I Learned the Pythagorean Theorem(danq.me)
danq.me
I Learned the Pythagorean Theorem
https://danq.me/2025/11/13/pythagorean-theorem/
8 comments
I have explained the Pythagorean Theorem to many people. It is part of an explanation that I give that simple does not mean easy. Math is simple in a way that we are not wired for. And so a key to not being frustrated is to realize that there is nothing wrong with us that we sometimes struggle to understand simple things.
As part of it, I show how simple the Pythagorean theorem is to prove. The same proof as https://etc.usf.edu/clipart/43500/43501/pythag3_43501.htm. It can literally be drawn on the napkin.
You start with two squares of size a+b. You cut one into a square of size a, a square of size b, and 4 right-angled triangles a-b-c. You cut the other into 4 right angled triangles and a square of size c. When you eliminate the triangles (that have equal area), we're left with a^2 + b^2 = c^2.
The point being that it can be very hard to come up with such a simple thing. And it can sometimes take a while to truly accept it. Because we messy humans are wired for certain kinds of complex - like recognizing voices - and not for always getting simple right.
As part of it, I show how simple the Pythagorean theorem is to prove. The same proof as https://etc.usf.edu/clipart/43500/43501/pythag3_43501.htm. It can literally be drawn on the napkin.
You start with two squares of size a+b. You cut one into a square of size a, a square of size b, and 4 right-angled triangles a-b-c. You cut the other into 4 right angled triangles and a square of size c. When you eliminate the triangles (that have equal area), we're left with a^2 + b^2 = c^2.
The point being that it can be very hard to come up with such a simple thing. And it can sometimes take a while to truly accept it. Because we messy humans are wired for certain kinds of complex - like recognizing voices - and not for always getting simple right.
The other old 'proof without words' is this one
https://en.wikipedia.org/wiki/Xuan_tu
https://en.wikipedia.org/wiki/Xuan_tu
I still have difficulties understanding on a high level why lengths in triangles can produce irrational numbers.
I guess once you accept that area in two dimensions involves multiplication, it is a necessary consequence.
I wonder what it means for projects such as wolfram physics where space is discrete. Do truly right angled triangles even exist in nature?
I wonder what it means for projects such as wolfram physics where space is discrete. Do truly right angled triangles even exist in nature?
The day I learned the Pythagorean theorem, I also learned the Triangle inequality. From then on every corner parking lot or diagonal sidewalk through a park became a shortcut to be taken, and enjoyed.
Little wonder that the way I got through the more boring homework assignments in 100 and 200 level CS classes later on was to turn in the most efficient version of the answer instead of the most expedient.
Little wonder that the way I got through the more boring homework assignments in 100 and 200 level CS classes later on was to turn in the most efficient version of the answer instead of the most expedient.
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I had a similar experience when I wanted to draw a circle on my Atari 800XL. There's no prebuilt drawing function for circles, so I had to learn about sine and cosine with the help of my dad. I then figured out from there how to do 3d graphics. It was a great learning experience (for me at least!).
I've seen many proofs of the Pythagoras theorem, both visual and formulaic. I've found this to be the best explanation to date
https://www.youtube.com/watch?v=tTHhBE5lYTg
https://www.youtube.com/watch?v=tTHhBE5lYTg
I would be happy to know if others had a similar experience. I date myself though.
This kit was Russian made and had just excellent finish, tiny chrome plated nuts and bolts.I haven't thought about it in a while.
Now I need to look for it at my parent's house.