Hi Steemians. Most of you probably had this classmate in high-school which tried to convince you that 1 is not equal to 1. The "proof" usually takes the form
But I upgraded this proof using some other fake proof methods (see the How did I create this "proof"?- section below) to a version which does not require any algebra! You just have to draw some pictures and you can convince somebody that 3=1!
We start with a green line of length one:
This BB does not approximate the green line very well. I will scale BB by a factor 1/2 . Let us lazily denote the resulting line by BB/2. Then I can cover the green line with two BB/2:
To approximate the green line even better I am going to scale BB by factor 1/3. The resulting line will be denoted by BB/3. Then I can cover the green line with three BB/3:
The edges of BB/3 are length 1/6. Three BB/3 have 18 straight edges so in total it has length 18/6=3. So we again get the same length as BB.
To get an even better approximation we can scale BB by a factor 1/n. The resulting line will be denoted by BB/n. Then I can cover the green line with with n pieces of BB/n. A single edge of BB/n is length 1/(2n). Observe that n pieces of BB/n consist out of 6n straight edges. So the total length is 6n/2n=3. Which means that the total length is always 3. In case you don't believe this I have drawn n=4,5,6 below:
If you look in the above figure then the baby blue curve is approximating the green line better and better. But if I let n go to infinity then BB/n will approximate the green line. The green line has length 1 and the total length of n pieces of BB/n is always 3. But then 3 = 1.
There is a perfectly fine reason why you cannot conclude that 1 is 3. Mathematics did not just break. But I hope you can find the reason by yourself ;). Leave a reply below if you think you have found the answer.
Hint: Collect all the horizonal and vertical lines separately.
This post is based on the famous pi =4 "proof" which I first encountered through a lovely Vi Hart video (click here). In her video she also gives a "proof" that square root 2 is equal to 2 which she then extends to a "proof" that a line of length 4 is equal to two times square root 2. The difference between my method and her method is that she uses a zigzag curve on the line . This zigzag curve requires some preperation to introduce so I think my version is a bit more elegant. Vi Hart does not really explain why the method yields an incorrect proof. However, Vi Hart seems to suggest a reason but it is incorrect (as you will hopefully also find out!).
Figures made with inkscape. All the equations written using a phyton latex extension for inkscape. Both are free-ware!
I actually wanted to do a post on a fun model/program I made for spreading infectious diseases but I found out in the weekend that even as a simple model it is really really really bad. On Monday I had the idea for this post. It took me a few nights to make this idea concrete so I don't have time for a second math post this week. Hopefully, I can create a math story about an infectious disease model by next week.
Thanks for being so kind to read my post. You are awesome! Please follow me if you enjoyed it. If you have any questions just post them below and I will answer them. Or if you might have a nice topic you want me to cover also let me know below. :o)
