A classic and basic problem in finance is getting people to realize that percentages don't add. This may seem trivial, but if something goes up 10% then down 10% it has gone down, overall, 1%. The problem, of course, is that after it goes up 10%, the -10% is a bigger number. This is usually called the markup/markdown problem.
Algebraically this is simple: $(1+r)(1-r)=1-r^2<1$, for all $r$ other than zero.
I know this is trivial, but I am trying to think of why (or if) this can be related to some deeper or more profound insight or theorem from analysis. But it is so obvious arithmetically that I keep drawing a blank.
Displacements are what people are usually used to, and to put two displacements together we use addition: "walk forward one step and then back two" will clearly leave you one step behind where you started.
Ratios of course are composed using multiplication, which is hard. Logarithms and exponentials can change this into addition instead. Imagine a world where "increase the price by 10 percent" actually meant multiplying the price by $e^{0.1} \approx 1.1052$. Then "decrease the price by 10 percent" would be multiplication by $e^{-0.1}$, and would correctly cancel. I've just written down my operations using different numbers, so I can add the numbers instead. (Aside: this is pretty much what decibels are: turning the volume up 2 decibels is actually something like multiplication by $10^2$)
Notice how $e^{0.1}$ is pretty close to $1.1$ in value. This will work for any small numbers, since the first order Taylor approximation of $e^x$ is $1+x$. I think the fatal flaw is conflating this exponential with its approximation, which works pretty well for small $x$, and few operations. It breaks outrageously for larger $x$ though: the operation which undoes increasing a price by 100% is decreasing it by 50%, not 100%.