B Understanding the Pythagorean Theorem

[wrong thunder]

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!

 

[.Scott]

Let's say I have two points, x1,y1 and x2,y2.
The distance between them will be: $\sqrt{(x_1-y_1)^2 + (x_2-y_2)^2)}$
In other words, the square of the distance is equal to the sum of the squares of the differences of each Cartesian coordinate (because the Cartesian coordinates are at right angles to each other).

Now let's say we want to know the absolute value of a complex number:
Absolute value of $a+bi$: $\sqrt(a^2 +b^2)$
Or the unit vector associated with x,y: $\frac{x} { \sqrt{x^2+y^2}}, \frac{y} { \sqrt{x^2+y^2}}$

 

[PeroK]

• 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

The Pythagorean theorem appears everywhere in physics. It's fundamental to the notion of Euclidean space.

A couple of years ago, a couple of high school students produced a new, elementary geometric proof:

https://www.scientificamerican.com/...ve-pythagorean-theorem-heres-what-that-means/

There are an infinite number of integer Pythagorean triples: that is three whole numbers whose squares satisfy the equation. Your example of 3, 4 and 5 is the smallest example.

 

[Hornbein]

The Pythagorean theorem appears everywhere in physics. It's fundamental to the notion of Euclidean space.

A couple of years ago, a couple of high school students produced a new, elementary geometric proof:

https://www.scientificamerican.com/...ve-pythagorean-theorem-heres-what-that-means/

There are an infinite number of integer Pythagorean triples: that is three whole numbers whose squares satisfy the equation. Your example of 3, 4 and 5 is the smallest example.

The ancient Babylonians inscribed on a clay tablet a table of a number of Pythagorean triples.

The Pythagorean theorem is used to find the variance of independent random variables. The volume of a sound can be considered a random variable. So the average volumes of two uncorrelated sounds add like a^2+b^2.

 

[fresh_42]

I like to consider it as a special case of the law of cosines. And I do not like its name. I mean, it wasn't as if this numerologist Pythagoras had found something new.

 

[mathwonk]

I like the version in Euclid. it has nothing to do with lengths, just comparing the "size" of some geometric figures. I.e. one can define equality of figures, via decomposition and reassembly, without assigning a number to that size. So the question of whether two things have the same size is more primitive than saying what number measures that size.

Euclid's version is that the square constructed on the hypotenuse of a right triangle has the same size as the two squares on the other two sides taken together.

The law of cosines, also in Euclid, says the square on any side of any triangle, differs in size from the two squares on the other two sides, by twice the size of a rectangle whose sides are (either) one of the other two sides, and a segment formed by dropping a perpendicular on that side from its opposite vertex. In the case of a right triangle, and its hypotenuse, that last segment has "length zero", i.e. is a single point, so indeed "Pythagoras" is a special case.

 

[robphy]

Check out:
https://www.cut-the-knot.org/pythagoras/ (so far, 122 proofs)
and
A Mathematical Fable - Numberphile

and
The Moral of the Scale Fable - Numberphile2


and

Visualising Pythagoras: ultimate proofs and crazy contortions - Mathologer

 

[robphy]

The Pythagorean theorem appears everywhere in physics. It's fundamental to the notion of Euclidean space.

A couple of years ago, a couple of high school students produced a new, elementary geometric proof:

https://www.scientificamerican.com/...ve-pythagorean-theorem-heres-what-that-means/

There are an infinite number of integer Pythagorean triples: that is three whole numbers whose squares satisfy the equation. Your example of 3, 4 and 5 is the smallest example.

https://maa.org/news/groundbreaking...eatured-in-the-american-mathematical-monthly/

Ne'Kiya Jackson and Calcea Johnson, students from New Orleans, Louisiana, have authored the lead article for the November issue of The American Mathematical Monthly, titled “Five or Ten New Proofs of the Pythagorean Theorem.”

Jackson, N., & Johnson, C. (2024). Five or Ten New Proofs of the Pythagorean Theorem. The American Mathematical Monthly, 131(9), 739–752. https://doi.org/10.1080/00029890.2024.2370240

 

[FactChecker]

1) The beliefs and life of Pythagoras are interesting. He thought that natural numbers were like Gods.
2) It was used to show that $\sqrt 2$ is not a rational number. The story is that it upset Pythagoras so much that he killed the person who discovered it.
3) It can be used to determine if a surface is flat or has some curvature.

 

[robphy]

2) It was used to show that $\sqrt{2}$ is not a rational number. The story is that it upset Pythagoras so much that he killed the person who discovered it.

Could it be said that
Pythagoras acted irrationally to
the discoverer of irrational numbers?

 

[TensorCalculus]

This is slightly more well known but...
Pythagoras and his group allegedly really, really had a thing against beans.

One of the stories describing Pythagoras's death includes him dying due to his reluctance to step on beans - whether it's actually true or not, it's definitely hilarious.

---

"Apparently, he was being chased, and he saw a field of beans, which he refused to cross. His assailants got to him and he was killed (but take all of this with a very large pinch of salt)

 

[DaveC426913]

Problem: Given a right-angled triangle with legs of lengths 3 and 4, what is the length of the hypotenuse?

5.

By the theorem of rote memorization.

 

[fresh_42]

5.

By the theorem of rote memorization.

No, by that Babylonian clay table @Hornbein has mentioned.

 

[Hornbein]

No, by that Babylonian clay table @Hornbein has mentioned.

Citations are key for career success.

 

[fresh_42]

Citations are key for career success.

Here we go:

 

[Hornbein]

This Babylonian clay tablet played a key role in my mathematical career. I didn't and still don't know how to generate the Pythagorean triplets. I realized that my skills lagged those of the ancient Babylonians. That's when I knew I wasn't cut out for a career in mathematics.

Though it is true they may have generated that table via trial and error.

 

[fresh_42]

Though it is true they may have generated that table via trial and error.

I don't think so. I rather agree with what is written on Wikipedia:

The cuneiform tablet Plimpton 322 contains 15 different Pythagorean triples, including $(56,90,106)\, , \,(119,120,169)\, , \,(12709, 13500,18541)$ which suggests that a method for calculating such triples was already known more than 3500 years ago.

The trick is using $(a+b)^2=a^2+2ab+b^2=a^2-2ab+b^2+4ab=(a-b)^2+4ab.$ Then we only need $a,b$ to be squares such that $4ab$ becomes a square.

 

[Hornbein]

I don't think so. I rather agree with what is written on Wikipedia:

The trick is using $(a+b)^2=a^2+2ab+b^2=a^2-2ab+b^2+4ab=(a-b)^2+4ab.$ Then we only need $a,b$ to be squares such that $4ab$ becomes a square.

I'm glad the someone noticed and could recognize the meaning of these numbers. Old school Austrian and USA education. I'll assume the archeologist solicited their advice on these curious number. It wasn't the usual tally of grain.

 

[fresh_42]

I'm glad the someone noticed and could recognize the meaning of these numbers. Old school UK education....

I'm sorry to disappoint you: (also Wikipedia)

In 1945, the researcher Otto Neugebauer discovered that these were Pythagorean triples.

Otto Eduard Neugebauer (born May 26, 1899 in Innsbruck, Austria-Hungary; died February 19, 1990 in Princeton, New Jersey) was an Austrian-American mathematician and astronomer.

And the table is now at Columbia University (NYC).

You beat me while I was searching.

 

[Hornbein]

You beat me while I was searching.

It happens. I almost always correct my writings after posting. I can't help it.

 

[DaveC426913]

It happens. I almost always correct my writings after posting. I can't help it.

Totally guilty.

 

[renormalize]

The trick is using $(a+b)^2=a^2+2ab+b^2=a^2-2ab+b^ 2+4ab=(a-b)^2+4ab.$ Then we only need $a,b$ to be squares such that $4ab$ becomes a square.

Did you perhaps mean to say:
"Then we only need $a,b$ to be positive integers such that $4ab$ becomes a square."?

 

[mathwonk]

probably he meant, we only need a,b to be squares, since then 4ab becomes a square. E.g. a=4, b=1, or a=9, b=4, or a=81, b=25, or a=144, b=25, or a= 15,625, b = 2916..... cool!

 

[Klystron]

...
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

...

I like to consider it as a special case of the law of cosines. ...

To be sure. About 50 years ago while teaching miltary radar operators methods of acquiring and tracking flying aircraft from ground based radars out in the field, I used the law of cosines as a rough method* to acquire aircraft cells based on partial position information.

Constructing right triangles simplifying the equations to Pythagoras theorem smoothed worried faces with the familiar expression. A group began chanting a popular refrain from a musical comedy, "The square of the hypotenuse of a right triangle is equal to the sum of the squares of the other two sides.". No reason math cannot be fun.

* For various reasons including curvature of the Earth and shape of EM fields, hyperbolic and/or spherical geometry provided more accuracy than Euclidean, but I liked to begin with a familiar formula.

 

[Hornbein]

Constructing right triangles simplifying the equations to Pythagoras theorem smoothed worried faces with the familiar expression. A group began chanting a popular refrain from a musical comedy, "The square of the hypotenuse of a right triangle is equal to the sum of the squares of the other two sides.".

The Scarecrow said that after having been awarded a diploma by the Wizard of Oz. Musical dramas can further education.

 

[Charles Link]

The Scarecrow said that after having been awarded a diploma by the Wizard of Oz. Musical dramas can further education.

To be more precise, I believe what the Scarecrow says is an incorrect take-off on the Pythagorean theorem that basically is never the case: "The sum of the square roots of any two sides of an isosceles triangle is equal to the square root of the remaining side." LOL

 

[Hornbein]

To be more precise, I believe what he says is an incorrect take-off on the Pythagorean theorem that basically is never the case: "The sum of the square roots of any two sides of an isosceles triangle is equal to the square root of the remaining side." LOL

Well by darn you are correct.

 

[pinball1970]

It's not the first proof I learned but it should have been. The one I know is the square inside a square but rotated.

The proof is nice but you get to see how you can mess around with squares, triangles, areas and see the end in sight as you are doing it.13 or 14 that would have made me think a little bit.

I think mathematics is great even though I struggle with the concepts. Aliens get here and tell us how crappy we are to each other and we are making a right mess we can always say, "Ok but we have this."

 

[Klystron]

The Scarecrow said that after having been awarded a diploma by the Wizard of Oz. Musical dramas can further education.

To be more precise, I believe what the Scarecrow says is an incorrect take-off on the Pythagorean theorem that basically is never the case: "The sum of the square roots of any two sides of an isosceles triangle is equal to the square root of the remaining side." LOL

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:

Danny Kaye sang about the Pythagorean theorem in the movie "The Court Jester" (1956). In the film, he performs a humorous song titled "The Vessel with the Pestle," which includes references to the theorem in a comedic context.

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 Oliver Stone's film "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.

 

[Albertus Magnus]

Carpenter often use it to check that their work is "square", i.e. joined at a right angle. The idea is to make a mark at three feet along one board followed by a mark at four feet along the other board, then check that the straight line distance from one mark to the other is exactly five feet.

 

[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.