In a recent paper it was stated (and maybe proved) that we can solve any polynomial equation with nested radicals. Here “nested radicals” means expression such as: $$ \sqrt[n]{a+b\sqrt[n/p]{a+b\sqrt[n/p]{a+b \cdots}}} $$ i.e. with infinite radicals nested each other. This means that every algebraic number can be expressed by a sort of ”generalized radicals” and, since […]

Given the following equations: $$a=\frac{py+qx}{2pq}$$ $$b=\frac{py-qx}{2pq}$$ Where p and q are some real constant number. And $(x, y)$ are some arbitrary real number. Any number can be inputted as $(x, y)$ but only those which produce whole integers for $a$ and $b$ respectively, are considered valid. Given these conditions, how can I find the valid […]

There’s the usual mapping $2^\mathbb{N}\mapsto\mathbb{R}_{[0,1)}$ (where $2^\mathbb{N}$ denotes the powerset of $\mathbb{N}$) given by $x\in2^\mathbb{N}\mapsto\sum_{i\in x}\frac{1}{2^i}\in\mathbb{R}$. I’m interested in the inverse $\mathbb{R}_{[0,1)}\mapsto2^\mathbb{N}$, i.e., given $r\in\mathbb{R}$, what’s the corresponding $x_r\in2^\mathbb{N}$ that generates it? (Note: I’m just guessing appropriate tags) Is that $x_r\in2^\mathbb N$ explicitly constructible? Or its characteristic function, i.e., for $r\in\mathbb{R}_{[0,1)}$ and $x_r\mapsto r$, $\chi_r(i)=\left\{{1\ […]

I realize that when a matrix is symmetric, then it must have all real eigenvalues. However, I am doing research on matrices for my own pleasure and I cannot find a mathematical proof or explanation when a matrix will have all real eigenvalues except for when it is symmetric. I am dealing with matrices such […]

Problem Assume that $a,b\in\mathbb{R}-\{0\}$ and that $a+b\not=0$. Prove that $\frac{1}{a}+\frac{1}{b}\not=\frac{1}{a+b}$. My Proof Let’s assume that $\frac{1}{a}+\frac{1}{b}=\frac{1}{a+b}$, then it follows that $$ \begin{equation} (a+b)^2-ab=0 \end{equation} $$ Let $x=a+b$ and $y=ab$. Now $b=x-a$ and so $y=a(x-a)=ax-a^2$. The previous equation can be written as $$ x^2-y=0 $$ Substituting $y=ax-a^2$ in this equation gives $$ x^2-ax+a^2=0 $$ The discriminant […]

Can anyone prove here that for $a,b\in\mathbb R\setminus\mathbb Q$ we have that the set $$ \{(m+ka,n+kb) : m,n,k\in\mathbb Z\} $$ is dense in $\mathbb R^2$? I know that the projections onto the coordinate axes both are dense in $\mathbb R$, but I cannot prove the density of the set in two dimensions. EDIT: It was […]

Prove: $ 0 \le a \lt b$ implies $ 0 \le a^2 \lt b^2 $ and $0 \le \sqrt{a^3} \lt \sqrt{b^3}$. Now show that the statement is false if the hypothesis $a \ge 0$ or $a \lt 0$ is removed. EDIT: Someone mentioned I should give context. I know how to disprove the statement. I […]

The inverse of $+$ is $-$, of $\times$ is $/$ and of $\text{^}$ is Log. Continuing upwards hyperoperationally, what is the inverse of $↑↑$? Whats more somtimes values that are fixed are given names, some being labels such as: $2\times = \text{double}\\ \text{^}2 = \text{squared}$ And some that are also other functions, such as: $\text{^}0.5 […]

What does it mean for a function $f(x)$ to be differentiable at $x_0$? I need this to understand more concepts in real-analysis and calculus. Thank you.

Without workin in a rigorous formal system, how can one intuitively establish that multiplication and addition are associative operations on the real line (including negative numbers)?

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