Timeline for Examples of interesting false proofs
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
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| Dec 1, 2023 at 10:52 | comment | added | Andrea Marino | @TimothyChow: I'd like to reformulate of your argument in terms of finite coverings of the punctured plane (or sheaf theoretic). Would be a nice example for some undergrad courses! :) | |
| Sep 2, 2021 at 23:53 | comment | added | Qfwfq | (p.s. of course also Timothy Chow's explanation is perfectly right, and equivalent) | |
| Sep 2, 2021 at 23:40 | comment | added | Qfwfq | @Elliott: I think it's more about domains of functions than about powers. Starting from $f:\mathbb{R}\to [0,\infty)$, $x\mapsto x^2$, restrict to $f_{+}:=f|_{[0,\infty)}$ and $f_{-}:=f|_{(-\infty,0]}$. Let the inverses be $g_{+}:=(f_{+})^{-1}$ and $g_{-}:=(f_{-})^{-1}$. Now $-1\in (-\infty,0]=\mathrm{dom}(f_{-})$ and $-1=g_{-}(f_{-}(-1))$. So far nothing strange. Then the false proof of my comment is explained by: $-1=g_{-}(f_{-}(-1))=g_{-}(1)\neq g_{+}(1)=1$. That is, the $(\ldots)^{1/2}$ in the notation is actually $g_{-}$ but $g_{+}$ is applied instead in the last passage. | |
| Sep 2, 2021 at 13:09 | comment | added | Timothy Chow | @Elliott I would rather phrase it this way: $x^y$ is (or can be, if $y$ is not an integer) multi-valued. So $(a^b)^c$ represents a set of possible values, as does $a^{bc}$. These sets will overlap but they may not be equal, unless we are careful to specify (or adopt a convention) which of the multiple values we're selecting. The simplest example is that $1$ has two square roots, and by convention we usually interpret $1^{1/2}$ to be the positive square root, but when we apply the "law" $(a^b)^c = a^{bc}$, we must carefully select the correct value out of the multiple possible values. | |
| Sep 2, 2021 at 12:45 | comment | added | Elliott |
@Qfwfq, I'm embarrassed to learn, after having done a math degree, that $$(a^{b})^{c}$$ doesn't always equal $$a^{bc}$$ I then googled and watched famous youtube videos that introduce the equation (for high school kids), and none of them mentioned that either a should be non-negative or b, c must be integers. I'm shocked that this hasn't caused havoc for my math/programming life. I need to go back and prove some of my basics.
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| Apr 5, 2019 at 18:51 | comment | added | Qfwfq | This is not conceptually different from $-1=(-1)^{2/2}=((-1)^2)^{1/2}=1^{1/2}=1$ | |
| Nov 27, 2013 at 16:20 | comment | added | Newb | Oh, this is really good. | |
| Jan 15, 2013 at 4:51 | comment | added | Noam D. Elkies | Math departments have the best bathroom graffiti. | |
| Apr 22, 2012 at 18:37 | history | answered | Timothy Chow | CC BY-SA 3.0 |