the probability becomes more than 1/2 that ALL of them share a common birthday.

"And what if we only wanted to have two people who came close. What is the number of people that must be present to make the probaiblity that at least two have birthdays no more than one day apart?" Well, a little research led me to this, "It turns out, for example, that it takes just 14 people in a room to have even odds of finding two birthdays that are identical or fall on consecutive days. Among seven people, there is about a 60 percent probability that two will have birthdays within a week of each other. Among four people, the probability that two will have birthdays within 30 days of each other is about 70 percent. " I was a little surprised that it took so few to have either a match or adjacent days. I would think that means that with the traditional 25 person classroom, you might not get a match, but you would almost certainly get a "near-match."

"...suppose a group (several hundred) are entering an auditorium, and you know that the first person to enter having a birthday that matches someone already inside will win a prize. Ok, you know not to go first, and if you go too late, it will already be gone.... so where in the line do you insert yourself to have the highest probability of being the first to enter and have a birthday match in the room? " Well if you look at the change in the probability of a match after each person enters the room. This requires only that we calculate p(n) − p(n − 1) and find its max value. It turns out to be twenty, so if you are the twentieth person in line you have a slightly higher chance of being the first match than people before or after you.

## Wednesday, 26 November 2008

### The Birthday Problems (Yes, there is more than one)

"Happy Birthday to YOU!!! ... ok, it's my mom's birthday and I was thinking about that old birthday problem. You've probably seen it. How many people must be in a room before the probability of at least two sharing a birth date (just month and year) is greater than 1/2?. Yeah, that one.

It is one of those problems that is best attacked by answering a different question, (there seem to be a lot of those in probability) what is the probability that NO two have a common birth date. If you begin by thinking of people entering a room one at a time, and calculate the probability that no two have a common birth date, then the probability after the nth person enters is given by P(n)=P(n-1)*(366-n) /365. We are assuming a 365 day year here.. so P(1) = 1 (if you are the only one in the room, there is no match)..
Now the next guy has 364/365 probability of preserving the NO MATCH and so on for each new body. After each calculation we check to see if the probability of NO MATCH is less than 1/2, in which case the proabability of a match would be more than 1/2. It turns out that at n=23 the probability of at least one match is slightly over 1/2, about .507.

Actually, it only seems like the problem has been around forever. I saw a quote that it was created in 1932 by Richard von Mises, the Austrian born scientist whose work spans several areas of engineering science, however, the earliest citation I can find is " Mises, R. von. "Über Aufteilungs--und Besetzungs-Wahrscheinlichkeiten." Revue de la Faculté des Sciences de l'Université d'Istanbul, N. S. 4, 145-163, 1939.", so you decide which is right.

After you play with that for a little while, you might want to tackle one of several different variations, that can take it well beyond a high school classroom exercise. For example, the classical problem ask for when there is at least one match, but if you ask for the number so that the probability is 1/2 of exactly one matched pair, it gets a little tougher (OK, a lot tougher). With modern technology, it can be examined by students withour some of the formal probability training if they are good at simulations. I gave this problem to one class and asked them to describe a way to simulate the event using a graphing calculator. The best answer I got was this one: a) Put n (for example 23) random integers from 1 to 365 in a list 1. b) sort the list in ascending order c) copy this list 2 and then replace the last number with zero and resort just this list (effectivly this moves the zero to the top and each number down one in the list from its natural position). d) Now do a logical L1=L2 store L3. This produces a list of ones when the n th term in the list is equal to the n+1 st term, ie, for each match. Sum this list to get the number of matches. e) repeat like crazy.

I thought it was a great way to solve it, although if you have Fathom or other good statistical software for simulations, it can be done much easier, but sometimes the software hides some of the understanding in the early stages of learning simulations. When I tried it (with Fathom) and using 23 for the sample size, and 5000 trials of the experiment, there were no matches in 2431 or 48.6% of them, there was exactly one match in 1829 or 36.6% of them, and to my surprise, there were 604 or 12% that had either two seperate matches, or three of a kind.... and if you do the arithmetic, that leaves another 136 or almost 3% that had multiple matches beyond these (I used the unique value function of the program so I only knew there were less than 21 unique numbers in those cases). With the student method, adjacent ones on the third list would identify triplets etc, while seperated ones would indicate multiple pairs.

A couple of examples you might want to explore, and I will try to get back to solutions if they are not submitted sooner:
1) What happens to the birthday problem if you switch to a different number of days in the year (say 400 or 1000) or what if we use weeks or months... in short, is there a general formula for the number of people to reach the p=1/2 that there is at least one match.

2) what is the number of people needed to make the probability that EVERYONE has a match equal to or greater than 1/2?

3) and what if we only wanted to have two people who came close. What is the number of people that must be present to make the probaiblity that at least two have birthdays no more than one day apart?

4) and finally, back to the original problem, suppose a group (several hundred) are entering an auditorium, and you know that the first person to enter having a birthday that matches someone already inside will win a prize. Ok, you know not to go first, and if you go too late, it will already be gone.... so where in the line do you insert yourself to have the highest probability of being the first to enter and have a birthday match in the room?

More on all of these later..good luck, and have fun..

## Sunday, 23 November 2008

### Whither the Schoolhouse

Recently reading Andrew Robinson's The Last Man Who Knew Everything, about the polymath Thomas Young, I came across a quote he wrote to his brother, "Although I have readily fallen in with the idea of assisting you in your learning, yet (there) is in reality very little that a person who is seriously and industriously disposed to improve may not obtain from books with more advantage than from a living instructor....Masters and mistresses are very necessary to compensate for want of inclination and exertion, but whoever would arrive at excellence must be self taught."

Yikes!!!. I am a teacher, one who struggles to "compensate for want of inclination and exertion" and I am finding I agree with him. I mean, REALLY strongly agree with him. And in the age of the internet, is not the availability of information more than ever before available to those who have the "inclination"? When you can independently peruse many of the great courses at MIT on line for free, is ignorance not a self-inflicted predicament?

All of which leads me to wonder what is the role of the typical school in the future. In many areas they have already been reduced to day-care for those intended to be held out of the workforce until a later date. My daughter in law teachers language and literature classes with more students than she has desks or textbooks, and this in a moderately wealthy community which is just outside Detroit. What educational atrocities will she face with the impending economic crisis in that area? Can the schools descend to such a state that home-schooling becomes the predominant choice of college bound students.

I can almost imagine a rebirth of that approach that was common here in England when knowledgeable people would support themselves by sitting up small instruction programs in a single topic where students would come and pay for instruction until they reached the level of competence they desired. An instruction guided mostly by independent study guided by the mentor with a few choice insights of the "masters." And if the public schools do not fade away completely, they will serve a purpose much like the gladiatorial combats of ancient Rome, to entertain the festering masses and keep them mostly off the streets while the talented few, with inclination and exertion, begin to widen the gap ever more greatly between the educational (and probably financial) haves and have-nots. I think the system will prevail, at some level for my grandchildren, but wonder at the education of their children.

Opinions are welcomed.

## Saturday, 22 November 2008

### Harmony, and Harmonic Problems

Things happen in threes according to the old myth, and in this case it was true. I was doing some research on the early history of a mathematical problem often called the "cistern" problem. You probably know the type; "If one pipe can fill a cistern in 6 hours and another can fill it in four hours, how long would it take both pipes working together." While I was working on that, I got a nice article sent to me on the first proof that the harmonic sequence diverges... and then, I was reading a blog by Dave Marain Math Notationsin which he posed a problem that asked, in its general form, given a square inscribed in a right triangle (with one corner at the right angle of the triangle), what is the length of a side of the square in terms of the legs of the triangle.

So what do all these have in common with each other. dare I say what makes them in "harmony"?.... the answer is Harmony, or at least the mathematical relationship of the harmonic mean.

To the early Greeks, if Nichomachus can be believed, all the means were descriptive of musical relations. Much is often made of the Harmonic Mean in relation to a musical sense, but this may not represent the Greek view. Euclid used the word enarmozein to describe a segment that just fits in a given circle. The word is a form of the word Harmozein which the more competent Greek Scholars tell me means to join or to fit together. Jeff Miller's Web site on the first use of Mathematical terms contains a reference to the very early origin of the harmonic mean, 'A surviving fragment of the work of Archytas of Tarentum (ca. 350 BC) states, 'There are three means in music: one is the arithmetic, the second is the geometric, and the third is the subcontrary, which they call harmonic.' The term harmonic mean was also used by Aristotle. "
My search for the early roots of the cistern problem had taken me back to Heron's Metre'seis around the year fifty of the common era. The problem became a staple in arithmetics and problem books and was used by Alcuin (775) and appears in the Lilavati of Bhaskara (1150). I found the illustration I used on the blog for The First Illustrated Arithmetic a few days ago, from the 1492 arithmetic, Trattato di aritmetica by Filippo Calandri.
The solution to a cistern problem is the harmonic mean of the times taken by each pipe. For example, one problem asks "If one pipe can fill a cistern in three hours, and a second can fill it in five hours, how fast will the two pipes take to fill the cistern if both are opened at once. The solution is given by finding the average rate of fill of the two rates, the harmonic mean of three and five, which is three and three-quarter hours. But as the name "mean" suggest, that's the average rate of the two so working together, they would take one-half the time, one and seven-eighths hours, or about an hour and 53 minutes.

The Harmonic mean is the reciprocal of the mean of the reciprocals of the values, so for values a and b, the harmonic mean is given by which for two numbers can be simplified to the more economical
Heron might have been the first recorded example of a cistern problem, but a problem calling on the reader to use the harmonic mean occurs even earlier in the Rhind Mathematical Papyrus, now located in the British Museum, in problem 76. The problem involves making loaves of bread with different qualities, but the solution is still the harmonic mean. (I have learned from David Singmaster's Chronology of Recreational Mathematics that the cistern problem appeared, perhaps 300 years before Heron's use, in China by Chiu Chang Suan Shu (around 150 BC).

The series of terms formed by the reciprocals of the positive integers is a common torment for college students in their first introduction to analysis. The sequence in which each number gets smaller and smaller seems to very slowly approach some upper limit. Even after adding 250,000,000 terms, the sum is still less than twenty, and yet... in the mid 1300's, Nichole d'Oresme showed that it will eventually pass any value you can name. In short, it diverges, slowly, very, very slowly, to infinity. Even when warned, it seems like students want to believe it converges. A well-known anecdote about a teacher trying to get student's to remember that it diverges goes:
"Today I said to the calculus students, “I know, you’re looking at this series and you don’t see what I’m warning you about. You look and it and you think, ‘I trust this series. I would take candy from this series. I would get in a car with this series.’ But I’m going to warn you, this series is out to get you. Always remember: The harmonic series diverges. Never forget it.”<\blockquote>

By the way, each number in the harmonic series is the harmonic mean of the numbers on each side of it (so 1/2 and 1/4 have a harmonic mean of 1/3), and in fact, of any numbers equally spaced away from it such as 1 and 1/5 also have a harmonic mean of 1/3.

And then, I came across that little problem of a square inscribed in a right triangle. If the two legs are a and b, then the sides of the square will have a length equal to the one-half the harmonic mean of a and b .  More generally, a square inscribe in any triangle with one side along a base will have sides equal to one half the harmonic mean of the base and the altitude to that base.  There are lots of other interesting problems that yeild to the use of the harmonic mean, and I mean to write again on that collection.

So I guess things do come in threes, unless I come across another one, but whether it comes in threes or fours, it all seems to work together, in perfect harmony.

## Wednesday, 19 November 2008

### Problem Solving?

Here is a neat little inspirational video my beautiful wife forwarded to me. You can draw your own meaning. For me it was a reminder of what I tell kids is the first rule of problem solving; "When you don't know what to do, do something!". Enjoy.

## Monday, 17 November 2008

### The First Illustrated Arithmetic

I was researching problems related to the harmonic mean (more of which I hope to share in a later blog or blogs) when I came across a note in David E. Smith's "History of Mathematics" (There are actually used copies for a nickel!) about Filippo Calandri's 1492 arithmetic, Trattato di aritmetica. Smith cites it as the first "illustrated" arithmetic, and checking around, David Singmaster seems to agree.
An actual copy is in the Metropolitan Museum of Art in New York, and they have some images from the woodcuts in the book posted here . The cut above was the one of interest to me as it describes a "cistern problem" which was one of the common recreational problems since the First Century, and one of the problems I was researching when I came across this. The book has another first, it seems to have been the first book to publish an example of long division essentially as we now know it.
Here are some additional notes from my web notes on division that pertain to the long division algorithm and five early methods that were used.

..... is the true ancestor of the method most used for long division in schools today, and was called a danda, "by giving". In his Capitalism and Arithmetic, Frank J Swetz gives “The rationale for this term was explained by Cataneo (1546), who noted that during the division process, after each subtraction of partial products, another figure from the dividend is ‘given’ to the remainder.” He also says that the first appearance in print of this method was in an arithmetic book by Calandri in 1491. The method was frequently called “the Italian method” even into the 20th century (Public School Arithmetic, by Baker and Bourne, 1961) although sometimes the term “Italian method” was used to describe a form of long division in which the partial products are omitted by doing the multiplication and subtraction in one step.

The early uses of this method tend to have the divisor on one side of the dividend, and the quotient on the other as the work is finished, as shown in the image below taken from the 1822 "The Common School Arithmetic : prepared for the use of academies and common schools in the United States" by Charles Davies. Swetz suggests that it remained on the right by custom after the galley method gave way to “the Italian method” in the 17th century. It was only the advent of decimal division, he says, and the greater need for alignment of decimal places, that the quotient was moved to above the number to be divided. In a recent Greasham College lecture by Robin Wilson at Barnard's Inn Hall in London, he credited the invention of the modern long division process to Briggs, "The first Gresham Professor of Geometry, in early 1597, was Henry Briggs, who invented the method of long division that we all learnt at school." (I assume he means with the quotient on top.)

I recently found a site called The Algorithm Collection Project. where the authors have tried to collect the long division process as used by different cultures around the world. Very few of the ones I saw actually put the quotient on top as American students are usually taught. In one interesting note, a respondent from Norway showed one method, then explained that s/he had been taught another way, and then demonstrates the common American algorithm, but adds a note that says, “but ‘no one’ is using this algorithm in Norway anymore.” I might point out that the colon, ":" seems to be the division symbol of choice if this sample can be generalized as it was used in Norway, Germany, Italy, and Denmark. The Spanish example uses the obelisk (which surprised me as it was used almost exclusively in English and American textbooks, and even then seldom beyond elementary school), and the other three use a modification of the "a danda" long division process. The method labled "Catalan" is like the "Italian Method" shown above where the partial products are omitted.

## Friday, 14 November 2008

### The Math Check Joke

My loving little sister from Ft. Worth got this one in the e-mail and forwarded it to me. It seems to have been going around for awhile under the title "How to Tell if you Pi**ed off a Mathematician", and mostly, with a mis-answer for the amount. The copy my sister sent to me had this explanation for the check amount:

Unfortunatly, if you look closly at the amount, it is NOT e2 pi, which would give the above amount (more or less). The exponent of e, as most mathematicians would expect, is in fact i * pi, where i is the imaginary constant. Leonhard Euler showed (although Cotes did a lot of the spade work for this) that ei pi is actually equal to -1. It is often called the most beautiful theorem in mathematics, and it is certainly one of the more useful as it allows us to tie the real and complex values together. I used the theorem not too long ago in a blog I Don't Get It! about a tongue in cheek quote from De Morgan.

Using the correct expression, the check turns out to be written for \$0.002, which is 2 mills, or two tenths of a cent. To me that makes the whole problem a little funnier and a little more interesting, but I'm not sure part of that is not a little intellectual snootieness (I get the joke and you probably won't); which I also spoke of not quite as recently Prime Time Fun.

## Thursday, 13 November 2008

### Some additional History on the Problem of Points

After my recent post on the probability history I got a nice note from Jim Kiernan advising me about an article he posted in the March, 2001 issue of The Mathematics Teacher. He is (or at least was according to the article) a teacher at the Edward R. Murrow High School in Brooklyn, NY. His interests were listed as math history, and in particular, the origins of probability and statistics.

The entire article is well worth the read, and if you are also interested in math/stats history, it is worth the trip just for the references. In particular, and I suspect that a stats-history buff like Jim already knows, but one of the references, A. W. F. (Tony) Edwards, not only wrote a really neat book on "Pascal's Arithmetical Triangle, The Story of an Idea", but he was also the last student under the tutelage of the great statistician, R. A. Fisher at Cambridge. My personal appreciation to Professor Edwards include thanks for a guided tour around the great hall at Gonville and Caius ( pronounced "keys") to view the stained glass tribute to John Venn (and several other math/science people), and for giving me directions to the (totally hidden in vines at the time) grave of John Venn (photo at top). In his book on Pascal, Professor Edwards points out that it was the problem of points that prompted Pascal to write his famous treatise.

I have taken the liberty of copying a few key remarks from Mr. Keirnan's article that compliment the previous post, and am grateful to Jim for the note.

As early as the twelfth century, the Arabs were acquainted with the binomial triangle and used it to solve problems that involved combinations. Islamic tradition also deals with problems of dividing inheritances. Tartaglia repeated the tradition that Leonardo of Pisa (ca. 1200), commonly known as Fibonacci, was responsible for bringing the practice of algebra to Italy from Arabia. "Although no mention of the "problem" is attributed to Leonardo, its origins apparently also lie with the Arabs. Oystein Ore refers to an Italian manuscript, dating from approximately 1380, that is probably of Arab origin and that contains the "problem." The Arabs seem to have had all the right tools, but no record of a solution exists.

Tartaglia and Cardano both tried (and failed) to solve the problem and he includes their wrong answers in the article. Then, Several other futile attempts were made to solve the problem before it fell into obscurity. Galileo wrote about probability, but no extant version of the problem appears in his papers. Widespread knowledge of the binomial triangle existed throughout Europe. It appears in the works of Cardano, Tartaglia, and Mersenne. Yet no record exists of anyone's applying it to the problem. Finally, during the summer of 1654, the problem was solved in three different ways as the result of a correspondence between two of the greatest French mathematicians of the seventeenth century: Blaise Pascal and Pierre de Fermat.

The correspondence began in response to a pair of questions submitted by the Chevalier de Mere. The second of these problems would be the catalyst for the founding of probability theory. The first letter from Fermat "on division" is missing, but Pascal (1952, p. 475) responded on 29 July that the "method is very reliable and is the first that had occurred to me." Pascal claims to have found a "different method much shorter and simpler." The letter ends with the heartwarming observation that "truth is the same at Toulouse and at Paris."

Pascal's first method can best be explained using the ideas of recursion and weighted averages. When a total of three games is required to win, he considers three cases: (2, 1), (2, 0), and (1, 0). The first case, (2, 1), is a simple example; the second, (2, 0), gives the answer to Pacioli's problem; and the third, (1, 0), gives the answer to de Mere's problem. In each case, a total of 32 pistoles is wagered by each player. This number seems to have been selected so that the solution would be a simple ratio.

Analyzing the simple case, "they now play a game ... if the first player wins, he wins all the money ... if the second player wins each should withdraw his own stake" (Pascal 1952, p. 475). The result is a split of either [64; 01 or [32; 32]. The first player is "sure of having 32... as for the other 32 ... let us share equally." So if the game is interrupted before the next round, the correct split should be [48; 16]. The second case reverts to the first case when the second player wins the next game. If the game is interrupted at (2, 0), the player who has two games should get 48 pistoles plus half of 16. So the correct split for this case and Pacioli's problem would be [56; 81, or 7 : 1 in simplest form.

De Mere's problem requires finding "the value ... when two players are playing for three games and ... one player has only one game and the other none" (Pascal 1952, p. 475). Using the process of recursion developed so far brings the situation back to the previous case. If the first player wins, the status becomes (2, 0), which entitles him to 56 pistoles. If the first player loses, the status is even, (1, 1), which entitles him to 32. So in the case of an interruption, the first player should get 32 plus half of (56 - 32). The correct split is [44; 20].

Pierre de Fermat's solution, dated 24 August, depended on determining the number of games required to declare a winner. If player 1 needs m games more and player 2 needs n games more to win, then a winner must be declared after m + n - 1 more games. Fermat then listed all possible outcomes for four more games

and formed the ratio of wins by each player where a is a win for player 1 and b is a win for player 2: aaaa 1 abaa 1 baaa 1 bbaa 1 aaab 1 abab 1 baab 1 bbab 2 aaba 1 abba 1 baba 1 bbba 2 aabb 1 abbb 2 babb 2 bbbb 2

"Therefore, they must share the sum in the ratio of 11 to 5" (Pascal 1952, p. 475).

This result is equivalent to Pascal's solution [44; 20]. Gilles de Roberval, a member of Pascal's intellectual circle in Paris, was not pleased with this means of listing outcomes. He criticized the use of four games when two or three would determine a win.

The last of the three methods used to solve this problem is contained in Pascal's Treatise on the Arithmetic Triangle, which was written in 1654 but not published until 1665.

Thanks again Jim...

## Tuesday, 11 November 2008

### White Rabbit Mathematics

One of the things that amazes me, and I think most people who are attracted to math, is the mysterious way that different parts of math come together in unexpected ways. I tried to explain this to someone once using a literary analogy..."It is as if you were reading along in some great drama, or trying to understand the message in some grand poem, and suddenly the White Rabbit from Alice in Wonderland comes running through muttering, "Oh dear! Oh dear! I shall be too late!"

It is not the White Rabbit you see in math, but the effect is the same. Euler must have felt that feeling after he struggled to find the value of the series .. and finds that it turns out to be . Wait.... Pi is the ratio of the circumference to the diameter of a circle, but there are no circles in the sum of the squares of the reciprocals of the integers; and yet, there it is, the mathematical white rabbit coming seemingly from nowhere. Certainly none of the many mathematicians of great repute who had worked on the problem found (or expected) Pi to appear.

The normal distribution is another example; De Moivre takes the binomial probability distribution for flipping a coin and generalizes it toward an infinite number of flips, and POW, the normal or bell-shaped curve that is ubiquitous in intro stats. And what happens? Right there in the middle, the height of the normal curve at Z=0 is .39894... No, NO, NO, NOT JUST .39894.. but the .39894... that is exactly equal to .

Ok, so what brought this sudden rebirth of excitement about mathematical interrelationships? Well recently I came across a blog that referred to another blog that (as these things sometimes do) led me to a paper on just such a mathematical "white rabbit". The paper was about partitions of numbers as powers of two (1, 2, 4, 8, 16, etc..)

It began with a simple question, what is the number of ways to write a number n as a sum of powers of two if each value can be expressed no more than two times. For example, we could express 4 as 4, or as 2+2, or as 2 + 1 + 1 since each value is a power of two, and none appears more than twice. You couldn't use 1+1+1+1 since it appears more than twice. For n= 4 it turns out that the number of partitions, as shown above, is three. If we assume that there is one way to express zero, and one way to express one, and figure out the others we get a string like this

1, 1, 2, 1, 3, 2, 3, 1, 4, 3, 5, 2, 5, 3, 4, 1, 5, 4, 7,..

Ok, you don't see a white rabbit yet... but then someone ask you a different question. Is it possible to write out ALL the rational numbers in simplified form without repeating any of them. The answer is "Yes, of course, see the list above."

"What?", you ask, "How?", but there it is... The sequence of rational numbers is formed by taking each of the numbers to be the numerator, and using the number behind it to be the denominator. 1/1; 1/2; 2/1; 1/3; 3/2; ... and you never get a repeat, never get an unsimplified form, and you eventually get them ALL, the entire Infinite Set.....

No way you would expect that partitions of powers of two should give you the rational numbers in their entirety... there is (it would seem) nothing to relate the two questions... and yet... there it is. I think that is what makes math the most exciting area of study in the world.

Prove it you say? Nope, In truth I ain't man enough, but you can find the entire paper
Recounting the rationals, by Neil Calkin and Herb Wilf. Read their proof and Enjoy.

## Saturday, 8 November 2008

### Some Early Probability History Notes

So we make a fair bet, I roll one die, you roll the other, and who ever gets the highest scores a point. If we tie, we just redo the roll, and the first one to five points wins. Easy enough, but then, when the score is three to one my favor, you get an emergency phone call and have to leave. How should we distribute the stakes?

It was just such a problem that formed the foundation of early probability, and when it was solved, it sparked a rapid development of problems, and applications of probability.

I would tell you more, but I just read a neat blog by Keith Devlin that covered just such a development, so here, in part, are the words of a master:

"The Unfinished Game,
The problem of the unfinished game, also known as the problem of the points, was described in a book on arithmetic and geometry written by the Italian mathematician Luca Pacioli in 1494, [The text was Summa de arithmetica, geometrica, proportioni et proportionalita, and you can view it here PAT] though it is known to predate that mention. It asks how the pot should be fairly divided when a multi-round tournament has to be abandoned before it is finished. For instance, suppose two players are rolling a pair of dice and agree to playa best of five rounds tournament. Three rounds are played, leaving one player ahead 2 to 1, at which point they must abandon the game. How should they divide the pot?

Pacioli was unable to solve this problem. So too were a number of other mathematicians (and gamblers) who tried, including Girolamo Cardano, Niccolo Tartaglia, and Lorenzo Forstani. The consensus was that the problem could not be solved.

Then, early in 1654, a gambler by the name of Antoine Gombaud, more often referred to in modern history books by his French nobleman's title of the Chevalier de Mere, asked his friend the mathematician Blaise Pascal. Pascal produced a complicated argument that can be made to work, but was not happy with it, so at a friend's urging he wrote to Fermat about it. Fermat quickly found a simple solution.

There are two rounds left unplayed, argued Fermat. In each round, either player can win, so there are in all four different ways the game could continue to its five-round completion. The player who has won one round to the other's two must win both those final rounds in order to win the contest; in the other three possible endings, the player who is ahead after three rounds will win. Therefore, said Fermat, the player who is ahead when the game is abandoned should take 3/4 of the pot, with the other player taking 1/4.

To anyone who sees this solution today, it seems simple enough. (The solution assumes the tournament is thought of as a "best-of-five" rounds, as opposed to a "first-to-three". You need a slightly more complicated argument in the latter case, but the answer is the same, a 3 to 1 division of the pot.) But no one before Fermat saw it, including Cardano who did work out all of the basic rules we use today to combine probabilities. Moreover, when he did see Fermat's solution, Pascal could not accept it, and nor could various of his colleagues he showed it to. What was their problem?

Since the computation is trivial, indeed no different from the calculation of the odds in any game of chance (and actually much simpler than many), the only thing that could be holding everyone back was the fact that what Fermat was counting were "possible futures." Something that two thousand years of received wisdom said was not possible.

Once word got out about Fermat's breakthrough, however - presumably through the highly mobile network of gambling European noblemen - it did not take long for others to jump into the "future prediction" act. Within a single lifespan, modern future prediction and risk management were in place.

The speed of developments that followed the solution to the problem of the unfinished game is staggering.

1657. Christian Huyghens writes a 16-page paper that lays out pretty well all of modern probability theory, including the notion of expectation, which he introduces.[This one is LIBELLUS DE RATIOCINIIS IN LUDO ALEAE and can be found here

1662. John Graunt, an English haberdasher, publishes an analysis of the London mortality tables, and in so doing establishes the beginnings of modern statistical inference.

1669. Huyghens uses his new probability theory to re-compute Graunt's mortality tables with greater precision.

1709. Nikolas Bernoulli writes a book describing applications of the new methods in the law. One problem he shows how to solve is how long must elapse after an individual goes missing before the court can declare him dead and allow his estate to be divided among his heirs.

1713. Jakob Bernoulli writes a book showing how the new probability theory can be used to predict the future in the everyday world. This is the first time the word "probability" is used in the precise, mathematical sense we use it today. He also proves the law of large numbers, of which more in a moment.

1732. The first American insurance company begins in Charleston, S.C., restricted to fire insurance.

1732. Edward Lloyd starts the precursor of what in 1734 becomes Lloyd's List, and eventually gives birth to the insurance company Lloyds of London.
1733. Abraham de Moivre discovers the bell curve, the icon of modern data collection.

1738. Daniel Bernoulli introduces the concept of utility to try to get a better handle on human decision making under uncertainty.

1760s. The first life insurance companies begin.

"

A pretty concise History for one blog... If you have additional notes to offer, please do.

## Tuesday, 4 November 2008

### Use Your Math Education, Count Fish!

Earn Big Bucks, Count Fish

Ok, maybe not BIG bucks, but it seems they need more people in the fisheries and wildlife areas to count things, and they are looking for mathematicians who can fill the bill. From an article on "Counting Fish," by Karen Kaplan. Nature, 16 October 2008.

"The Fisheries Service of the U.S. National Oceanic and Atmospheric Administration (NOAA) is looking for a few good fish counters." So beings this brief article in the Career View section of Nature. Scientists with a background in mathematics, computer science, and/or conservation are needed---and are in short supply---to fulfill positions as "stock-assessment scientists." Reports say that U.S. institutions will graduate only 160 such scientists to fill at least 340 positions. Stock assessment scientists "gather data on species populations, on the basis of catches and aerial surveys. The data inform mathematical models that help design monitoring programs and predict populations under different managemant scenarios. This in turn helps regulators to set quotas." One such scientist, Larry Alade, says "he's now in a job he loves---contributing to sustainability."

## Monday, 3 November 2008

### more on margins of error from NPR

Keith Devlin did a thing on NPR explaining how Polling Margins of Error work (the accepted view) but messed up on a couple of points in the history.....as pointed out by Peggie Lewis:
"Keith Devlin conflated two events when describing the polling errors in the Dewey Truman race in 1948. The main problem with that year's polling was that it stopped several weeks before the election and missed the late swing to Truman (See Scholastic News, 2008) The previous major polling error was made in 1936 and reported by the Litery Digest on the FDR Alf Landon race. The Digest mailed out straw ballots drawing names from telephone books and DMV records. Among many other things wrong with their polling methods was the biased sample this reflected (In the depression year of 1936, a sample drawn from those who owned autos and/or who had phones was necessarily a biased sample."

Thanks to Shelli Temple from the AP Stats group for bringing this to everyones attention.

## Sunday, 2 November 2008

### Political Polls and Margins of Error????

Well, we are almost to the election, which means an end, finally, to the interminable projection polls. Ok, I actually like statistics, but I'm not sure I accept that political polls are not playing a little fast and loose with the assumptions that are needed to compute confidence intervals. I love it when the election goes the wrong way and they have to come up with scenarios for WHY they blew it. Of course with so many of them out there making 95% confidence intervals, about five percent of the ones you hear SHOULD be wrong... but I think there is more to the problem than just that.

I came across a blog from Iowahawk ( I didn't provide a link because my students come here and some of his language is not the sort of thing I display for my students..they know all the words anyway, but they won't hear them from me) that had a nice expression of what I felt, so I stole parts of it shamelessly...

Statisticians love balls and urns. A typical Stats 101 midterm, for example, usually includes a question along these lines:

"You take a simple random sample of 1000 balls from an urn containing 120,000,000 red and blue balls, and your sample shows 450 red balls and 550 blue balls. Construct a 95% confidence interval for the true proportion of blue balls in the urn."
From this the typical Intro stats student can deduce that they are 95% certain the real proportion of blue balls in that urn is 55%, plus or minus 3.1% .

"This is, for all intents and purposes, how political pollsters compute the mysterious "margin of error," which has everything to do (and only to do) with pure mathematical sampling error. If you look at the formula above and round it just a smidge, you get a simple rule of thumb for the margin of error of a sampled probability:
Margin of Error = 1 / sqrt(n)

So if the sample size is 400, the margin of error is 1/20 = 5%; if the sample size is 625 the margin of error is 1/25 = 4%; if the sample size is 1000, it's about 3%.

"It works pretty well if you're interested in hypothetical colored balls in hypothetical urns, or survival rates of plants in a controlled experiment, or defects in a batch of factory products. It may even work well if you're interested in blind cola taste tests. But what if the thing you are studying doesn't quite fit the balls & urns template?"

What if 40% of the balls have personally chosen to live in an urn that you legally can't stick your hand into?

What if 50% of the balls who live in the legal urn explicitly refuse to let you select them?

What if the balls inside the urn are constantly interacting and talking and arguing with each other, and can decide to change their color on a whim?

What if you have to rely on the balls to report their own color, and some unknown number are probably lying to you?

What if you've been hired to count balls by a company who has endorsed blue as their favorite color?

What if you have outsourced the urn-ball counting to part-time temp balls, most of whom happen to be blue?

What if the balls inside the urn are listening to you counting out there, and it affects whether they want to be counted, and/or which color they want to be?

If one or more of the above statements are true, then the formula for margin of error simplifies to

Margin of Error = Who the heck knows?

## Thursday, 30 October 2008

### I Will STILL Derive

OK, after all those years of teaching calculus, it seems I've been saying it all wrong, or at least that is the opinion of several people on the AP calculus list who think that the verb "Derive" is not acceptable for "differentiate" which they suggest is better. I have even posted videos "I will Derive" parodies of the old "I will survive" song.

Ok, maybe they have a point...but I'm a little confused. If they want to get pedantic about not using "derive" then they certainly must avoid "derivative" (that which is derived, to obtain or receive from a source)..If you differentiate, then you get a differential; yes??? So if you want to find the derivative... well, you get my drift.

Derive is kind of a great word, it comes directly from the word river (which once meant the banks and not the river, hence the Riviera is the coast). In days of yore the draining of water out of the river and into the fields for irrigation was something like "derivering" and then worked its way into "deriving" (get it, to draw from a source). We use derive in that sense a lot in math for any type of deductive reasoning. To me it seems perfectly logical to talk about deriving the function for the slope from the original function, and calling the result of that derivation, the derivative.

Sorry guys, but I'm afraid that I've been doing it way to long to break the habit now, so I will continue to derive, but if I can remember I will try to tell my students that they may encounter a future professor who may dislike the usage, and they should be prepared for such an event... but honest, I don't much care which word they use, I just wish they would remember to apply the chain rule.

## Monday, 27 October 2008

### What! You Don't Believe in Aliens?

Casey, one of my Stats students, didn't believe in the evidence of alien intelligence on Earth. I decided that I would have to convince him with the one most certain piece of evidence imaginable, so I asked, "You know what a thremos bottle is, right; so what happens if I put hot coffee in a thermos?"

Casey: "It stays hot." almost as a question

ME: "Yes.... and what if I put cold milk in the thermos,,,, what then?"

Casey, uncertainly: "Ummm... well, ..it stays cold."

ME: " Un Huh... (in triumph) and HOW does the thermos know which I put in????" with a slow wink and a finger tapping the side of the head

I'm sure that Casey now believes, but if you are one of those people who STILL doesn't believe there is other intelligent life in the universe, here is the incontrovertable proof you have been waiting for from John Hodgman. A story about aliens, physics, time, space and the way all of these somehow contribute to a sweet, perfect memory of falling in love.

## Sunday, 26 October 2008

### Searching for Snarks

The happy band of mathematical warriors above joined me on Thursday night as we set out to the Center for Mathematical Sciences at Cambridge. Our quest was to discover all that Emeritus Gresham College Professor of Geometry Robin Wilson might know about "Lewis Caroll in Numberland." The lecture, which can only be described as "math-lite" was entertaining none the less, and made even better by the good companions who joined me.

Wilson is also, not by coincidence, the author of a book on the topic that is soon to be released in the US (and can be purchased in advance from Amazon at a healthy discount) entitled, Lewis Carroll in Numberland: His Fantastical Mathematical Logical Life.

Wilson explained that if Dodgeson (the real name of Lewis Carroll) had not written the "Alice" stories for which he is so well remembered, he might well be remembered for being one of the pioneering child photographers of the 19th century. And if he had not done either, he might be remembered as an accomplished mathematician and teacher who made contributions in the areas of Logic, algebra, geometry, and the mathematics of elections. Wilson points out that:
"Yet another interest of his was the study of voting patterns. Some of his recommendations were adopted in England, such as the rule that allows no results to be announced until all the voting booths have closed. Others, such as his various methods of proportional representation, were not. As the philosopher Sir Michael Dummett later remarked:

It is a matter for the deepest regret that Dodgson never completed the book he planned to write on this subject. Such was the lucidity of his exposition and mastery of this topic that it seems possible that, had he published it, the political history of Britain would have been significantly different."

He also credits Carroll with the invention of the modern method of seeding tennis matches:
Another interest of Dodgson's was the analysis of tennis tournaments:"At a lawn tennis tournament where I chanced to be a spectator, the present method of assigning prizes was brought to my notice by the lamentations of one player who had been beaten early in the contest, and who had the mortification of seeing the second prize carried off by a player whom he knew to be quite inferior to himself.
Let us take sixteen players, for example, ranked in order of merit, and let us organise a tournament with 1 playing 2, 3 playing 4, and so on. Then the winners of the first round will be 1, 3, 5, and so on; those of the second round will be 1, 5, 9 and 13; the final will then be won by player 1, defeating player 9 who wins the second prize but actually started in the lower half of the ranking.
To avoid this difficulty, he managed to devise a method for re-scheduling all the rounds so that the first three prizes go to the best three players, which presaged the present system of seeding.

For a brief discription of some of his work in Logic diagrams see My page here and for A different version of the Wilson's talk on Lewis Carroll given at Gresham College, look here

## Wednesday, 22 October 2008

### More on Price matrices for Primitive Pythagorean triples.

For the lack of a better term, I will call the 2x2 matrices used to describe Primitive Pythagorean Triples in my last blog as "the matrices".

One of the consequences of the way they are constructed is that the two numbers in the right column represent the difference and sum of the two numbers in the left column.

This allows you to construct the full 2x2 matrix when any two of the elements are known. This, in conjunction with the knowledge that each offspring on the Barning tree has an incenter that was one of the three excenters of the parent, allow us to produce the tree without the use of matrix transformations.

If we start with the 3,4,5 triangle, which has a matrix ) we have already established that the products of the bottom row and the two diagonals give the radii of the three excircles (1x3=3, 1x2=2, 2x3=6)... So we can find the three offspring of the 3,4,5 triangle by promoting each of these excircles to an incircle, as shown here:

Now by using the fact that the right hand column is made up of the difference and sum of the left hand values, we can complete the two missing values to find the descendants of the parent matrix.

But we don't really want to know the matrices, we want the Pythagorean triples. Fortunately it is easy to find the triple associated with any of the matrices. Multiply down the left column and double to get the even leg; multiply down the right column to get the odd leg, and then sum the products of the two diagonals to get the hypotenuse. So the matrix on the lower left with rows of 2,1 and 3,5 will have an even leg that is twice 2x3, or 12. Its odd leg is 1x5 = 5 and the hypotenuse is 1x5 + 1x3 = 13. That means one descendant of the 3,4,5 triangle is the 5, 12, 13. You can work out the other two.

If you are paying careful attention, we just added the excenters of the two legs to get the hypotenuse of the triangle... did you know that worked that way? How about this, you also get the hypotenuse if you subtract the top row product from the bottom one. Geometrically that would be subtracting the incenter from the excenter on the hypotenuse... Now go convince yourself that makes perfect sense.

If you find the determinant of a matrix, but ignore the signs and add everything (like we did when we added the diagonals of the 2x2) it is called the Permanent of the matrix. I had never seen this word before. Jeff Miller's web site on the first use of math words provided a little detail. The term seems to have been created by Cauchy. "In his book Permanents [9] H. Minc mentions that the name permanent is essentially due to Cauchy (1812) although the word as such was first used by Muir in 1882. Nevertheless a referee of one of Minc's earlier papers admonished him for inventing this ludicrous name!"