Archive for the 'Mathematics' category

Infinite series are weird -- redux!

May 25 2010 Published by under ... the Hell?, Mathematics

A bit over a year ago, I wrote a blog post about the mathematics of infinite series, and how weird such series can be, considering in particular the behavior of "conditionally convergent series".  A recent post at Built on Facts covered similar oddities and gave a nice and different perspective on them.  In the comments of that post, though, an even more bizarre result from the theory of infinite series was introduced, namely the argument that

$latex displaystyle 1+2+3+4+ldots = -1/12$.

This result, if true, is enough to shake one's faith in mathematics, and is completely non-intuitive for no less than three very big reasons:

  1. The sum of an infinite series of increasing positive integers should not converge to a negative value,
  2. The sum of an infinite series of increasing positive integers should not converge to a fractional value,
  3. The sum of an infinite series of increasing positive integers should not converge at all to a finite value!

So is the equation above correct?  Not exactly; it is based on a valid bit of mathematics centered on the Riemann zeta function, but that mathematics is being somewhat misinterpreted to get the paradoxical equation.  An explanation of what went wrong is interesting in itself, however, and allows me to describe a rather difficult concept in the theory of complex analysis known as analytic continuation.

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The Time Axis by Henry Kuttner

Aug 03 2009 Published by under Adventure fiction, Mathematics, Weird fiction

The more I read of Henry Kuttner, the more ashamed I am that I didn't read all of his works long ago!  Henry Kuttner (1915-1958) was a versatile writer of pretty much every genre of weird fiction imaginable: science fiction, horror, fantasy, adventure, and things that defy ready classification.  His is undeniably one of the most influential science fiction writers of the early 20th century, and many of his short stories are beautiful and classic.

I've been using this blog as an excuse to read more of Kuttner's work, though I don't really need one!  I've previously written about his foray into sword-and-sorcery fiction, in his Elak of Atlantis stories, and his exploration of adventure stories, with Thunder Jim Wade.  All of these are short stories, however, so I finally got around to reading one of his novel-length works, The Time Axis (1948):

thetimeaxis

(picture of an early edition via Fantastic Fiction.)  The book is currently available in an excellent quality albeit rather plain edition by Wildside Press, and can also be read online.

It's great!  Like a lot of Kuttner's work, it manages to blend a pulp adventure tale together seamlessly with a science fiction story, and gives the reader a sense of awe and wonder that is altogether rare in fiction.

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Abramowitz and Stegun online!

Jun 29 2009 Published by under Mathematics

Abramowitz and Stegun is a classic reference book which contains all sorts of information about special functions and their integrals.  If you've ever needed to reference something on the road and don't have your copy with you, you will be happy to learn that the book is accessible online.

I happened across this by pure dumb luck while looking for some sort of obscure Bessel function integral some time ago...

7 responses so far

Infinite series are weird!

Mar 18 2009 Published by under Mathematics

I'm in the mood to do something a little more 'math-y'!  A few weeks ago, Tyler at PowerUp did a nice post about the divergence of the harmonic series, and that got me thinking about the weirdness of infinite series.  Since I've been working on a book chapter on infinite series anyway, as a part of my upcoming 'math methods' textbook, I thought I'd talk a little about infinite series and some unusual results associated with them!

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Spot the math errors!

Dec 12 2008 Published by under Mathematics

Via StumbleUpon, I came across this short text page which lists three mathematical 'proofs' which seem to violate common sense, listed below.  The first is:

2equals1

The second one is:

piequals3

The third one is:

neg1equals1

Each of these proofs is (intentionally) wrong!  They highlight classic fallacies in mathematical thinking.  See if you can figure out where, in each of them, the proof goes wrong, and then look for the answers below the fold...

(Note: the third proof involves the 'imaginary number' $latex i = sqrt{-1}$.  If you're not familiar with it, you can safely skip that problem, as it is closely related to one of the others.)

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65 responses so far

Very odd odds: Unusual dice

Jul 03 2008 Published by under Mathematics, role-playing games

I'm still in the midst of a massive move into a new house, but everything has at least been moved from point A to point B; now the unpacking, organizing and fixing of things begins. I'll hopefully get back to some normal blogging next week.

In the meantime, I happened across (well, 'Stumbled Upon') a few sets of very interesting dice for sale: Sicherman Dice and non-transitive dice. Both of these have some rather surprising and interesting aspects, and are new to me, anyway, so I thought I'd do a post!

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5 responses so far

Mathematicians on Mortgages

Jun 09 2008 Published by under Mathematics

In a nice little coincidence, recently two mathematics bloggers have decided to give a bit of a description of the subprime mortgage market crisis.  Neither is an economist, but that's probably okay, even preferable, considering it was the economists who got us into this mess.

Ben Allen of Plektix has written a two-part discussion here and here, and

Mark Chu-Carroll of Good Math, Bad Math has written a two-part discussion here and here.

All of these articles are worth a read!

6 responses so far

Event horizons in water flow: the math!

Mar 15 2008 Published by under Mathematics, Physics

In a previous post, I discussed recent research which demonstrated the creation of an artificial 'event horizon' in a fiber optic cable. In that post, I described how a river speeding up as it goes towards a waterfall has an event horizon: waves that are created past the horizon have no possibility of escape. This was illustrated by the figure below:

As you can see, I've drawn the wavefronts created by rocks dropped in the water as ellipses, which seems like the obvious solution: waves will be 'stretched out' along the flow of the river, while they will spread normally perpendicular to the flow. Being a nitpicky sort of guy, though, I wanted to demonstrate that this is the case mathematically, which I do below the fold... (warning: algebra and calculus follow!)

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Math fonts in LaTeX

Feb 22 2008 Published by under Mathematics

Over at The Daily Photon, Andrew Dawes has put up a nice post outlining how to use different fonts in LaTeX: including finding a math font that matches the text.

I find this especially helpful because, in writing Powerpoint talks, I often run into a conflict between using TeXPoint for my LaTeX equations and a pretty font for my regular text.  It gets rather annoying having to juggle several fonts in order to make certain that the (inline) equations and variables are comparable to LaTeX's standard fonts.

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A LaTeX how-to (for windows)

Jan 13 2008 Published by under Mathematics, Physics

There was a nice post on Good Math, Bad Math about Donald Knuth's classic scientific typesetting software, TeX. In the comments section, a number of people asked about how to learn to use the software. I thought I'd write a little introduction on how to install and use LaTeX, with a nice sample file to play with. Since I often have to do this for my students anyway, it'll be nice to have it all in one place.
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7 responses so far