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 Dec 11 revised Good books and lecture notes about category theory. Replace dead link Dec 10 awarded Excavator Sep 24 awarded Autobiographer Aug 31 awarded Yearling Jun 10 comment 'Obvious' theorems that are actually false @AnonymousPi $i = \exp\left(\frac{(4n + 1) \pi i}{2}\right)$ where $n \in {\Bbb N}$. Hence $i^i = \exp\left(\frac{-(4n + 1) \pi}{2}\right)$. Apr 16 awarded Nice Answer Oct 9 awarded Critic Sep 27 revised Good books and lecture notes about category theory. fixed link (old link 404s) Aug 31 awarded Yearling Jun 11 awarded Nice Question May 6 comment Prove $a+b+c+d$ is composite Note this does not work if you include 0 in the natural numbers. (Obvious counterexample: a = 0, b=3, c=0, d=4, so ab = cd = 0, but a+b+c+d=7 which is prime). Feb 11 answered 2013th derivative of a trigonometric function Jan 11 revised How to make two vectors orthogonal? added 227 characters in body Jan 11 revised How to make two vectors orthogonal? Very sloppy bad math on my part. Jan 11 revised How to make two vectors orthogonal? added 112 characters in body Jan 11 answered How to make two vectors orthogonal? Jan 10 comment How to solve this calculus problem? @MaoYiyi - There an infinite number of potential functions $f(x)$ that satisfy the given property (e.g., if you assume a form of Ax^4 + Bx^3 + Cx^2 + Dx + E; then A=-134/2475, B = 3622/2475, C=-31924/2475, D=106937/2475, E= -7369/165; or you could assume Ax^10+Bx^8+Cx^6+Dx^4+Ex^2 and get a totally different answer that meets the criteria). E.g., you can give a rough estimate of f'(2.5) based on f(2), f(3) -- even if its more of an average value of f'(x) between 2 and 3. Jan 10 comment How to solve this calculus problem? @proximal - I was trying to not suggest evaluate the limit, but to see f'(x) = lim h->0 (f(x+h) - f(x-h))/(2h) we can approximate f'(x) ~ (f(x+h) - f(x-h))/(2h) at several points, exactly as you said. Just was trying to leave it as "hints". Jan 10 answered How to solve this calculus problem? Dec 11 comment Does multiplying polynomials ever decrease the number of terms? Generalizing; let a,b,c,d be non-zero. (x^2 + a x + b)(x^2 + c x + d) = x^4 + (a+c)x^3 + (ac+b+d )x^2 + (ad + bc) + b d. So if a+c = 0, ad + bc + 0, and ac + b+ d = 0 it works. The first eqn gives c = -a; which reduces the second equation to (a)(d-b) = 0, and since a is non-zero means b=d. Hence the last equation becomes 2b - a^2 = 0. So this works for any non-zero a of the form: (x^2 + a x + a^2/2)(x^2 - ax + a^2/2).