B Understanding the Pythagorean Theorem

[mathwonk]

I remember my friend, a college educated, successful building contractor, asking me why the 3,4,5, rule that his carpenters used works, and his amazement at the simple explanation.

 

[mathwonk]

What do you know, I found the following among my old notes for one of my own number theory classes (that I taught).

"Theorem: Every primitive solution of (*) x^2+ y^2 = z^2 either has the form x = 2ab, y = a^2 - b^2, z = a^2 + b^2, or the form x = a^2 - b^2, y = 2ab, z = a^2 + b^2, where a > b > 0 are natural numbers, one of a, b is odd, the other even, and gcd(a,b) = 1. (In particular, for any primitive solution, exactly one of x and y is odd, and z must be odd.)

The idea of the proof is to divide through (*) by z^2, changing the problem to one of finding all positive rational points on the unit circle, and then compute that all such points are obtained by taking all lines through the north pole of the circle having rational slopes between -1 and 0. The formula transforming the rational slope, - b/a of such a line, into the coordinates (u,v) = (2ab/{a^2+b^2}, {a^2 - b^2} / {a^2+b^2}) of the corresponding point on the circle shows where the expressions in the trick come from. "

One good thing about gradually losing ones memory in old age, is one can continue to enjoy the same things over and over, as if they were new. The upshot is that the trick fresh gave in post #17 is essentially the only way to construct pythagorean triples, (i.e. up to multiples).

 

[PeroK]

Before I retired and got back to studying maths and physics, I was telling someone about the trick for generating the integer squares by adding the odd numbers in increasing sequence:

0, 0+1 = 1, 1 + 3 = 4, 4 + 5 = 9 ...

I then realised that here was an elementary proof that there are infinitely many distinct Pythagorean triples:

Any odd number is the difference between two consecutive squares. If we start with an odd number, $k$, then $k^2$ is odd and is the difference between $(n+1)^2$ and $n^2$ for some $n$. In particular where $k^2 = 2n +1$.

In any case, $(k, n, n+1)$ is a Pythagorean triple. And, as $n$ and $n + 1$ have no common factor, these triples are all non-trivially distinct.

 

[Junior12342007]

I like the notion that that you can deviate from The original difference of two squares with the custom expansion of ( a^2 + b^2 ) = a^2 + 2ba + b^2, with the removing of the 2ba factor as it's multiplied by a cosine which is fascinating

TL;DR Summary: A discussion on the Pythagorean Theorem, its applications, and interesting problems.

Hello everyone,

I wanted to start a discussion about the Pythagorean Theorem. It's one of the fundamental concepts in mathematics, stating that in a right-angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.

I'm curious to know about:

• Different applications of the Pythagorean Theorem in real life
• Interesting problems or puzzles related to the theorem
• Any historical insights or lesser-known facts about the theorem

Here's a basic problem to get us started:Problem: Given a right-angled triangle with legs of lengths 3 and 4, what is the length of the hypotenuse?

Looking forward to your insights and discussions!

 

[Mark44]

I like the notion that that you can deviate from The original difference of two squares with the custom expansion of ( a^2 + b^2 ) = a^2 + 2ba + b^2

The equation above is not true, so I'm not sure what you're doing. Did you mean $(a + b)^2 = a^2 + 2ab + b^2$?

 

[eddy1946]

The USAF 1970s firecontrol radar operators -- christened the "Crow Killers" after WWII radar intercept units in UK -- sang an excerpt from an old Danny Kaye comedy "The Court Jester" often shown on television when we were children.

From an online AI search:


As we were all "TV babies", raised in front of televisions when musical comedies were the rage among adults, we watched "Wizard of Oz" and "Court Jester" repeatedly.

For confirmation see the scene in Stanley Kubrick's "Full Metal Jacket" where young marines leaving a devastated Hue City in 1968, sing the Disney theme for "The Mickey Mouse Show" in unison. I borrowed the term "TV babies" from Steven Spielberg's "Natural Born Killers" with the connotation that watching TV as children desensitized us to excessive violence.

Back on topic, @Hornbein is correct that movies (and TV shows and, by extension, video games) can teach as well as entertain.

Close, but no cigar. The Danny Kaye film featuring "The square on the hypotenuse" was "Merry Andrew" (1958). See:

 

[renormalize]

I borrowed the term "TV babies" from Steven Spielberg's "Natural Born Killers" with the connotation that watching TV as children desensitized us to excessive violence.

Huh? IMDB.com nowhere lists Steven Spielberg in the credits for "Natural Born Killers" which was directed by Oliver Stone from a story by Quentin Tarantino.

 

[Klystron]

Huh? IMDB.com nowhere lists Steven Spielberg in the credits for "Natural Born Killers" which was directed by Oliver Stone from a story by Quentin Tarantino.

Thanks for the correction. Meant to credit Tarantino as writer or Stone as director with "TV babies" expression. If memory serves (poorly by the evidence), the line is not spoken but appears written in English in Grandfather's hogan during the eponymous characters' surreal peyote hallucinations.

 

[Klystron]

Close, but no cigar. The Danny Kaye film featuring "The square on the hypotenuse" was "Merry Andrew" (1958). See:

Good catch. Judging from your handle you precede me on this blue planet by 6-7 years, about the age of my eldest sister who chose the movies we watched on TV when I was a tyke. No way I knew the name of the film but I never forgot Kaye's fanciful rendition of Pythagorean equation. Besides, distributors often change the name of movies released in different countries. My first attempt at 'AI' lookup flopped.

The pedagogical point is that my students, most younger, remembered the same song from TV and correctly applied the mathematical relationship. This was before electronic calculators. So we basically applied trig functions on the fly.