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 Aug 17 comment About the cardinality of natural numbers [Solved] The set of all natural numbers is defined to be countable. The concept of countability is, when $A$ is countable, we can label the elements of $A$ as a number and can count them; We observe that we have used the natural numbers when enumerating and numbering something. Aug 17 accepted Relation between topological denseness and denseness over poset Aug 16 comment Proving or disproving of a homomorphism between two groups The fact that $\Bbb{Q}^\times$ does not give the direct proof that there is no non-trivial homomorphism between $\Bbb{Q}$ and $\Bbb{Q}^\times$, but it gives an motivation. I describe a proof in preceding comment. Aug 16 comment Proving or disproving of a homomorphism between two groups You might know what divisible group is. $\Bbb{Q}^\times$ is not divisible, i.e. there is $q\in \Bbb{Q}$ and a natural number $n$ s.t. $q \neq p^n$ for all rational $p$. Aug 16 comment Proving or disproving of a homomorphism between two groups $\Bbb{Q}$ is divisible, whereas $\Bbb{Q}^\times$ is not. Let $f:\Bbb{Q}\to\Bbb{Q}^\times$ be a non-trivial homomorphism. Whataver $f(1)$ is, you can find some sufficiently large $n$ s.t. $f\left(\frac{1}{n}\right) = f(1)^{1/n}$ is irrational. (You might find that $f(1)\neq 1$. In that case $f$ should be trivial.) Aug 16 comment for which positive integer $m$ does $(ab)^{2015} = (a^2 + b^2)^m$ have positive integer solutions If $m< 2015/2$ then $(a^2+b^2)^m\le (ab)^{2m} < (ab)^{2015}$ so it has no positive integral solution in that case. Aug 16 comment How to expand $(x_1 + x_2 + x_3 + x_4 + x_5 +\cdots+x_n)^{2}$ That is the generalization you might want. Aug 16 comment Proving or disproving of a homomorphism between two groups It seems to me that there is no general way to prove the existence of non-trivial homomorphism between two groups. One can consider that if $f$ is a homomorphism between given two groups then show that $\ker f$ is whole domain or $f=0$, but its proof depends on the properties of the two groups. Aug 16 comment How to expand $(x_1 + x_2 + x_3 + x_4 + x_5 +\cdots+x_n)^{2}$ Have you ever heard about the multinomial formula? Aug 16 revised $\lim_{x\to 0}\frac{x-\int_{0}^{x}\cos u^2 du}{6\sin^{-1}x-6x-x^3}$ edited title Aug 16 revised Help with Set Problem added 2 characters in body Aug 16 comment Arc Length Parametrisation It seems hard to integrate, although the result is elementary; see the result from WolframAlpha. Aug 16 revised Arc Length Parametrisation added 9 characters in body Aug 16 revised Are the hyper-reals countably transitive? edited tags Aug 16 reviewed Looks OK calculating costs of manufacture Aug 16 answered Countable product of first countable Spaces is first countable Aug 16 asked Relation between topological denseness and denseness over poset Aug 14 revised Ultrapower construction of the Hyperreal numbers added 11 characters in body Aug 14 comment Numerically Calculating the solution of very complex equations @Houtarou Asking on SE communities is sometimes difficult due to unawareness of writing style or syntax of questions on SE, so we can edit questions written by others! Aug 14 comment Numerically Calculating the solution of very complex equations Hello, Houtarou-san! Please use the triple-space when you type a code source.