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 Apr 19 comment Problem with basic definition of a tangent line. @AaryanDewan You move the second point towards the first without ever actually touching them together. The distance just gets smaller and smaller without actually reaching 0. (In other words, you take the limit as the distance approaches 0.) Or something like that, I haven't actually thought about tangent lines in a while. Mar 24 comment Why are turns not used as the default angle measure? Are degrees truly artificial? I always assumed they were based on an approximation of the number of terrestrial rotations per revolution. Which actually makes them quite natural, albeit very biased towards the terrestrial year. Jun 23 awarded Commentator Jun 23 comment Are older mathematics textbooks still “valid”? @HagenvonEitzen That last one is still true, though. Sep 24 awarded Autobiographer Jun 10 comment 'Obvious' theorems that are actually false @MJD Of the failure case, I meant. Jun 9 comment 'Obvious' theorems that are actually false Should it be assumed that "point" means "region on the dart board such that the area is 0" (or perhaps "0-dimensional region on the dart board", though the property would hold true for a 1-dimensional line as well), which would not necessarily coincide with the layman's definition of "point"? Jun 9 comment 'Obvious' theorems that are actually false What about higher-dimensional "edges"? (e.g. for $n=8$, there are no shared $(n-1)$-cubes, but what about $(n-2)$-cubes and so forth?) Or does the failure of the conjecture for $n>7$ imply that edges need not be shared either? Jun 9 comment 'Obvious' theorems that are actually false Would the logic equivalent be "This statement is false"? Jan 31 comment Is conditional probability also probability? @nomen Leaving aside Zado's (correct) comment, any (real, complex, or imaginary) number can be considered a ratio of itself out of 1. It's just that such treatment normally has no meaning on its own except as an intermediate step in an operation. Jan 31 comment Is conditional probability also probability? @smwikipedia To elaborate on Goos' comment, % is (usually) read as "percent", or more clearly, "per cent", where cent = 100. Thus, it defines an implicit ratio, in this case 80 out of 100 (which of course simplifies to the 4 out of 5 Goos mentions). In short, $P(X) = (100 * P(X))\%$. Jan 17 comment Why do logarithms produce such difficult problems As an extension, the reason the integer 2 (as opposed to a non-integer real value) is an exact solution is that this is a case of the more general equation $\log_{a}\left(x+u\right)=1-\log_{b}\left(x-v\right)$ where $x-v=1$ and $x+u=a$, which reduces the problem to $\log_{a}\left(a\right)=1-\log_{b}\left(1\right)$, $1=1-0$, $1=1$. Jan 14 comment Splitting a sandwich and not feeling deceived @user1729 Engineering students, not graduated engineers. It'd be their final design project. Jan 14 comment Splitting a sandwich and not feeling deceived @user1729 Engineering students would design the machine to ensure accurate proportioning to start with, you'd only need one button to start the machine (and to stop it, for safety reasons). Jun 25 awarded Informed May 30 awarded Supporter