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 6h answered Definite integral with cube roots of trig functions 21h comment Is there $n$ such that $n,n^2,n^3$ start with the same digit ($\neq 1)$ Here's one way to work it out. You want the fractional parts of $\ln n,\,\ln n^2,\,\ln n^3$ to be approximately equal, so $n$ should be near a power of $10$. We're now allowed to go just over, so go just under instead. If $n=10^k-1$ then $n^3\approx 10^{3k}\left(1-3\times 10^{-3k}\right)$, so $k=1$ won't help but $k=2$ will. 1d comment how can we define closed set or open set for a set of matrices? @jaggu Let $S$ denote the desired set of $\Delta<0$ matrices. Andre's comment noted an inner product on $\mathbb{R}^{2\times 2}$, which we can use to define a distance metric. Let $B\left(A,\,r\right)$ denote the radius-$r$ open ball in $S$ of centre $A$. Since $\Delta$ is continuous in all entries of $A$, for sufficiently small $r>0$ we have $B\left(A,\,r\right)\subseteq S$. 1d comment how can we define closed set or open set for a set of matrices? @jaggu You're considering the set of $A$ satisfying $\Delta\left(A\right)<0$. Can you see why such a set would contain a neighbourhood of each element? Feb 11 revised Can an infinite sum of irrational numbers be rational? deleted 38 characters in body Feb 10 answered How do I determine if A is diagonalizable Feb 10 comment What are some surprising appearances of $e$? To make $e$ out of this, take determinants with $t\text{tr} A = 1$. Feb 10 answered number of real solution of $\sin x+2\sin 2x = 3+3\sin 3x\;\forall x\in \left[0,\pi\right]$ Feb 10 answered Can an infinite sum of irrational numbers be rational? Feb 9 answered Find the derivative of $y=\frac{\tan(x)}{1+\tan(x)}$ Feb 9 answered Propositional-Calculus/ Set Theory Proof using Identities Feb 8 answered Contradiction! Any Symbol for? Feb 7 answered Prove that $\frac{7}{12}<\ln 2<\frac{5}{6}$ using real analysis Feb 6 answered Show the given series is a solution of $y''-xy'-y=0$ Feb 6 comment solution of differential equation $\left(\frac{dy}{dx}\right)^2-x\frac{dy}{dx}+y=0$ @NoChance That's the special case $a=0$. Feb 6 revised solution of differential equation $\left(\frac{dy}{dx}\right)^2-x\frac{dy}{dx}+y=0$ added 29 characters in body Feb 6 answered solution of differential equation $\left(\frac{dy}{dx}\right)^2-x\frac{dy}{dx}+y=0$ Feb 5 answered Solve $x^2 = 2^x$. Feb 5 answered Catherine is now twice as old as Jason but 6 years ago she was 5 times as old as he was. How old is Catherine now? Feb 5 revised How to get this result of integral? deleted 104 characters in body