The 0'Factor, episode 10

(Click on the cartoon to see the entire image.) (C)Copyright 2009, C. Burke. All rights reserved.


It's that Law of Gravity that's really keeping me down...

By the way, I'm aware that Rich Bond isn't really a money guy. But c'mon! It's just too good a name not to use.

Previous episodes of The 0'Factor: 01, 02, 03, 04, 05, 06, 07, 08, 09.

Plotting

(Click on the cartoon to see the entire image.) (C)Copyright 2009, C. Burke. All rights reserved.


You know they are, but you have to wait for Geometry for the proof.

Sunday Comic: Super-Stick-Man 52: Super-Stick-Man and the Avalance

Dateline: Stick-World!
In the mountains north of The City, there's always snow, which means that there's always skiing. And where there's skiing, there are skiers in trouble. And where there's trouble, there is Our Hero!

Click on the link to the comic and follow the exciting story!

Motivational Math Poster

I saw this one, thought about it, decided to post it:

Geocities closing down, closing down, closing down...

Yahoo has announced that they have decided to get rid of all the free Geocities web sites and encourage users to fork over a lot of money for one of the premium webhosting sites. Not right away, of course, but later this year.

How does that affect my blogger account?

Well, it so happens that my first school website was a geocities website. It had a minimum amount of success with my then-current students. However, one of the problems was that there was an advertising bar on the side of the page. It could be closed, but it reopened with every click. It was useful or appropriate for a school website.

However, the page offered 15 Mb of free storage, which didn't have advertising attached. Most of the images that are on this website are actually stored on geocities, which is also why, on occasion, they all disappear for about an hour or two when the bandwidth is too high. (That doesn't happen too often because this page doesn't have that kind of popularity.)

What it means for my page is that I'll have to find someplace else -- cheaper than $10/month -- to store all the images. I seem to have problems with linking directly to the xwhy.comicgenesis.com images, so that isn't a solution. But I need to find a solution soon, because I have to edit more than half the posts on this blog.

I will also have to hurry up and copy all the current book reviews on my other book blog Real Soon Now.

UPDATE: Well, two choices have emerged. First, I tried linking an image from Flickr and that seems to work, or I can use a Google page. Yahoo, which owns Geocities, also owns Flickr, which I've been using for a while and already has some of my images. Google, on the other hand, owns Blogger, where this blog resides. Are you thinking what I'm thinking?

I tried using Blogger for my images before and it was horrible, but I believe that was before Google bought it. So I have a decision to make. Any input from anyone that uses either one, or just from my legion of readers (who number almost as many as the Legion of Super-Heroes ... at least, the version from the late 70s) who want to give an opinion.

Besides, I really, really appreciate comments, even the ones from people who aren't related to me or owe me money.

Tesseract

(Click on the cartoon to see the entire image.) (C)Copyright 2009, C. Burke. All rights reserved.


You are traveling through another dimension, a dimension not only of sight and sound but of bad puns . . .
Yes, the fourth-dimension is a long way to go for a bad pun, but this comic knows no upper bounds (or lower bounds, for that matter).

For more on Henry's and Ian's adventures, click on the following links:

Through a Window Darkly: Part I,II,III, IV, V, VI, TVH, TUC.

Identity Property

(Click on the cartoon to see the entire image.) (C)Copyright 2009, C. Burke. All rights reserved.


You've never heard of a talking horse?
Well, listen to this.

Pythagorean Triples: An Easier Way

Almost done. Promise

There Has to Be an Easier Way

Continued fractions are a pain (to be kind), and my first attempts resulted in errors that were compounded in later steps. (I had 84, 85, 119 at one point. Not quite.)
However, I noticed that the numerators and denominators of the convergents of the continued fraction of follow a pattern

P Q 1 0 2 1 5 2 12 5 29 12 70 29 169 70 408 169 985 408 2378 985

P1 = 1, Q1 = 0, Qn+1 = Pn, Pn+1 = 2 X Pn + Qn To follow my method, calculate 1 + Q/P as an improper fraction. As in my prior post, this will give you a triple in the form (a+b)/c, where b = a + 1. This method was "easier" for me than what Kraitchik (1953) wrote, but it required more iterations to get each Pythagorean Triple. Kraitchik offers a shortcut, which is essentially: a = 2PQ and b = P2 - Q2, which yields numbers such that | a - b | = 1 For example: a = 2(2)(1) = 4, b = (2)2 - (1)2 = 3, c = 5 a = 2(5)(2) = 20, b = (5)2 - (2)2 = 21, c = 29 a = 2(12)(5) = 120, b = (12)2 - (5)2 = 119, c = 169 etc.

The same triples were reached in half the steps. The downside is that these numbers get so big as to be unusable. We didn't use calculators when I was in high school, so it is not surprising that I never saw these:

a = 2(29)(12) = 696, b = (29)² - (12)² = 697, c = 985
a = 2(70)(29) = 4060, b = (70)² - (29)² = 4059, c = 5741
a = 2(169)(70) = 23,660, b = (169)² - (70)² = 23,661, c = 33,461
a = 2(408)(169) = 137,904, b = (408)² - (169)² = 137,903, c = 195,025
a = 2(985)(408) = 803,760, b = (985)² - (408)² = 803,761, c = 1,136,689
a = 2(2378)(985) = 4,684,660, b = (2378)² - (985)² = 4,684,659, c = 6,625,109

Last two notes about patterns:
First, notice that every value for c later shows up as P and Q. Keep expanding the list if you don't believe me. I dare you.
Second, notice that the value of b - a alternates between -1 and +1 for each step. What does that mean? I don't know. I just found it curious.

References:
Kraitchik, Maurice (1953). Mathematical Recreations, Second Revised Edition, Dover Publishing.

Pshaw!

(Click on the cartoon to see the entire image.) (C)Copyright 2009, C. Burke. All rights reserved.


They apparently caught a showing of George Bernard Shaw's Getting Married.

Sunday Comic: Super-Stickman 51: Super-Stick-Man and the VILLIAN

Dateline: Stick-World!
The Evil Villian is at it again. Find out more!

Click on the link to the comic and follow the exciting story!

Pythagorean Triples: Consecutive Legs

Almost done now. Honestly. Maybe one more entry after this one. Maybe two.

On page 100 of Mathematical Recreations, by Maurice Kraitchik, Second Revised edition, Dover Publishing, 1953, the author poses this question:

Find a right triangle whose legs are consecutive integers

The answer is not the 3 - 4 - 5 triangle (again!), but an infinite set of such triples (of which 3-4-5 is a member). I have to admit that I didn't follow the logic the first time through and had to re-read the entire section, but I could sum it up with my own method once I discovered that was involved, along with the fraction:
, which converges on .

It made sense to me that would be involved with Pythagorean triples because of the numbers association with right isosceles triangles. As the legs of the triangle increase in size, the ratio of two consecutive numbers comes closer to 1:1.

Followed by: 99/70 and 239/169

If you look at the fractions with the odd denominators, you have a Pythagorean Triple in the form of (a+b)/c, where a and b are consecutive numbers (i.e., b = a + 1) 7/5 gives you 3, 4, 5. 41/29 gives you 20, 21, 29 239/169 gives you 119, 120, 169

There has to be an easier way to find these triples!

There is ... and that's on the next page of my journal.

Casting Out Nines

(Click on the cartoon to see the entire image.) (C)Copyright 2009, C. Burke. All rights reserved.


What's the name of the guy who owns the boat?
No, What's the name of the guy who sold the bait.

I'm still mad at myself. After finishing the picture, I remembered that I wanted to give Ernie a goofy fishing hat. Oops.