Show that the perimeter of an octagon is $8(\sqrt{2} -1)$
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Show that the perimeter of an octagon is $8(\sqrt{2} -1)$

My question is as follows: The top of a table is made in the shape of a regular octagon by cutting the congruent isosceles triangles from the corners of a...

Published: Aug 26, 2026

Finding the limit when denominator = 0
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Finding the limit when denominator = 0

I'm having trouble solving this limit: $$\lim_{x \to -2^-} \frac{1}{(x + 2)^2}$$ I can't find a way to rationalize the denominator. Also, is the...

Published: Aug 26, 2026

Prove that if A and B are mutually exclusive then $P(A)\leq P(B^c)$.
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Prove that if A and B are mutually exclusive then $P(A)\leq P(B^c)$.

Suppose $\textbf{P}(A\cap B) = 0$. Prove $\textbf{P}(A) \leq \textbf{P}(B^{c})$ using the axioms of probability. I know the axioms are: $\textbf{P}(A)\geq...

Published: Aug 26, 2026

Find the composition of a piecewise function
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Find the composition of a piecewise function

Let $$f(x)= \begin{cases} 5 &\text{, x $\geq$ 0}\\ x^2 &\text{, x < 0} \end{cases}$$ and $$g(x)= \begin{cases} \sqrt{x} &\text{, x > 3}\...

Published: Aug 26, 2026

Show that the dimention of the intersection of projective linear sub-spaces of dimentions $d_1$ and $d_2$ of $\mathbb{P}^n$ is bigger than $d_1+d_2-n$
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Show that the dimention of the intersection of projective linear sub-spaces of dimentions $d_1$ and $d_2$ of $\mathbb{P}^n$ is bigger than $d_1+d_2-n$

Proposition: Let $L$ and $M$ linear projective subspaces of $\mathbb{P}^n$ of dimention $d_1$ and $d_2$ respectively. Prove that $\operatorname{dim}(L\cap...

Published: Aug 26, 2026

Difference between imaginary part and imaginary number
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Difference between imaginary part and imaginary number

Is there a difference between imaginary part and imaginary number? Because our teacher says there is. I mean in $5+3i$ , is it right to say the imaginary ...

Published: Aug 26, 2026

Find a circle perpendicular to two other circles.
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Find a circle perpendicular to two other circles.

Here are two circles $C_1 = [ x^2 + y^2 = 1]$ and $C_2 = [ (x-2)^2 + y^2 = 3 ]$. The radii are $1$ and $3$ and the two circles are orthogonal to each othe...

Published: Aug 26, 2026

Finding the local trivilization of canonical line bundle
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Finding the local trivilization of canonical line bundle

I am going through Nakahara's "Geometry, topology and physics" by myself and I got stuck on a calculation. I have the total space $$L=\{(p,v)\in\math...

Published: Aug 26, 2026

Express $\cos(\tan^{-1}(x))$ in terms of $x$.
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Express $\cos(\tan^{-1}(x))$ in terms of $x$.

Express $\cos(\tan^{-1}(x))$ in terms of $x$. My Attempt: Let $\tan^{-1}(x)=A$. $$x=\tan (A)$$ Then, $$\cos(\tan^{-1}(x))=\cos (A)$$

Published: Aug 26, 2026

Composition of function with itself
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Composition of function with itself

For the set of all functions $f:\{0,1, \dots,2014\} \to \{0,1,\dots, 2014\}$ such that $f\left(f\left(i\right)\right)=i$, for all $0 \leq i \leq 2014$. Co...

Published: Aug 26, 2026

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