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 Oct 18 comment Is there a way to get trig functions without a calculator? If there weren't a way to evaluate trig functions without a calculator, then how would anyone be able to program the calculators? Jun 24 comment Why do calculus students learn to think of the derivative as a limit? How else would you define it? Jun 20 comment How to solve $2^x+e^x=400$ @HenningMakholm: As an extreme example, consider Fermat's Last Theorem. Easy to state, but took 358 years to prove. Jun 20 comment Why isn't the empty set an element of $A \times B$, while it is a relation from $A$ to $B$? If you're a computer programmer (in a statically-typed language with templates/generics), it may help to think of the × operation as the function Set< Pair > CartesianProduct(Set a, Set b). The return value can be an empty set, but it can't contain an empty set because then the type would have to be Set instead of Set. Jun 4 comment Name of the highest power of 2 smaller than or equal to a given number "Two to the Power of the Floor of the Base-Two Logarithm" = TPFBTL of x, pronounced kinda like "top of bottle". Jun 2 comment Show that the question “Is there life beyond earth?” is decidable The question can straightforwardly be answered by the Python (a Turing-complete language) expression any(obj.location != Earth and obj.alive for obj in universe()). The only hard part is iterating every object in the universe. But if you assume a finite universe, it's theoretically possible. Jan 26 comment Can you be 1/12th Cherokee? @Keavon: Technically, the protagonist of that song is his own step-grandfather, not his own biological grandfather. Dec 30 comment Why rationalize the denominator? @Ooker: Note the "floor" brackets. Sep 12 comment In plain language, what's the significance of a field? @Hal: Not always. Another thing that would keep a field from having less/greater than is the numbers being conceived of two-dimensionally. Think of $\mathbb{C}$. Jul 25 comment inverse trigonometric equation $\displaystyle \tan^{-1}{x}+\cot^{-1}{x}=\frac{\pi}{2}$ Hint: $\cot x = \tan(\frac{\pi}{2} - x)$. Jul 16 comment Are all existence proofs by contradiction? Jul 4 comment How to distinguish walking on a sphere or on a torus? @dotancohen: Even if they could look up, people living on the outer parts of Planet Doughnut wouldn't have a view of the hole. Apr 20 comment Kid's homework: 4 equations 5 unknowns? Going crazy! @metacompactness: Not necessarily. For example, there could be half-siblings on this trip, with 1 father and 2 mothers. Or cases where one parent is unable to attend. Apr 5 comment Derivatives of exponential functions There's no $x$ in your function. Did you mean dt instead of dx? Apr 5 comment How can a piece of A4 paper be folded in exactly three equal parts? It would be interesting to see an answer that relies on the $\sqrt{2}:1$ ratio of ISO paper sizes. Mar 29 comment Why is there a different button for 'minus' and 'negative' on a calculator? @Rahul: Based on your comment, I assume that you have an RPN calculator. However, I've also seen distinct - and ⁻ buttons on infix-notation calculators like the TI-8x series, on which there's no obvious need to distinguish them. Mar 29 comment Why is there a different button for 'minus' and 'negative' on a calculator? This doesn't really answer the question. Although unary and binary minus are indeed distinct operations, a lot of programming languages, spreadsheet programs, etc., do manage to use the same symbol - for both of them. Mar 29 comment Can a piece of A4 paper be folded so that it's thick enough to reach the moon? Assuming a typical 0.1 mm thickness, how large would a sheet of paper need to be so that it could be folded 42 times? Mar 28 comment Can a piece of A4 paper be folded so that it's thick enough to reach the moon? Just to verify, I measured a package of 500 sheets of paper, and it was about 5 cm thick. So, 0.1 mm per sheet is a good estimate. Mar 28 comment Can a piece of A4 paper be folded so that it's thick enough to reach the moon? A piece of A4 paper cut in half 42 consecutive times would be A46.