general
Is the sum of two rationals or two irrationals irrational?
1. I know this statement is false (if I am correct) but how to prove it's false? "The sum of two rational numbers is irrational." 2. I know this stat...
Apr 14, 2026
Archive: General
general
Is the Polynomial Factor Theorem true over all commutative rings?
I'm reading Fraliegh's text on abstract algebra, and his statement for the factor theorem is, "An element $a\in F$ ($F$ is a field) is a zero of...
Apr 14, 2026
general
Finding two roots of unity which their sum is also root of unity
Could we find two complex numbers such as $z_1$ and $z_2$, where both of them are roots of unity (de Moivre numbers) and also when we define $$s = z_1 + z...
Apr 13, 2026
general
What does the term "unbiased estimator" mean?
In my textbook for my statistics class, it says that $s^2$, sample variance is a "unbiased estimator" for population variance, $\sigma^2$. Does ...
Apr 13, 2026
general
Green's theorem, double integral over a triangle
The original problem is given as thus Find $$\iint_Dx\,dxdy $$ where $D$ is a triangle with vertices $(0,2),(2,0),(3,3)$. Green's theorem says that $...
Apr 13, 2026
general
Why a linear transformation is invertible if it is **both** right and left invertible?
My intuition on invertible linear transformation $A:V \rightarrow W$ is that there is another linear transformation that kind of sends the vectors back so...
Apr 13, 2026
general
Angle between two 3D lines
I know for given 2 vector $\vec{u},\vec{v}$ the angle between them achieved by - $$\cos{\theta} = \frac{\vec{u} \cdot \vec{v}}{\|\mathbf{u}\|\|\mathbf{v}\...
Apr 13, 2026
general
$\mu$-recursive definition of ulam (3n+1) function
$\newcommand{\ulam}{\operatorname{ulam}}$ The ulam function is defined as $$ \ulam(x) = \begin{cases} 1 & x = 1 \\ \ulam\left( \frac{x}{2}\right) &...
Apr 13, 2026
general
Why is a triangle with edges identified according to $aab$ a Mobius strip?
Suppose we've got a fundamental polygon with edge-word $aab$. It's not too hard to use classification of compact surfaces to show that this is a...
Apr 13, 2026
general
integrating $\int_{x_0}^{x}\arccos(a-\cos(x'))dx'$
I've been looking at an integral of the form $$\int_{x_0}^{x}\arccos(a-\cos(x'))dx'.$$ We can set $x_0$ as to give no contribution from the...
Apr 13, 2026
