Caleb Jares
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 Dec3 comment Is it possible to combine the formula for $f_0$ and $f_{n\ge1}$ to get $f_{n\ge0}$? Oops, you're right. I thought I was cancelling out the $0$ case, but it was already cancelled. However, if we were actually considering the case where $A(x)=-1+x+6x^3+...$, would we be able to find $f_{n\ge0}$? Nov17 comment On irreducible polynomial What is $\mathbb{Q}(\sqrt2,\sqrt3)$ again? Is that the polynomial ring over the rationals modulo $\sqrt2$ and $sqrt3$? Nov7 comment What are some examples of notation that really improved mathematics? That's why I write $\varepsilon \in X$ in my delta-epsilon arguments :) Nov7 comment What are the most overpowered theorems in mathematics? Whoops, just went through another round in my head. 0 is indeed less than 1. Sep30 comment How to prove that $\frac{(mn)!}{m!(n!)^m}$ is an integer? Interesting. I worked out that those are equal algebraically, but I'm interested to know if there is a combinatorical argument to show they're equal. Sep29 comment How to prove that $\frac{(mn)!}{m!(n!)^m}$ is an integer? Why exactly is $\binom{kn}{n}$ divisible by $k$? Jul26 comment Find the sum of series $\sum_{n=0}^\infty\frac{(4n)!}{(4n+4)!}$ Forgive me for asking, but what college course would I take to learn how to answer this question in the same way? Apr17 comment Showing a polynomial $f\in\mathbb Q[x]$ is irreducible if it has rational coefficients? Ok I see. Thanks Mar10 comment In polar/cylindrical coordinates, does $r dr d\theta=r d\theta dr$? The way you visualize it makes more sense for cylindrical, thanks! The way I've been visualizing is a little different. For example, consider a triangle bounded by $y=0,x=1,y=x$. If I integrate over $y$ first, then in my mind I'm getting rid of the $y$ dimension, so now I just have a line from $x=0$ to $x=1$. Mar7 comment What was the first bit of mathematics that made you realize that math is beautiful? (For children's book) Which proof of $\sqrt{2}$ being irrational? Feb28 comment In a group $G$ with operation $\star$, can I apply $\star$ to both sides of an equation? @GerryMyerson I meant that if you went from $a=b$ to $a^2=b^2$, you can don't lose any information because you still have the two equations. Feb28 comment In a group $G$ with operation $\star$, can I apply $\star$ to both sides of an equation? @GerryMyerson Isn't going from $a^2=b^2$ to $a=b$ the same as $a^2=b^2\implies a=b^2/a$, and because $a=b$, $\implies a=b^2/b=b$. Feb14 comment Proving $4^{47}\equiv 4\pmod{12}$ Thanks, this is a great way to prove it without induction! Jun22 comment When representing a base-n number in decimal ($\frac{x}{n^l}$), will there be a different set of terminating representable numbers than base-10? Both mean the same thing. If there's a different set of terminating numbers there will also be a set of non-terminating numbers. Correct? Jun22 comment When representing a base-n number in decimal ($\frac{x}{n^l}$), will there be a different set of terminating representable numbers than base-10? You confirm my revised suspicion: the set of representable non-terminating numbers does depend on the base. Thanks! Jun22 comment When representing a base-n number in decimal ($\frac{x}{n^l}$), will there be a different set of terminating representable numbers than base-10? Yes, this is what I mean. Jun8 comment Is it generally accepted that if you throw a dart at a number line you will NEVER hit a rational number? @QiaochuYuan what languages are better at making these distinctions? Dec12 comment How many unique pairs of integers between $1$ and $100$ (inclusive) have a sum that is even? Oh why thank you very much :) Nov15 comment Evaluating $\sum\limits_{n=0}^{20} \frac{(-1)^{n}2^{n+1}}{3^{n}},$ Thanks! That worked. Nov15 comment Evaluating $\sum\limits_{n=0}^{20} \frac{(-1)^{n}2^{n+1}}{3^{n}},$ Yes, just the partial sum. I know it's wierd, but my professor put it on our test and I had no idea how to do it then. Now he offered test corrections because the average was below 60 and I still don't know how to do it.