A Visual Proof
I found a strange piece of paper in an old pile. I believe that it is a visual proof of the following statement:
If ∞ = 1/0, then 0 = 1/∞.
Proof. Assume ∞ = 1/0. Rotate each side of the equation counterclockwise 90 degrees. We get 8 = −10. Subtract 8, getting 0 = −18. Then rotate both parts back: 0 = 1/∞. QED.Share:
When I was in school, at the very beginning of calculus in the study of limits, the notions of 1/∞=0, 1/0=∞ were introduced, alongside ∞/∞ and 0/0 being undefined. We passed over them as with any other definition, and used them without difficulty to prove hosts of theorems and results.
Later in university, 1/∞ and 1/0 were treated like religious concepts. In modern secular calculus courses they are simply not discussed at all, or for the most part avoided completely. More advanced topics treat them like theological mysteries, dancing about them with carefully choreographed reasoning, only ever reaching conclusion for the case of a particular series or limit. Still others treat them like outdated superstitions, and mock their careless application.
For a long time, I essentially dismissed these old concepts from my mind. Recently however, while reading a book on complex functions, the authors suddenly defined the function J(z)=1/z on the Riemann sphere to satisfy J(0)=∞ and J(∞)=0. I paused; a fog descending on my thoughts. And out of the mists came my old acquaintances 1/∞=0 and 1/0=∞. They passed me by, with only a brief nod and the words “He’s talking about us”. They disappeared once more, the fog lifted, and I returned to the book, which subsequently became much clearer as I learned to recognise the concepts behind the rituals that the authors had been obliged to invoke.
If anyone is interested in seeing this pair being invoked without formality, there’s a paper, “Leonhard Euler’s Integral: A Historical Profile of the Gamma Function”, which describes some of Euler’s crazy manipulations of the above which we would not even bring ourselves to consider nowadays. I found it invigorating, but then again I am someone with only one shirt and no ties, so YMMV.1 June 2012, 5:47 pm
I love things like this, this one especially because the beginning and end actually *kind* of make sense (at least to a limit).
The other math/words/symbols joke I’ve heard was that for any n, (sin x)/n = 6. Cancel the n and the answer is clear.6 June 2012, 7:34 am
hahaha, hahaha! Very funny! I am now a fan!9 June 2012, 5:17 pm
Note that the converse is also true ! 🙂18 June 2012, 6:29 am
Hahaha, this is funny. This goes into my collection of funny math pictures. Thanks!5 July 2012, 10:49 pm
I’m confused surely by logic 1/infinity = 0 (to 0 d.p.) rather than actually being 0 as it should be 0.00000….for infinite length17 December 2019, 5:59 am
Note this converse also true!30 May 2022, 3:22 am