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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.