Welcome to The Nonlinear Library, where we use Text-to-Speech software to convert the best writing from the Rationalist and EA communities into audio. This is: Explaining a Math Magic Trick, published by Robert AIZI on May 5, 2024 on LessWrong.
Introduction
A recent popular tweet did a "math magic trick", and I want to explain why it works and use that as an excuse to talk about cool math (functional analysis). The tweet in question:
This is a cute magic trick, and like any good trick they nonchalantly gloss over the most important step. Did you spot it? Did you notice your confusion?
Here's the key question: Why did they switch from a differential equation to an integral equation? If you can use (1x)1=1+x+x2+... when x=, why not use it when x=d/dx?
Well, lets try it, writing D for the derivative:
f'=f(1D)f=0f=(1+D+D2+...)0f=0+0+0+...f=0
So now you may be disappointed, but relieved: yes, this version fails, but at least it fails-safe, giving you the trivial solution, right?
But no, actually (1D)1=1+D+D2+... can fail catastrophically, which we can see if we try a nonhomogeneous equation like f'=f+ex (which you may recall has solution xex):
f'=f+ex(1D)f=exf=(1+D+D2+...)exf=ex+ex+ex+...f=?
However, the integral version still works. To formalize the original approach: we define the function I (for integral) to take in a function f(x) and produce the function If defined by If(x)=x0f(t)dt. This rigorizes the original trick, elegantly incorporates the initial conditions of the differential equation, and fully generalizes to solving nonhomogeneous versions like f'=f+ex (left as an exercise to the reader, of course).
So why does (1D)1=1+D+D2+... fail, but (1I)1=1+I+I2+... works robustly? The answer is functional analysis!
Functional Analysis
Savvy readers may already be screaming that the trick (1x)1=1+x+x2+... for numbers only holds true for |x|
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