What is Abstract Algebra? (Modern Algebra)
Why Mathematics Stack Exchange recommends it
Mathematics Stack Exchange users link it as a quick, friendly first look at abstract algebra for curious beginners and as the start of a recommended video series.
Recommended by 3 people across 3 different months, 2018 to 2020.
What it is
A three-minute introduction from Socratica to abstract algebra, explaining how it differs from high school algebra and previewing structures such as groups, rings, fields and modules. It opens Socratica's Abstract Algebra series and mentions applications across mathematics and the sciences.
What people said
Strong rec user5389726598465 · 2020-09 · Mathematics Stack Exchange · on “List of interesting math videos / documentaries”
Socratica [LINK] 's youtube Abstract Algebra [this video] series is the best! Of the playlist, simple groups [LINK] was my favorite. I also like mathew salomone [LINK] , mathdrbob [LINK] and Michael Penn [LINK] and everyone likes blackpenredpen [LINK] .
Recommends AlkaKadri · 2018-01 · Mathematics Stack Exchange · on “How can a curious high-schooler to go about doing mathematical research?”
Unfortunately, I don't think there's much research in the realm of "calcluus" that you can get involved in until you've studied a few more course's worth of real/complex/numerical analysis and linear algebra. One thing I will say, however, is that I wish I knew about "Abstract Algebra" when I was in HS. Those two words may not sound like much, but it's this vast, wondrous world that you somehow never hear about in HS. Here's a start: [this video] It doesn't take the most amount of knowledge in Abstract Algebra to find your way to a (very difficult) open problem.
Recommends Stanislav Bashkyrtsev · 2020-12 · Mathematics Stack Exchange · on “Understanding Multiplication/Dot Product of Two Numbers/Two Vectors”
I think learning a bit of Abstract Algebra ( here's an introduction [this video] ) will answer some of these questions. An operator (addition/multiplication/etc) and type of objects (integers/reals/vectors/etc) it acts on - come together. You need to define what a multiplication is for this type of objects. For instance: Multiplication of 2 integers can be represented with repeated addition as you mentioned (Scalar) Multiplication of a geometrical vector (an arrow) is stretching squeezing that vector. Imagine a physical stretching of an arrow without multiplication of its components. You just have an instruction to stretch 2x and that's what you do with a pen and paper. You can come up with any m