Simply put, a basis in mathematics is the simplest way to represent something, be it a set, object, anything! However, in looking into how several fields of mathematics define a basis, it can turn ...

I’ve been working on Beachy and Blair’s Abstract Algebra Book, specifically their chapter on Group structure (7), and I wanted to summarize what I’ve learned about group structures. Finite groups...

My Notes Since I have a bit of time before I have to leave for college (Riju’s already left), I went and polished my Abstract Algebra notes from this year. You can find the PDF link to my notes h...

I’ll give an intuition for how mathematicians define what graph ends are, and explain their use in geometric group theory. However, before I can do that, I’ll explain what ends are in a graph theor...

DeMorgan’s Laws are one of the most important laws in basic set theory and come up time and time again. The proof of DeMorgan’s law provides a nice introduction to proofs and reasoning to why it wo...

Perfect numbers are a highly studied subset of the positive integers that relate to our understanding of number theory. I will show a proof of the structure of all even perfect numbers, and a bonu...

With groups, it’s easy for us to find finite ones that have the same order, but are not the same (up to isomorphism). The simplest example I can give is $\mathbb{Z}_4$ and $\mathbb{Z}_2\times\mathb...

Field Extensions and Linear Algebra I hope to show the relationship between field extensions and linear algebra, and how a proof using the latter can extend to a useful property of the former. Ex...

Algebraic Elements Which number is “further” from the integers, $\sqrt{2}+\sqrt{3}$ or $\sqrt{2}$? It intuitively feels like it would be harder to get to $\sqrt2+\sqrt3$ using the integers and the...

Llamas Hello! This is my first real post, so this is going to mostly just be a test of my Markdown abilities. Anyways I thought it was interesting how repetitive the introduction of linear algeb...