Day 6: Pythagorean Triples... Again
If you're a math teacher, and you like numbers, there's no way to avoid Pythagorean Triples, set of three rational numbers which can form a right triangle. I started musing about them when this blog was young -- and blog writing still new to me -- back with this post more than ten years ago. What started that was a curiosity born out of desire to find new numbers to use in math problems. Seriously, almost every triangle was 3-4-5 (or multiple), with the rare exception being 5-12-13 or 8-15-17.
Not that I've ever corrected my main line of reasoning, but I was originally caught up with primitive triples where the hypotenuse was either one or two more than the longer leg. I had Kraitchik's book back then -- there's a page scribbled with notes that's been a bookmark since then. In fact, this is probably where the idea to investigate triples with consecutive legs came from. I hadn't (to my memory) encountered any examples of those before. Surprisingly, but not too much, those pairs were related to the square root of 2. This much I reasoned: the legs are nearly congruent, and if they actually were the same, then the hypotenuse would be radical 2. However, because there are only close to being equal, the hypotenuse would come from an approximation to radical 2. With that in mind, I was able to get through the section on using expanded fractions for the square root of two to get possible triples.
Chapter 4: Arithmatico-Geometrical Questions
I read this chapter (and another book in my collection) back then, but at the time, the notation -- one again -- got to me. In particular, it uses x2 + y2 = z2, instead of the now familiar a2 + b2 = c2. (Ironically, the older version would have made the realization about the formula of a circle so much more obvious, once upon a time.) Moreover, a and b are used for calculating values of x and y. (I've seen m and n used in different sources.) And u and v get toss in for good measure to show the relationship between a and b.
On my own, a long time back, I discovered that instead of looking at the two longest sides, I should have been looking at the two odd numbers in the triples. And once I did that, I saw that the leg was always the hypotenuse minus 2n2. While that didn't give me the triples themselves, it told me that I needed to be looking at pairs of numbers with a difference of 2, 8, 18, etc. (I also came to the conclusion that part of the way I looked at the problem came from the fact that the triples I generally used were relatively small, so I just didn't come across some of them.)
At some point -- as this was me, not any book -- I worked out that for a counting number n, 2n + 1 is the leg of a triple and 2n + 1 + 2n2 is the hypotenuse. The other leg, was the square root of c2 - a2, and I was fine with that. And then I realized b2 = (c - a)(c + a), so I don't have to subtract bigger numbers in my head if I don't want to. And then when I started to work out a table:
| n | 2n + 1 | 2n2 | 2n + 1 + 2n2 | c - a | c + a | b2 |
| 1 | 3 | 2 | 5 | 2 | 8 | 16 |
| 2 | 5 | 8 | 13 | 8 | 18 | 144 |
| 3 | 7 | 18 | 25 | 18 | 32 | 576 |
| . . . | ||||||
... I started seeing some of the same numbers showing up. It reminded me of a quiz I gave (special ed class, if I recall correctly) where one student got four out of five problems correct when he used an incorrectly-remembered formula. The four problems he got "right" were all primitive triples. The only that was incorrect was a multiple.
Kraitchik gets into more of the calculation, which I won't repeat here beausee they're likely in so many places on the web (as "helpful" people told me back in 2009), and there, of course, are copyright concerns. After this, there are discussions of Trigonometric Ratios and Heronian Triangles. Thankfully, both are brief and the chapter short. I don't think I would have gotten through something more complex, particularly if it moved in higher dimensions.
Moving on...
I'll likely skip commenting on the next couple of chapters. The chapter on the Calendar was moderately interesting but calculating the day of the week for any day in history isn't something I'm likely to work out with a ruler. There are perpetual calendars and simple apps for that type of thing. The chapter on Probability just isn't all that interesting. There's history, and then there's coin flipping and gambling and Gambler's Ruin. (Side note: it is the one source that says you could win at Roulette, but only because it leaves out the possibility that you will run of money or hit the table limit before you finally win one, which you inevitably will.)
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