Wednesday, 30 July 2008

Math Symbols Are NOT All Created Equal

I have a friend named Dave Refro who writes and edits questions for one of those high stakes tests that is used for admission into certain graduate programs and uses his job as an excuse for his fascination with archiving old math journal articles. Some folks garden, Dave archives. He spends hours pouring through journals and abstracts and fits together articles with a common theme. If you read almost any math discussion on line, you will probably have come across one of Dave's responses to a question with numerous links to how the question was addressed, discussed, and argued over through history.

Fortunatly for me, Dave sometimes finds an article that he thinks might be of interest to me, and when he gets a stack of them, I get a big present in the mail and my wife knows I will be taking my meals in the den for a few days. In a stack of journal articles he sent recently, (THANKS Dave!) there was one particularly interesting article by Florian Cajori from 1923. In the article Cajori points out two interesting things about the equal sign that every one uses; and that is one of the interesting things he points out, is that EVERYONE uses it. Even in 1923, it was one of the most ubiquitous math symbols in the world and today there are still only about four math symbols that you could write and they would not only be understood, but written exactly the same way whether you found yourself in darkest Africa, the Far East, or downtown Los Angeles. It seems like the perfect symbol, and as Robert Recorde said when he created the symbol in 1557 in his "Whetstone of Witte", the use of "a pair of paralleles, or Gemowe(twin, from the same root as Gemini) lines of one length ... bicause noe 2 thynges can be moare equalle." In fact, Recorde's equal sign had much longer lines than is common today, sort of like == but longer .

Indeed, one wonders why it hadn't been thought of years before, and assume that it immediatly became the most common of mathematical notations....ahhh, but not so. The other thing Cajori commented on that I think would surprise young students is that it took over a hundred years for the symbol to become accepted. So what symbol did mathematicians use before the good old == signs? Well, many of them used nothing. The early development of algebra occurred with a very rhetorical approach. When people wanted to write 7x+5 = 26, they would say," the product of seven and some quantity when added to five will equal twenty-six." Ok, they probably said it in Latin, and sometimes they did write numbers in place of the words for numbers, but for equals, they often wrote out the Latin aequales or some variation of it. Frequently they used abbreviations instead of full words and so "p" would stand for plus and "m" for minus...and they would shorten aequales as "aeq" or just "ae". By the time that Recorde had his inspirational stroke, lots of other people had decided THEY had a really good symbol. A pair of vertical lines, ||, was used by Xylander (Wilhelm Holzman) in his translation of Diophantus, Arithmetica only a few years later, and Regiomontous had used a single horizontal line for equality almost a century earlier. Descartes used a which was probably drawn from the "ae" abbreviation for aequalis. Descartes symbol became a popular competitor on the continent, finding favor with Huygens and the Bernoulli's, while many of the he English mathematicians, Wallis, Barrow, and Newton, followed Recorde's lead. Others used the "gemowe" lines of Recorde for other meanings, Descartes used it to mean +/- in his Geometrie, and Johann Caramuel used them where we would use a decimal point, so Pi would be 3==1415 etc.

So what brought the divided world into a common accord? It took a brand new idea, a reveolutionary idea, the calculus. As if by divine providence, the two great minds that created the calculus, almost in unison, tended to publish their versions with a common symbol for equality, Recorde's "gemowe" lines. They disagreed on almost every other symbol they used, but in the last half of the 1600's and the early 1700's the = sign rose to world dominance. In Cajori's words, "The fact that both Newton and Leibniz used Recorde's symbol led to its general adoption."

If I can get my students to understand how long and difficult it is to get mathematicians to accept a symbol, perhaps they will not be too surprised if their College Prof goes into a rant when they use the symbol "ln" for the natural log... and if they accept that the symbol exists (honest, they don't all accept it's use), I can't begin to imagine how they will react if you pronounce it differently than they would. Wait for them to say it first!

Friday, 25 July 2008

Proof Without Words

I have stated previously how much I like "napkin" techniques that give quick calculations or estimates of a problem. I also like things like visual displays which essentially prove some mathematical idea. The one at the top of the page is from the cover of Roger Nelsen's "Proofs without words II.." which is way too expensive for a paperback, but I will probably break down and buy it.

The problem, is that, except for mathemtaiticans who already know the proof, it seldom convinces "without words". Most of my high school students will not look at this image and be able to explain easily and clearly why it shows that the limit as n goes to infinity of 1/4 + (1/4)2 + (1/4)3+ ... +(1/4)n is equal to 1/3. If I'm wrong, not generally, but in your particular case, then stay with me and read what's written, and tell me if you see it the way I do.

I think they will be able to see that the triangle is divided into thirds... by the colors, 1/3 purple, 1/3 orange, 1/3 white. But I don't know if they can see the series of powers of 1/4 going off to infinity. That's why they have high school teachers... and so here are some words to help make it more "visual"

Look at the largest white triangle... can you see it is 1/4 of the largest (outside) triangle?... Ok, now look at the line across the top of the biggest white triangle, it connects the midpoints of two legs of an equilateral triangle, so the triangle above this medial segment, the one with multiple smaller triangles in it, is exactly congruent to the Biggest white triangle and is 1/4 of the total area of the outside triangle also. This upper triangle is a scale model of the original outside triangle,with all the same colors in the same positions and the white triangle in it is 1/4 of the area of this upper replica. So the second largest white triangle is 1/4 of the area of the Largest white triangle.... its area is 1/16 or (1/4) 2. Now the line above the second smallest white triangle is a medial segment of the upper triangle, and so the triangle above it, which is also a scale model of the original biggest triangle, is also 1/16 of the total area... and the third smallest white triangle, is 1/4 of that, so its area is 1/4 of 1/16 or (1/4)3... OK, now you see it, and as you move out each white triangle is 1/4 of the previous one... and sure enough, all the white triangles add up to 1/3 of the total area.

If you are one of my students, next year remind me to say "similar triangles" a lot...if you are going to be in calculus, don't worry, we will

Tuesday, 22 July 2008

More Stuff That is “Easier Than You Think”

A couple of days ago, I mentioned that I had been shown how to test divisibility by seven, and by using the same method, I could develop a similar approach for any of the small (but difficult primes).. So today I want to show you how that works, and if you read carefully, you will be able to mentally determine if any number is prime up to 9000 ....go ahead, impress yourself.

first, how the seven rule works…. I’m going to talk about what big people call modular congruence, but don’t let that scare you. It just means that two numbers have the same remainder when divided by some specific quantity (in the first case, seven). So 9, 16, 23, etc are all congruent mod seven because they all have a remainder of 2 when we divide them by seven… see…easy..and if they are congruent to zero mod seven, that means they have no remainder…they are divisible by seven

Now a rule that should make sense if you think about it; is that any two numbers that have a congruence of zero, when added together will have a sum that is congruent to zero… In mod seven we could think of 14 and 21. Both are divisible by seven, so if we add then together we get another number divisible by seven. I know it is hard to believe you can come up with much from such a simple rule, but watch.

Lets think of some number n, and for a moment let us break it into two parts, the number made up of all the digits except the units digit, call that one A, and the units digit, call it B. Then N = 10A + B… As an example with 324 we would say A is 32 and B is 4 so 324 = 10( A) + B…. got it?

Now we don’t know if N is divisible by seven, but we notice that if we take 2 times N it will equal 20 A + 2B…. yeah, I hear you saying “So, what?”… patience. Now 20 A is equal to 21 A – A…. you know that… so we can write 2N =20 A + 2B = 21 A – A + 2B. NOW, we have something, because 21 A is divisible by 7, and so if –A+2B is divisible by seven, then N must be also by the rule above in italics. Now it doesn’t matter if –A+2B is positive or negative, so since A may be a multi-digit number it would be easier to do A-2B. So we want to know if 2947 is divisible by seven (it is). We take 294 (A) and subtract 2 x 7 and get 280. Still a big number so we can do it again. 28 – 2x0= 28…and HEY, I know that is divisible by 7, so the original number, 2947 must be.

Ok, can we make that trick work for other primes. We can skip 11 because it already has an easy divisibility rule… (see casting out nines and elevens). What about 13? Well 3x13 is 39, which is almost 40. That is the secret of these prime tests; we want a multiple that is one more or one less than an even multiple of ten (one less is easier as we will see, no subtraction). So we go back to N=10A + B, and we want to know if it is divisible by 13… Well, set 4N = 40 A + 4B and that is the same as 39A + A + 4B. Now the 39A is a multiple of 13, so if the A+4B is a multiple of 13, then 4N, and hence N is a multiple of 13. So we just use the rule A+4B to test a number…. Like 403. 40 + 4x3 = 52… and if you didn’t recognize that as a multiple of 13, you could do 52 as 5 + 4(2) = 13..hey, we ALL recognize that is divisible by 13.

Here are some tests for other primes I worked out by the same approach, and they are pretty easy to remember by just two rules of thumb. Find a multiple of the prime that is one more or one less than a multiple of ten. If it is one more then you have to subtract some multiple of B from A, and if it is one less, then you add some multiple of B to A. And the multiple of B, it is just the number of tens that we are one more or one less than (don’t get confused, it is NOT how many times we multiply the prime to get there). Try making sense of how these were done, then you can make your own for even higher primes (although they start to have some big multiples, but 39 should be an easy one)…

Test 7 by using A- 2B
Test 13 by using A + 4B
Test 17 by using A- 5B
Test 19 by using A+2B (see, it is already just one less than twenty)
Test 23 by using A+ 7B (OK, that is the hardest one in the group, sorry)
and test 29 (one less than 30) by using A+3B.

Any number less than 9000 must have a prime factor less than 30 if it is not prime, since 302 = 9000, so if you come across some big number like 8531 and wonder if it is prime, just try the rules one at a time.

3 sevens is 21 one more than two tens so the rule is A-2B… 853-2=851… 85-2 = 83 and 7 won’t go into that evenly so it is NOT divisible by 7.

Not divisible by 11 so we try 13, 3x13 = 39 which is one less than 40, so we add A + 4B… 853+4 = 857… 85+4(7)= 109… 10 + 4(9) = 46…nope, 13 won’t go into that… on to 17

3 x 17 is 51 so we need to subtract A – 5B…… 853-5(1) = 848… 84-5(8)= 44, nope 17 won’t go into 44 evenly… let’s try 19.

19 is already one less than 20 so we test with A + 2B… 8531.... 853+2(1)= 855 .... 85+2(5)= 95 ... 9+2(5) = 19…hey, it is divisible 19, and in fact, 8531 = 19 x 449. ….”easy peasy” as the British folk say.

Sunday, 20 July 2008

Collateral Damage in the Math Wars

I'm a victim of friendly fire in the math wars...wait, scratch that... I'm a victim of used-to-be friends fire.... you apparently have to be on one side or the other, and recently I am catching hell from both sides.

I guess it is my own fault. For the last ten-plus years I simply refused to enter into any dialog involving the "New Math/intuitive math/discovery math" camp on the one side and the "gimme that old time religion/long division or die/calculators are evil" camp on the other. The problem, at least as I see it, is that each group is assuming that it has to be all one or all the other. As one blogger put it, it is a choice between "why" vs. "how".(No one apparently envisions that a teacher might actually try to teach both how AND why.) California has fallen aside as the chief battleground after a victory, more or less, by a coalition of anti-reform elements (many of whom would not be caught dead championing the excess of the most conservative among them), and now Washington state seems to be the new battleground..(see A University View)

I can understand parents getting upset... as one observer of the process wrote, "Parents .... don’t understand what the specific tasks are, or how to perform them, or even why their kids are being taught this way instead of the older one. And again the proponents of this style (rehashing the why vs. how discussion) come off as arrogant. Their concerns aren’t for the parents’ ability to follow along with what their children are learning — and something tells me these are the same educators who insist that parental involvement is key — but just that the parents aren’t screwing up all their hard work." I admit I was shocked to read the following from the teachers resource packet for a program called "Everyday Learning"...
"The authors of Everyday Mathematics do not believe it is worth students’ time and effort to fully develop highly efficient paper-and-pencil algorithms for all possible whole-number, fraction, and decimal division problems. Mastery of the intricacies of such algorithms is a huge endeavor, one that experience tells us is doomed to failure for many students. It is simply counter-productive to invest many hours of precious class time on such algorithms. The mathematical payoff is not worth the cost, particularly because quotients can be found quickly and accurately with a calculator." I am reminded of a quote from my wall at school,” For every problem there is a solution which is quick, easy, and wrong! "

For me the intuitive learning and process learning support each other; the kids learns the multiplication algorithm, then he sees in algebra that two digit multiplication is the same as the "foil" method he is learning, and the rigor and intuition feed each other. One guy calls it the Mr. Miyagi method (from the Karate Kid movies). OK, that means teach the algorithm, but don't expect every kid to achieve 100% accuracy on the most difficult problems in the fourth grade, and then show them the intuitive approach that helps them understand WHY the algorithm works

Strangely, most of the good HS math teachers I know support a mixture approach; learn the long division algorithm, learn to multiply fractions, then use the mastry to understand concepts and big ideas..(see"Easier Than You Thought")Education Professors keep pushing the idea of concept based learning without foundation training (I think that is wrong) and Ultra conservatives keep pushing the abstract calculation as if mentioning a use of math was a sin (I think that's wrong too, most of us learn math to use it, only a very few use pure math untainted by application, and then when the least expect it, we find an application for what they did.)

So now I have two ex-correspondents who have , me off, each because I endorse their view too weakly, or more likely, because I am too tolerant of the opposition viewpoint. But I will go on teaching long division and multiplication, factoring and all the other taboo subjects and trying my best to balance them with an intuitive understanding of why these ideas work, why the are actually esthetically beautiful ..at least until some administrative zealot of one or the other side of the math wars tells me to pack my bags and leave. But even them, look for me in your public park standing on a soap box selling my "evil" to anyone who will listen. But if you get caught in the Math wars, my best advice is from those old Uncle Remus stories by Joel Chandler Harris..."Ol’ Brer' Fox, he don't say nothing. He just lay low."

Friday, 18 July 2008

EASY????

Some calculations are easier than you think! I’m drawn to math that can be done on a napkin at a restaurant. Quick mental calculations and methods of approximation that give you the ball park quick answer…”It’s about …blah.” and then I try to look as if I don’t think I’m really clever. I had a couple of these come up recently, so I thought I would share…

One was at a backyard barbecue…an electrical engineer, a graduate physics student, and a math teacher, beers in hand as the hamburgers burned and the hot dogs turned to charcoal… One had just come back from New York City and pointed out that the Empire State bldg. was 1224 feet high to the upper observatory… then someone brought up the old conjecture that a penny dropped from the Empire State building would kill you. The engineer suggests that in truth the penny would reach some terminal velocity that would keep it from actually killing anyone. The physics student suggests that it would not be too hard to figure out using differential equations if we just ignored the air resistance, but he didn’t have his calculator (which apparently is his main resource for doing integration). The math teacher offers, “Well, with only high school math and physics, and a little mental calculation, it should be about 280 feet per second.”

A long silence, followed by “How did you get that number?” So I explained. In physics the big ideas often provide great simplifications…and in this case the big idea was the conservation of energy laws…. The total energy remains constant. The Penny on the 102nd floor has a potential energy equal to its weight times its height. When it hits the ground the potential energy has all been converted to kinetic energy (and a very small amount of heat due to air friction which I ignore) . Epot= m g h and Ekin = ½ m v2. Cancel the mass on each side and we have gh = ½ v2. If we use the common value of 32 ft/sec2 for the gravitational force, then we can simplify this all down to v2= 64 h.. or taking square roots, we get v= 8 times the square root of h…. Now the square root of 1224 is about the same as the square root of 1225 which is 35 (one of those multiplication tricks I teach my students each year) and 8 x 35 = 280, so the answer is about 280 ft/second.

They both look incredulous… so I offer a second approach… OK, an object dropped from rest will fall 16t2 feet in t seconds. To fall 1224 seconds it would take t=sqrt(1224/16) and taking the square root of top and bottom we get 35/4 seconds… OK, so velocity is 32 times t, so 32 times 35/4 = (wait for it…. ) 8 x 35 = 280 again… “Easier than you thought… right

The second event corrected a misconception I have had for years. I have mentioned earlier that one of my preoccupations on the road is looking at mile markers and license plate numbers and trying to factor them. I know all the easy tricks for divisibility by small numbers, except seven…. The rules for divisibility by seven always seemed harder than just dividing by seven, so I was surprised to read a note from a guy who seems to have invented (I never saw it before) an easy way to test divisibility by seven

Without going into the derivation, here is the rule. Take the number, call it N, and think of it as 10A + B. For example if the number is 2345 the A = 234 and the B= 5. Now the method is just to calculate A-2B and if that answer is divisible by seven, so is the original. And if you don’t know, just reapply the same rule again. For example 2345 is divisible by 7 only if 234-10 = 224 is also….which can be applied again with 22-8 = 14… ohh….. I KNOW that is a multiple of seven, so 2345 is also a multiple of seven, in fact it is 7 x 335.

This is getting a little long, so on a later day I’ll explain how he found it, and even show you a method I found for testing 13 which is almost as easy, and another for testing divisibility by 17… as the tv folks say….stay tuned

Thursday, 17 July 2008

Two Problems

OK, take off your political blinders, whether you are conservative or liberal, and read this comparison like a mathematician/statistician. It is a comment about large multi-national companies and their relation with smaller nation-states. Be warned, there is a major flaw (and many minor ones) in this argument… find it :
”In Nigeria, a relatively economically strong country, the GDP is $99 billion. The net worth of Exxon is$119 billion. ‘When multi-nationals have a net worth higher than the GDP of the country in which they operate, what kind of power relationship are we talking about?’ asks Laura Morosini. [Source: ‘Impunity for Multinationals’, ATTAC, 11 Sept 2002”

Ok, little things that niggle (yes, that IS a word) at me; Exxon’s net worth includes the value of off-shore oil rigs in the Gulf of Mexico and office buildings in downtown Houston (or wherever). It would not be practical to think they could use all of that value to bring to bear in a competitive situations…. And GDP is the value of goods and services sold in the country…. You sell me a watermelon for $3.00 and it goes into the GDP, you spend that$3.00 for (part of ) the payment on a car, the $3.00 counts again…. so Nigeria couldn’t use all that in a conflict with Exxon (footnote, when the Exxon Valdez went aground and spread crude oil over large parts of Alaska’s waterfront, the GDP of Alaska went up… all those people cleaning up sludge, even if they were volunteers were eating food, renting motel rooms… driving up the GDP… see, disasters are GOOD for you)….. but neither of those are the BIG one…. Think… what is being compared??… OK, here is a similar comparison, I have$1000 in my bank account (not really, I’m practically broke, but hypothetically) and you make $8.50 an hour at McDonalds, so I have an economic advantage in trade with you…… see it, one is a rate and one is a fixed value. In the Exxon/Nigeria situation, the GDP is measured each year…just by custom, and the value of Exxon is just Dollars, a fixed value. What if it was the custom to measure GDP on a ten year basis. Now we would have a GDP of$990 billion for Nigeria, and poor old Exxon looks like a chump with only $119 Billion. The idea in Physics and math class was called units analysis. If you needed newtons, which is Kg meters/sec2, then you needed to multiply mass times distance and divide by the square of time.. and with comparisons, they don’t make sense if you are not comparing the same thing…. My car gets 26 miles to the gallon and the Eiffel Tower is about 324 meters high??? Yeah… and so what? The trouble is, when people like Ms. Morosini (a wonderful person I’m sure) make comparisons like this, you have to do more than just nod and smile…. THINK, ask questions….. OK, second problem… What is the least likely group to be hurt by the increased gasoline pump prices? Did you say “People who don’t own cars”? You would think that, but I just read in an AARP article (OK, I’m old, put your big girl panties on and deal with it) that gas prices are causing a big drop in volunteers to deliver meals-on-wheels programs to shut-ins. Without sufficient manpower, many have resorted to delivering one hot meal and several prepared frozen meals so that they don’t come as often. In addition, as the economy goes south, the people who donated money and food for the program and reducing their contribution. Ok, children; Ignore the signs…DO feed the old people. One of those fundamental rules of Economics is called the Law of Unexpected Consequences. It’s a good one to keep in mind, especially if you think you have the solution to any economic problem. Wednesday, 16 July 2008 BI-MODAL??? Several years ago at the AP Stats reading I remember Ann Watkins (Cal State, Northridge) talking about how rare bimodal distributions are, at least that lots of things we think might be bimodal, really aren’t. The classic example we always talked about in class (before enlightenment) was distribution of heights when both men and women were included. Turns out it just isn’t so… according to Ann’s speech . So I took some statistics from the National Health Service in the US that said the average height of white males over 20 was 70.2 inches, and for females it was 64.6 inches. Then I pulled out my trusty TI-84. It seems incredible that almost none of the pages which give average height pay any attention to the standard deviation; but since the distributions shown led me to believe it was around 2.5 inches for both groups, I used a generous 3.0 inches for the std. dev. of both groups. In the first image shown the two distributions are shown separately with a window ranging from 60 to 75 inches, with a cursor on the male curve at 67 inches . If we combine the two, (I simply added the functions and divided by two) as if we measured the nations adults without regard to gender, the distribution looks like this, So how much would we have to separate them to get a double mode? Well, a lot it seems. Remember that the two means, 70.2 and 64.6, are already almost two standard deviations apart. I decided to see what would happen if we made the males even taller, so I moved them up to THREE standard deviations above the female height, 73.6 inches (ever girls dream, all the guys are over six feet). The resulting combined distribution looked like this. OK, so why bring this up now? Well, I recently read a report about the entry level salaries of newly minted lawyers (and minted is right for most of them). Most lawyers start out in private practice and make a pretty good salary, (way more than teachers), but some start out in public law and often (almost always) make considerably less. The image posted on the distribution is shown below, and is one of the most striking bimodal distributions I have ever seen for real data. Yeah, you want to know.. the lower mode is a bridge between the$40,000 and $50,000 (the data seems to have been reported in$5,000 increments). This was 11% of the total data set. And the fat-cats???, there modal hump was at $135,000 to$145,000 (fresh out of the box... wow!) with about17% of the data set.

OK….. TRUTH in Statistics time… data like this is biased by selective reporting… Some of the folks who got low salaries just didn’t send it in…..”Yes, I have the lowest starting salary of anyone in my class.” Others may have lied ( Yes, people do that) and made them a little higher than the truth; but even with very accurate reporting, this looks like one of those true bimodal distributions. OK, keep your eye out for others like this, and when you find them, send me a link… (wow only a month till school starts… )

Friday, 11 July 2008

Picturing statistics

Statisitcs is about data in context, but great statistics, convincing statistics, requires more. In a world where nearly half the people are mathematically illiterate, and way more than half believe the old saw about, "Liars, damned liars, and statistics" you have to be able to present information in a dramatic way. A way that shakes people out of the cold dull stares we reserve for passive entertainment. I think much of what Chris Jordan does falls into that "great" statistics category.

The Ted Talk description says, "Artist Chris Jordan shows us an arresting view of what Western culture looks like. His supersized images picture some almost unimaginable statistics -- like the astonishing number of paper cups we use every single day." How about this for a shocker...we use 4,000,000 plastic cups a day on airlines alone... A Day! Other things that may surprise you...one out of every four people in prison in the world, are in the USA...2,300,000... "Land of the free?"

Remember 9-11? Of course you do... but on that day 3000 (or so) people died from a terrorist attack. Chris points out that on that day, and every day since, 1100 people die from cigarettes, over 400,000 people every year, (rule of thumb, ALWAYS check statistics.. people lie, so I checked with the Center for Disease Control,
.."Cigarette smoking is the single most preventable cause of premature death in the United States. Each year, more than 400,000 Americans die from cigarette smoking. In fact, one in every five deaths in the United States is smoking related. Every year, smoking kills more than 276,000 men and 142,000 women."... …
and 157,000 are JUST from lung cancer). When people think of cancers caused by smoking, the first one that comes to mind is always lung cancer. Most cases of lung cancer death, close to 90% in men, and 80% in women are caused by cigarette smoking. There are several other forms of cancer attributed to smoking as well, and they include cancer of the oral cavity, pharynx, larynx, esophagus, bladder, stomach, cervix, kidney and pancreas, and acute myeloid leukemia. The list of additives allowed in the manufacture of cigarettes consists of 599 possible ingredients. When burned, cigarette smoke contains over 4000 chemicals, with over 40 of them being known carcinogens. Kind of like the lottery of death, except there is a “winner” every time…I am reminded that I teach my students that in statistics we calculate binomial probability using the term p for the probability of success, even if the “success” in question is death.

I'm a teacher. I work with teenagers every day, and it scares me that 65,000 teenagers will begin smoking this month. Maybe a visual impact on their senses will stop one... or two?

A good conclusion, "How do we change?" WATCH !!

Thursday, 10 July 2008

Catalan Numbers

Here is a simple problem to try out. Draw a 4x4 array of dots. Now construct all the paths possible from the bottom left to the upper right corners by moving only upward or to the right, (think of rook moves on a chess board). Go ahead, there are not that many (only 6 choose 3). Now draw the diagonal from the start corner to the stop corner. How many of the paths never cross the line? You can touch it, but not cross. Now generalize the answer, what if the array was nxn. AFTER you have tried this, read on..

Catalan Numbers first appeared in disguise in a problem Euler first proposed to Christian Goldbach in 1751. The problem is now called "Euler's Polygon Division Problem", and asks, in how many ways may
a plane convex polygon of n+2 sides be divided into triangles by diagonals. Euler gave a solution that looked like The numbers in the sequence are now called Catalan Numbers

The numbers in the Catalan sequence also answer questions such as the one at the top of the page. In how many paths can you move from the origin of a coordinate axis to the point (n,n) if each move consists of either an upward move or a move to the right one unit between two lattice points and you cannot cross the line y=x . Another application (or the same application viewed differently) answers the question in how many ways can 2N beans (for an application, think votes) be divided into two containers if one container can never have less than the second?

The sequence of numbers is given by 1,2,5,14,42,132 ... or
in general by the function . It may seem hard to figure out where the 1/(n+1) comes from in the formula. I found a really nice explanation with graphics at this Wikipedia page(go down to the third proof).

The numbers also can be found from Pascal's triangle by taking (2n Choose n) - (2n Choose n+1). These are two adjacent numbers in each even numbered row

The function is named for Eugene Catalan of Belgium (1814-1894).

Wednesday, 9 July 2008

The Writer's Desk...

In Charlotte, N.C. there is a memorial in front of the ImagiOn Center to a writer named Rolfe Neill called “The Writer’s Desk” ..."Rolfe Neill, was a publisher of The Charlotte Observer and The Charlotte News, Larry Kirkland created whimsical sculptures of granite pencils, rubber stamps, a typewriter keyboard, and a tower of books to celebrate Neill’s passion for truth, the written word and the writing profession. Located on the Plaza of ImaginOn – a children’s learning center and partnership of the Public Library of Charlotte-Mecklenburg County and the Children’s Theatre of Charlotte, the artwork “serves as a place for people to gather, reflect, sit quietly or noisily, or create a spontaneous performance at an outdoor gathering,” said Kirkland. "

I especially loved one quote

In an article by John Brock, another Southern Newsman, I found an interesting quote from Neill..,

"Rolfe Neill, retired publisher of the Charlotte Observer, informed us one day that of the more than 600,000 words in the English language, only 43 words constitute 50 per cent of everything we say or write! This is not limited to Southerners but applies to the rest of Americans as well.

In fact, only nine words make up 25 per cent of all that we communicate. These words are: and, be, have, it, of, the, to, will and you. In this self-centered world, it is surprising that “I” is not in there somewhere.

Check it out. The frequency of the nine words in the following documents is: The Boy Scout oath, 25%; the Marines Hymn, 35.4%; The Miranda rights, 32.1%; the first five verses of St. Paul’s letter to the Galatians, 23.2%. You get the idea."

Tuesday, 8 July 2008

Triangular Numbers

A friend reminded me that I hadn't explained triangular numbers here or on my web site, so taking care of both in a limited way today...

The triangular numbers are known back to before Pythagoras (500ish BCE). They are simply the sums of the natural or counting numbers 1 + 2 + ... + n is the nth triangular number, so the sequence of triangular numbers is 0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210, etc. An explicit formula for the triangular numbers is Tn = n(n+1)/2. For example, T10= (10)(11)/2 = 55.

The triangular numbers appear as the third number in each row of Pascal's triangle.

This means that they are also expressible as (n+1 choose 2)

One of the interesting properties of the triangular numbers is that any two consecutive ones form a square number.... in the image below, T4 (in red) is added to T5 in blue, to form S5, the fifth square number or 25.

It is also known that any even perfect number, and all the ones we know of are even, is a triangular number. The first few are 6, 28, 496.

It is also known that the digit root in base ten of any triangular number is either 1 or 0. That means that all triangular numbers are either divisible by three, or have a remainder of one when divided by three. An easy test of any number to see if it is triangular is to multiply by eight and add one... if the result is a perfect square, then the number is triangular.

It is also interesting that the square of a triangular number Tn is equal to the sum of the cubes of the natural numbers from 1 to n,

....

And one final note on the triangular numbers, the infinite sum of their reciprocals is an integer, ... how sweet, and unexpected, is that? I think you can figure out what it is..but if you get stuck and need a hint, or the answer... just drop a note

Tuesday, 1 July 2008

More...Things I Learned on the Road

More Things I learned on the Road.... The picture above is for those who have always secretly wondered what I would look like with long flowing locks... Newsflash!! Spanish Moss is NOT Spanish...AND (wait for it.....) it is NOT moss... talk about bad names... The French gave it the "Spanish" name as an insult???? Guess you had to be there. Probably an old Monty Python skit about that, "Your Moss is Spanish and your Mother smells of Elderberries!!!". Ok, and the moss part, NOPE... its an epiphitic (gets its food from the air) bromeliad (cousin of the pineapple). Now you know

I also was reminded that people are really strange. In the middle of the road in Enterprise Alabama, there is a statue to the Boil Weevil . Now if you are really a city child and know nothing of the agricultural history of the south, the weevil wiped out many cotton farmers in the south around the beginning of the 20th century. The idea of a monument to the weevil in Alabama would be like a giant grasshopper in Salt Lake City, Utah.

I also continued my road heroics on Jekyll Island (which used to be called Jeykl Island, but someone thought the extra L would be better ). I saved a Loggerhead Turtle and set him free. Ok, I didn't do it all by myself. The Georgia Sea Turtle association was releasing a ten year old (about 150 pounds) that had been kept in the Georgia Acquarium since its egg-ship. I did happen to be in the crowd watching as Dylan, that was his name, went free. He seemed to need a little encouragement as he turned back to shore several times. Eventually prompted by the workers in the waters urging he swam out to sea. Perhaps he was also encouraged by the shouting on shore as the crowd yelled "DIll - Enn.. Dill Enn... repeatedly, especially if he was as confused as I was at first. I thought I had fallen in with gourmets of the turtle soup variety yelling "Kill him, kill him..' eventually I got it straight, and hope Dylan also knew we were NOT trying to have him for supper.

More stuff I learned, don't explain palindromes while driving through Elba, Alabama. As I came into town the name of the town reminded me of one of the first palindromes I ever leaned (OK, second, the first was Madam I am Adam.. read it backwards if you don't know what a palindrome is); Able was I ere I saw Elba. In math, we call numbers like 121 or 1331 or 14641 (surely you recognize the powers of 11) palindromic numbers. We even refer to a polynomial like 1x2+2x+1 as palindromic (look at the coefficients). As I was making sure my wife understood, I saw flashing lights in the rear view mirror... apparently the speed through town was only 25 mph and I was, as the kind officer explained holding thumb and finger an inch apart, "A little over." It was Sunday Morning, and in the spirit of Christian charity, he let me off with a warning.... Thank goodness I was not in Georgia.. I might still be rotting in Macon County Correctional Institute for Out of state drivers.