Friday, October 30, 2009

another origami ideal

This week I had a great origami day - I found Origami For the Connoisseur, by Kunihiko Kasahara and Toshie Takahama at a used book sale (for $3.00CDN), and then when I arrived home I found that my copy of the Between the Folds DVD had arrived.

The Between The Folds documentary is great - beautiful, inspiring, informative, and the accompanying short piece, Origametria, would make a great short video to play at a school math department meeting if you want to get teachers fired up about using origami with their students.

Between The Folds brings to light much of what draws people to origami, and I think that, for those not familiar with modern paper-folding, it will contain many surprises. The images of origami tessellations are particularly beautiful, and the more representational origami 'sculptures' that are presented are often incredibly detailed.

A main focus of the film was the tension between origami technique and artistry. For Robert Lang, there is no conflict between the two - technique sometimes dominates over artistry, but ultimately artistry assimilates and employs technique. At the same time, it seems that Eric Joisel fights an internal struggle to resolve his need for artistic expression and the temptation to focus on technique.

Although the conflict, or potential conflict, between the artistic and technical sides of origami is a concern for the practitioners who are at the summit of the art form, I find that neither artistic expression nor technical complexity are important to my own experience of origami. Perhaps that is because I lack both artistic and technical talent. I was encouraged, however, by how Paul Jackson, Erik Demain, and Tom Hull were all manipulating very simple (yet beautiful and interesting) models as they spoke about origami. Paul Jackson contemplated some 'single fold' models, and both Erik Demain and Tom Hull were seen manipulating a hyperbolic paraboloid (instructions for this simple model by Demain are here, and Hull has instructions on page 65 of the handouts for his book, found here).

Turning to Origami for the Connoisseur for further inspiration, I found this statement, which was a welcome counterpoint to all the artistry and technical prowess of Between the Folds:

Something that is simple and duplicatable too is another origami ideal.

A very simple modular origami model for "rotating tetrahedron," designed by Tomoko Fuse, shares the same pages with the quote above. This model is an intriquing toroidal structure of linked tetrahedrons that has been studied by engineers and chemists (see for example Simon Guest's symmetry page and paper). The rotating tetrahedron and hyperbolic paraboloid are examples of the surprising yet simple models that make origami an accessible and beautiful way of exploring mathematics (see this earlier post for other origami-math links).

Thursday, October 29, 2009

JUMP math

                                                         image from Everyday Number Stories

A couple of years ago I saw a presentation by John Mighton of JUMP math. He is an interesting and engaging speaker, and you should check out his short video at MAKE magazine's The Elements of Humanity. An example of how a community organization has been able to implement the JUMP program can be found here, at the Ottawa Centre for Research and Innovation (OCRI) site.

Tuesday, October 27, 2009

Harmonic Denominator Number Triangle

This number triangle is made up of the denominators of the Leibniz Harmonic Triangle. From the earlier post about that number triangle, you can see that there are two ways of generating the Harmonic Denominator Triangle:

1. take the reciprocal of the entries in the Leibniz Harmonic Triangle; or
2. multiply the entries of Pascal's Triangle by n+d (here we are using the 'triangular number style' indexing of Pascal's Triangle, rather than the usual 'binomial coefficient style' indexing).

The entries of the Harmonic Denominator Triangle ($g^d_n$) are given by:

\[g^d_n = \frac{1}{h^d_n}\]
\[g^d_n = (n+d)t^d_n\]
Where $h^d_n$ are the entries in the Leibniz Harmonic Triangle and $t^d_n$ are the entries in Pascal's Triangle. The connection to the triangular numbers gives us the general formula:

\[g^d_n = \frac{n(n+1)\cdots(n+d)}{d!}\]

I am looking forward to learning more about this number triangle (and the Leibniz Harmonic Triangle too). If you make use of OEIS, you'll see that the The HDT contains a lot of well-known sequences.

A quick look at the rows and columns and you'll find the a bunch of well-known sequences: A005430, A002457, A002378, A027480, and A033488, to name a first few.

The whole triangle has an entry in OEIS: A003506. One of the comments in the OEIS entry for the HDT points out a neat way to express the relationship between Pascal's Triangle and the Harmonic Denominator Triangle. The entries of the k-th row of the HDT are the coefficients of the first derivative of a polynomial whose coefficients are the entries of the (k+1)-th row of Pascal's Triangle. More specifically, the coefficients of $(x+1)^{k+1}$ give you the k+1 row of Pascal's Triangle, while the coefficients of $\frac{d}{dx}(x+1)^{k+1}$ give you the k-th row of the Harmonic Denominator Triangle.

Friday, October 23, 2009

Mesopotamian maths

On Notation. Whether one believes that mathematics is created or discovered, notation is certainly created. And notation can direct the course of mathematics. 

- Elisha Peterson, Unshackling Linear Algebra from Linear Notation

A common entry point into learning about the mathematics of ancient Iraq is through its notation. The sexagesimal (base 60) system written in cuneiform defines what the mathematics of ancient Iraq means to most people who have been exposed to it through textbooks and short articles. One way to get some exposure to this is to visit Wolfram Alpha, type in a number, and choose "other historical numerals" -  you'll be treated to a digitized version of what Babylonian cuniform sexagesimal numbers look like.

The Wolfram Alpha implementation of its sexagesimal cuniform  translator was not without its problems, but teachers are already using it in their lesson plans.

It is worthwhile to think about the long series of abstractions that have lead up to the Wolfram digitized cuneiform. As Eleanor Robson points out in her book Mathematics in Ancient Iraq: A Social History, the earliest 'writing' of numbers were actually the impressions of tokens into the clay tablets - originally the tokens were actually sealed into the clay in order to record counts. Later, it was determined that the tokens themselves could be dispensed with and that the impressions could stand in their place. Later still, the impressions became incised cuneiform markings on the tablets - one of the earlies forms of writing. For a long time since, we have been mimicking these incisions with pen and paper or typeset drawings, and now a  cuneiform simulacrum can be generated automatically for us in this new digitized form.

While Robson's book tells us a lot about cuneiform writing and the various systems of ancient Iraqi mathematics, it points out that we are severely mistaken if we understand the mathematics of ancient Iraq solely in terms of  notational differences between it and our more familiar ways of writing mathematics. The mathematics of ancient Iraq is not merely our familiar mathematics recast with a different base and different writing style, it is fundamentally different in ways that we have trouble appreciating. It seems that when westerners look at the mathematics of another culture or another era, we tend to view it through the fixed lense of our own mathematics, we wonder to what extent did they anticipate our current mathematics, what their "contribution" was, and in what areas they were limited. We tend to ask questions like,  did they have a concept of 'zero'?, did they have a 'Pascal's Triangle?' did they know 'Pythagoras's Theorem'? This limited way of viewing the math of others leads us to make a number of untenable assumptions about the history of mathematics.

Robson's analysis quickly undermines the simple stories that are often repeated in the margins of math textbooks that place Iraq at the starting point of what became, after elaboration by the Greeks, the "Western" tradition of mathematics. As she notes early on, 'the mathematical culture of ancient Iraq was much richer, more complex, more diverse, and more human than the standard narratives allow." (p. 2) Her book attempts to provide "a new look, and a new perspective" (p. 8) on a subject that has been glossed over far to much, often from an overly simplifying western vantage.

In ancient Iraq, numeracy, literature, the mechanics of cuniform, along with the intellectual, state, and educational cultures, grew together to shape each other and to give the mathematics of this period its distinct richness. The 3000 years examined in this book show a a surprisingly diverse history of mathematical practice that begins with emergence of a rudimentary numeracy that developed into applications in accounting and engineering, and that ultimately evolved into a baroque system of numerology and mathematical divination.

It is interesting that although the period explored is an ancient one, much of it has only come to light recently. Consequently, much of the non-scholarly writing about the mathematics of ancient Iraq is quite simplistic in its outlook. Robson argues that as examples of cuneiform mathematics came to light in the early and mid 20th century, scholars gravitated towards those texts that fit with their preexisting notions of Greek and Egyptian mathematics (to the exclusion of more representative ones), a process that helped only to reinforce received orientalist ideas about the mathematics of this period.

A striking example of how our preconceptions can influence our understanding of how others understand and use mathematics is provided by contrasting two translations of an ancient cuneiform tablet (table 9.1, page 277)- an early translation from the mid 20th century is full of modern mathematical terminology, while a more recent translation attempts to be closer to the original. The translation that uses a modern lens uses the convential terms like adding, subbtracting, multiplying, and dividing numbers to describe the content of the tablet - in sharp contrast, the contextualized reading uses images of lengths "holding" each other, of "turning back" within a calculation, and of "tearing" and "accumulating" surfaces.

Robson's approach is rooted in a view of mathematics that is essentially one of social constructivism - one that she admits may not be held by the majority of working mathematicians, but that provides what may be the best perspective for historical analysis.  Interestingly, she also argues that a philosophical outlook of mathematical platonism (a view held by many mathematicians and historians of mathematics) inevitably influences people to see more connections between the mathematics of different cultures than can be legitimately said to exist. If we believe that mathematical objects are "real" and "discovered" rather than invented, then we would expect different cultures to somehow discover the same mathematical truths. If we abandon this perspective, differences in mathematical practices and understandings are less surprising.

It is interesting to read about the prevelence of educational texts among the surviving cuniform tablets. Some of these tablets mimic "official" documents, but bear the tell-tale signs of being used for the education of scribes. For example, some tablets are identified as educational by "the unrealistic size of the numerical parameters, ... and the lack of credible contextual data" (p56). Creating math problems infused with realistic context has apparently plagued teachers for thousands of years. Robson suggest that there is evidence that scribal training, in some periods, made use of situated learning rather than purely rote schooling (p84).

One of the consequences of abandoning a more reductive and simplistic approach to understanding this period and its culture of mathematics is that we loose the "grand narrative" of mathematics that links ancient Mesopotamian mathematics, the Greek mathematics of Pythagoras and Euclid, and what eventually became the western tradition. The simple thread that united these mathematical traditions is cut, in favor of a much more complex weaving of influences.

Mathematics in Ancient Iraq is an important reference both for its subject and for its method. For anyone researching the history of mathematics, or this period, it provides an important resource and example. The detail and close readings of original sources and its commitment to situating these within their cultural context, essential for the method that Robson pursues, might, however, prove too much for the reader with only a casual interest in the subject. For this group of readers, hopefully the insights from this text will soon be reflected in more general works and texts. Fortunately there are many good web resources that reflect this emerging understanding of ancient Iraqi mathematics, many of them contributed or maintained by Robson herself (check out the links on her webpage).

Monday, October 19, 2009

whither blogger LaTeX?

If you have a Blogger blog, you may be using some $\LaTeX$ renderer to make your math look nice. A couple of days ago, the one I was using (from stopped working (hopefully it will be up again by now, or soon...).

My limited research has shown me that most Blogger use of $\LaTeX$ relies on a small piece of javascript embedded somewhere on the page that calls a larger piece of javascript (possibly located on another server) that in turn invokes some other hosted program that actually does the rendering (possibly located on yet another server): lots of potential points of failure. Another unfortunate aspect of this method is that it doesn't allow the rendered math to show up in readers - you have to look at the page itself to have the script run.

If you have lost your math rendering you can learn a little about these things by reading the replacemath.js docs and the mathtex docs.

For now, I am using the following bit of script to render the math:

<script src="" type="text/javascript"></script>
<script type="text/javascript">
replaceMath( document.body ); 

Place this somewhere on your page, and it will invoke the replaceMath function that is defined in 'replacemath.js' - see the documentation about this at the link mentioned above.

Although the rendering isn't doing quite as nice a job as the watchmath version did, it's not too bad. I'll look for something better when I have the time...

PS: This blog post provides a good overview of the LaTeX rendering options that are out there.

Thursday, October 15, 2009

Math Teachers at Play 17

There is plenty of humor, a few movies, some great activities and explorations, and lots of math in Math Teachers at Play 17. Hope you enjoy reading these as much as I did. Thanks to everyone who submitted posts or answered my requests to include their work.

First, a bit of administrativa: Please see this post at Walking Randomly about the rebirth of the Carnival of Mathematics, and some upcommng scheduling changes for Math Teachers at Play.

David Richeson at Division By Zero has brought us a great list of activities that are likely to inspire a few budding mathematicians in his post Kindergarten Mathematics.

In her post, Elementary Math isn't Easy, Joanne Jacobs directs us to a recent article on the importance of elementary mathematics and elementary math teachers by Hung-Hsi Wu in American Educator.

Rachel M encourages us to grow a Counting Garden posted at - an activity that helps reinforce basic counting and number recognition.

Mathematics is everywhere, particularly at the breakfast table. At Math with My Kids we are reminded of the challenges and rewards of mathematical discovery in Math with Bannanas. Rick Regan suggests an interesting and edible 100 days of school project in his post One Hundred Cheerios in Binary at Exploring Binary.

Kakie presents Teach kids how to make $1,000,000 in 30 days with pennies! posted at Bur Bur & Friends: Community Blog. Keeping with the economics theme, Kendra offers Pumpkin Patch: Piggy Bank Math Game posted at Pumpkin Patch.

Much has been written on math blogs lately about how to understand and explain "negative multiplied by a negative is positive." On this fruitful theme, Brent Yorgey presents Minus times minus is plus posted at The Math Less Traveled. Jason Dyer gathers many of the threads that this question has spawned together and offers his own explanation in Negative times negative posted at The Number Warrior.

Denise offers some guidance in How to Solve Math Problems II posted at Let's Play Math!.

Sue Van Hattum offers a way of tackling the problem of math anxiety in her post Math Relax: A Guided Visualization for Overcoming Test Anxiety in Math at her blog Math Mama Writes.

You may have heard of the "Jigsaw" method in collaborative learning, but what about the "Speed dating" method? Kate Nowak introduces this structure in her post Speed Dating at f(t).

John Golden presents a dynamic geometry exploration of the Pythagorean Theorem, considering all triangle types in GeoGebra: Triangle Tuning posted at Math Hombre. See his GeoGebra introduction in an earlier post here. You may also want to see Kate Nowak's instructions for putting your GeoGebra into your blog here.

Kimberly Lightle's post, Dynamic Math and Science Learning With Simulations, at Exemplary Resources for Middle School Math and Science and Maria Andersen's post, Interactive Simulations from PhEt, at Teaching College Math both provide some great links to online simulaiton software. Sue Vanhattum of Math Mama Writes tells us about an early pioneer of learning math through computer interaction in her post Mindstorms: Children, Computers, and Powerful Ideas.

Liz at STEM-ology has pointed us towards a new resource and recruitment program for mathematics teachers, Math for America, in her post X plus Y.

Another math resource has been provided by the Mathematics Department at BYU, which announced that their site When Will I Use Math? has launched. This perrenial question was also answered by Deb Russell in her post Math: When Will I Ever Use This Stuff?, which points to an article on mathematics and 3D animation.

In Pandemics and Their Numbers, Terese Herrera points us to an H1N1 inspired lesson created by The New York Times at the blog Exemplary Resources for Middle School Math and Science.

Humor and mathematics go hand in hand in several recent posts.

Pat Ballew offers a little tongue in cheek math talk in  You Might Be a Mathematician IF... posted at Pat'sBlog.

Xi at 360 presents some great math comics the posts All about A4, Another Math Comic, and How About Another Comic?. Luke Kane points us towards some other recent, and very slick, math comics in his post Comics and Math at Logic Nest.

At Komplexify, we have a bit of calculus poetry in the post Calculus Haiku - The Derivative.

Mistakes, ambiguities, the unexpected, and the impossible - math teachers encounter these, from time to time.

Jakie points out how mistakes often turn into teachable moments in her post When Sketchpad is Wrong at Continuities.

Glowing Face Man presents Ambiguities in Mathematics posted at Glowing Face Man.

Sam Shah shows us some examples of functions behaving badly in his post sin(1/x) at Continuous Everywhere but Differentiable Nowhere.

Vlad Alexeev shows us an impossibly small book of impossible figures in the post Mini Books of Anatoly Konenko at his blog Mathematical Paintings and Sculptures.

John Cook has posted on the math behind musical scales in his posts Circle of fifths and number theory and Circle of fifths and roots of two at his blog The Endeavor.

Alison Blank has put together an inspired and inspiring Prezi presentation, Math is Not Linear, and posted about it on her blog Axioms to Teach By.

Maria H. Andersen of Teaching College Math has put together another impressive prezi presentation  and linked to it from her post How can we measure teaching and learning in mathematics?

The Albany Area Math Circle lets us know about the math documentary Hard Problems: The Road to the World's Toughest Math Competition. To find out more about Math Circles, check out the National Association of Math Circles site.

Green Fuse Films had another math-film announcement on their blog: Between the Folds, their documentary about the math and art or origami, is now available on DVD.

Reidar Mosvold lets us know about what looks like a really neat event: Maths Week in Ireland on his blog Mathematics Education Research Blog.

Marjorie Morgan presents her thoughts on Outdoor Education and Mathematics teaching in her post Lindsay & Sharon - outdoor adventurers at GO! Girls Outdoors.

It surprises some that guessing and measuring can both fall under the mathematical umbrella.

Tom DeRosa presents The Very Exact Science of Guessing posted at I Want to Teach Forever.

Maria Miller presents 10/10 and the Metric Week posted at Homeschool Math Blog.

Staying with a metric theme, Austen Saltz, a senior high school student who is blogging at Talking Science, points us to Nikon's Universcale in his post The Size of the Universe. Universcale is very reminicent of the Powers of 10 film from the 70s, but much flashier (see the film here).

Two approaches to thinking about breaking sticks and making triangles are found at Bill the Lizard and Pat's Blog. Bill takes a simulation-based approach in his posts,  The Broken Stick Experiment and The Broken Stick Revisited, while Pat explains how to tackle the problem using limits in his post, A Limit Approach to a Classic Geometric Probability Problem.

Thanks to everyone. If you submitted an article that was not included, or if you have a post that you think would have made a nice addition, please consider submitting it to the next installment. The next Math Teachers at Play will be up at Math Mama Writes on October 30th.

Wednesday, October 7, 2009

Three number triangles, two telescoping series

There are so many relations present [in Pascal's triangle] that when someone finds a new identity, there aren't many people who get excited about it anymore, except the discoverer! - Donald E. Knuth (as quoted by Martin Gardner)

I was inspired by a post on Pat's Blog and by reading A.W.F. Edwards's Pascal's Arithmetical Triangle to look at summing the reciprocals of higher dimensional triangular numbers. It turns out that you can use the same telescoping series technique that allows you to sum the reciprocals of the 2-dimensional (i.e. the usual) triangular numbers, and that the 'telescoping' feature of these sums can be expressed in terms of some nice identities.

The statement of the sum is a nice one. For d > 1 we have:

\[\sum^{\infty}_{n=1} \frac{1}{t^d_n} = \frac{d}{d-1}. \]
Which gives you a result of 2 for d = 2 as mentioned in Pat's post. In the case where d = 0 or 1, the series does not converge.

In the notes below, none of the identities are new (the newest is, I think, about 400 years old) - the famous quote by Donald Knuth at the top of the post is intended as a caution for anyone who gets carried away and derives scads more.

In all the statements in this post, the index d ranges over the nonnegative integers (0, 1, 2, ...) while n ranges over the natural numbers (1, 2, 3, ...). This may seem somewhat inconsistent, but we like to start with dimension zero (d = 0) and with the first triangular number (n = 1).

The d-triangular numbers, $t^d_n$ (d for "dimension") are defined by $t^0_n = 1$, and

\[ t^d_n =\sum^{n}_{i=1}t^{d-1}_i \mbox{ for } d > 0. \]

The formula for $t^d_n$ can also be expressed as a difference, $ t^d_n - t^d_{n-1} = t^{d-1}_n $.

When d = 2 we get the usual triangular numbers, d = 3 gives the pyramidals, and d = 4 gives the triangulo-triangulars, etc. - each dimension up is visualized as a stacking of those below it. Note that this definition has the first triangular number (for all dimensions d) as 1 and not 0, as is sometimes preferred; in this context it makes sense to start at 1.

From this definition, and the Pascal identity, you can establish that

\[ t^d_n = \left( \begin{array}{c}n+d-1 \\d \end{array} \right). \]
If you are familiar with Pascal's Triangle and look carefully at the triangular number definition, you'll see that sum in the definition of the d-triangular numbers is the Pascal Triangle "hockey stick theorem" in disguise. This provides us with a direct formula for the d-triangular numbers:

\[ t^d_n = \frac{n(n+1)\cdots (n+d-1)}{d!}. \]
And this also suggests that we arrange the d-triangular numbers into Pascal's triangle, while remembering that we are not indexing them in the way we usually do for the binomial coefficients.

The formula also gives us the opportunity to generate a bunch of identities, like:

\[t^d_n = \frac{(n+d-1)}{d} t^{d-1}_n, \]
\[t^d_n = \frac{n}{d} t^{d-1}_{n+1}. \]
From here we flip each entry in the triangle to obtain a new triangle, the Leibniz triangle (so called by Edwards), whose entries are the reciprocals of d-triangular numbers.

There is a nice difference relationship in this triangle too, for > 1,

\[\frac{1}{t^d_n} = \frac{d}{d-1} \left( \frac{1}{t^{d-1}_n} -\frac{1}{t^{d-1}_{n+1}} \right) \]
To prove this identity, generalize the method that allows you to split up the fraction $\frac{1}{n(n+1)}$ into $\frac{1}{n} - \frac{1}{n+1}$ in the usual telescoping series example.

To get our last number triangle, we divide each entry in the Leibniz Triangle by $(n+d)$, which gives us the Leibniz Harmonic Triangle. In other words, we define $h^d_n = \left(\frac{1}{n+d}\right)\frac{1}{t^d_n}$, and create a new triangle with entries $h^d_n$.

Going back to the formula for $t^d_n$ we can obtain some other identities, like:

\[ h^d_n = \left( \frac{1}{d+1}\right) \frac{1}{t^{d+1}_n}, \]
\[ h^d_n = \left( \frac{1}{n}\right) \frac{1}{t^{d}_{n+1}}. \]
And, as with the other triangles, there is a nice difference relationship, for d < 0 :

\[ h^d_n = h^{d-1}_n -h^{d-1}_{n+1} . \]
This can be proved following the same ideas as was used to show the difference relationship for the inverse triangulars. It's worth contrasting this identity with the corresponding identity for Pascal's Triangle.

Now, if you've proved these identities, there are two easy ways to find the sum, $\sum^{\infty}_{n=1} \frac{1}{t^d_n}$.

First method: use the Leibniz Harmonic triangle.
Taking advantage of these two identities:

\[ h^d_n = \left( \frac{1}{d+1}\right) \frac{1}{t^{d+1}_n}, \]
\[ h^d_n = h^{d-1}_n -h^{d-1}_{n+1} . \]
We can combine them and observe that for d > 1:

\[\begin{array}{lll} \sum^{\infty}_{n=1} \frac{1}{t^d_n} &=& d\sum^{\infty}_{n=1} h^{d-1}_n \\ &=& d\sum^{\infty}_{n=1}\left( h^{d-2}_n - h^{d-2}_{n+1} \right) \\ &=& d\left(\sum^{\infty}_{n=1} h^{d-2}_n -\sum^{\infty}_{n=1} h^{d-2}_{n+1} \right)\\ &=& d\left(\sum^{\infty}_{n=1} h^{d-2}_n -\sum^{\infty}_{n=2} h^{d-2}_{n} \right)\\ &=& d\left(\sum^{\infty}_{n=1} h^{d-2}_n -\sum^{\infty}_{n=1} h^{d-2}_{n} + h^{d-2}_{1} \right)\\&=& dh^{d-2}_{1} \\ &=& d\left( \frac{1}{d-1}\right)\\ &=& \frac{d}{d-1}\end{array}\]

Incidentally, we also have $\sum^{\infty}_{n=1} h^d_n = \frac{1}{d}$ for d > 0.

Second method: use the reciprocal difference identity for the Leibniz triangle.
Here we see that everything comes directly from the identity:

\[\frac{1}{t^d_n} = \frac{d}{d-1} \left( \frac{1}{t^{d-1}_n} -\frac{1}{t^{d-1}_{n+1}} \right). \]
We have, for d > 1:
\[\begin{array}{lll} \sum^{\infty}_{n=1} \frac{1}{t^d_n} &=& \frac{d}{d-1}\sum^{\infty}_{n=1} \left(\frac{1}{t^{d-1}_n} - \frac{1}{t^{d-1}_{n+1}}\right)\\&=& \frac{d}{d-1}\left(\sum^{\infty}_{n=1} \frac{1}{t^{d-1}_n} - \sum^{\infty}_{n=1}\frac{1}{t^{d-1}_{n+1}}\right)\\ &=& \frac{d}{d-1}\left(\sum^{\infty}_{n=1} \frac{1}{t^{d-1}_n} - \sum^{\infty}_{n=2}\frac{1}{t^{d-1}_{n}}\right)\\ &=& \frac{d}{d-1}\left(\sum^{\infty}_{n=1} \frac{1}{t^{d-1}_n} - \sum^{\infty}_{n=1}\frac{1}{t^{d-1}_{n}} + \frac{1}{t^{d-1}_{1}} \right)\\ &=& \frac{d}{d-1}\left(\frac{1}{t^{d-1}_{1}}\right) \\ &=& \frac{d}{d-1}\end{array}\]

After working out all this, I came across the same "reciprocal of generalized triangular numbers" problem at Topological Musings - solution #3 is the same as the one here, but expressed in terms of binomial coefficients (and without extracting some general identities among the terms).

Monday, October 5, 2009

beautiful negatives

teacher (emphatically): A double negative makes a positive, but a double positive can never make a negative!
student (lazily, from the back of class): yeah, yeah...

I am not sure where this topic started, but The Number Warrior has collected up pointers to most of the "minus times minus is plus" posts that have been cropping up here.

So far I haven't read about what I thought was the usual visualization (but it has probably been mentioned, burried in some comments): multiplying by -1 is a counter-clockwise rotation by 180 around zero.

The problem with this image is that it takes us off the number-line and has us floating in space for a moment. But this is exactly right - the space we are floating in is actually the field of complex numbers, and seeing "multiplying by -1" as "counter clockwise rotation by 180" is the visualization that corresponds to "the most beautiful equation in the world" $e^{i\pi} = -1$. In the same way, a rotation by 90 degrees corresponds to a multiplication by $i$, and multiplying by $i$ twice gets us to the same place as multiplying by -1, which makes sense, since $i^2=-1$. It has to be counter-clockwise in order to give the imaginary axis its usual direction.

Saying that multiplying by -1 corresponds to a 180 degree rotation does not offer an explanation for why a negative times a negative is positive, but it provides a way of seeing it that is consistent with how we visualize other operations (complex multiplication). If you actually said "this is why a negative times a negative is a positive" someone could easily ask, "why does complex multiplication involve rotation at all?" If anything, this visualization is a way of providing a comfortable introduction to the idea that complex multiplication involves a rotation.

Although "negative times negative is positive" seems obvious once you've gotten used to it, putting double negation in a more general context is interesting. If we always expect not not a to equal a, we are relying on the law of the excluded middle, which not everyone accepts all the time.

Saturday, October 3, 2009

math teachers at play

Math Teachers at Play blog carnival, edition 16, is up at I want to teach forever.

The next MT@P blog carnival will be hosted here at mathrecreation in two weeks on Oct 16, 2009.

Please send in your submissions at the blog carnival site, by clicking on the icon below, or by email to me directly. The deadline is Wednesday Oct 14, 2009.
The focus of this carnival is preK-12 mathematics, with the emphasis (as the name suggests) on playfulness and exploration. This blog is not only for math teachers (as the name suggests), but is open to submissions from anyone who plays with math.

If you are new to MT@P, you should get started by browsing the first edition, at Let's Play Math, and move on from there.