updates
Why does this pattern occur: $123456789 \times 8 + 9 = 987654321$
I came across the following: $\begin{align} 1 \times 8 + 1 &= 9 \\ 12 \times 8 + 2 & = 98 \\ 123 \times 8 + 3 & = 987 \\ 1234 \times 8 + 4 &am...
Apr 18, 2026
Archive: Updates
updates
Finding the average height and average velocity of a ball dropped from height of $400$ ft
Rosanne drops a ball from a height of 400 ft. Find the ball's average height and its average velocity between the time it is dropped and the time it ...
Apr 18, 2026
updates
What is an algorithm for determining if a finitely presented group is finite
Suppose I am given a presentation of a group with a finite number of generators and a finite number of relations. Is there an algorithm for determining if...
Apr 18, 2026
updates
The Similarity Matrix of graph Laplacian Matrix has different names. What's the difference between these names?
The graph Laplacian is defined: $$L=D-W$$ Where $W$ is the Similarity Matrix of the graph and $D$ is a diagonal matrix whose entries are column sums of $W...
Apr 18, 2026
updates
Simpson's Rule for Double Integrals
Simpson's Rule for double integrals: $$\int_a^b\int_c^df(x,y) \,dx \,dy$$ is given by $$S_{mn}=\frac{(b-a)(d-c)}{9mn} \sum_{i,j=0,0}^{m,n} W_{i+1,j+1...
Apr 17, 2026
updates
What is a monoid in simple terms?
I encountered the term "monoid" but I didn't really understand what is it useful for or what's it about. If I understand correctly a &...
Apr 17, 2026
updates
Domain for index of Radical Sign
What is the domain of $\sqrt[x]{a}$, and is $\sqrt[x]{a}=a^{1/x}$ always true?? I was told that the domain of $\sqrt[x]{a}$ is natural numbers and the dom...
Apr 17, 2026
updates
How can it be proved that the limit definition and series definition of $e$ are equivalent, how do they model continuous growth?
The limit definition of $e$ gives the best intuition toward how $e$ can model continuous growth. $$\lim\limits_{n \to \infty} \left( 1 + \frac1n\right)^n$...
Apr 17, 2026
updates
Defining the Cosine Function from First Principles, intuitively
Throughout most of my mathematics education, the cosine function has been defined formally either using its series expansion, as $\mathfrak{Re}(\exp i\the...
Apr 17, 2026
updates
How to check if an image to image homography is "reasonable" using matrix properties
Intro - In case you don't know what a homography (computer vision) is I'm working with some image processing problems where I extract a bunch of...
Apr 17, 2026
