Beginning Complex Numbers ideas

[DrummingAtom]

I've been working through the book The Story of i(sqrt of -1). It's kinda like a story with a lot of Math. The first 2 chapters deal with cubics and geometry for solving cubics functions. I understand the algebra behind it but I'm getting lost with the big picture. I need a supplemental book or papers to help me along. Any recommendations?

Also, what branch of Math would I find these ideas in? It's pretty heavy Algebra mixed with Geometry. Thanks.

 

[Kreizhn]

I'm not certain what you mean by "big picture." Do you mean, "what imaginary numbers are used for?" Unfortunately, that's not easily answered because quite frankly, they're used everywhere. The concept of imaginary numbers (or constructs that are isomorphic) are prevalent throughout all areas of mathematics and physics.

The primary field that deals with analysis in complex spaces is called, unsurprisingly, complex analysis. Again however, this is just the "calculus" of complex spaces. For the algebra of complex numbers you'd need to look more at abstract algebra.

If you are specifically interested in how complex numbers relate to fundamental geometry and the solution of polynomials, then you should look into Galois theory. However, Galois theory is a very advanced area of mathematics that requires a comprehensive understanding of both group theory and ring theory. Nonetheless, if you are still interested, http://www.andrew.cmu.edu/user/calmost/pdfs/pm442_lec.pdf" is an introduction to Galois theory.

If you add together the algebraic and analytical elements of complex numbers, you'd want to start looking at differential geometry and Lie Theory.

Hope that helps.

 

[DrummingAtom]

Thanks, I guess what I'm asking is more along the lines of that Galois stuff. All the stuff I'm getting stuck on is the Cubic function solutions and how they're solved. My Math textbooks (Pre-Calc to Calc) don't ever talk about Cubic functions. There seems to be a lot of Algebra "tricks" involved with the Cubics.

 

[Kreizhn]

Those notes that I linked you too actually derives the cubic and quartic equation as well as some special cases. As you may notice, the analytic solution by radicals to such polynomial equations are generally very messy which is why they often require some trickery.

 

[CellCoree]

I would first like to commend you on your interest in exploring complex numbers. They are a fundamental concept in mathematics and have many practical applications in various fields of science.

To address your concern about understanding the big picture, I would recommend looking into the concept of complex analysis. This branch of mathematics deals specifically with functions of complex numbers and their properties. It will provide you with a deeper understanding of the geometric interpretation of complex numbers, as well as their algebraic properties.

In terms of supplemental materials, there are many books and papers available on complex analysis that can help you further explore the topic. Some popular ones include "Complex Variables and Applications" by James Ward Brown and Ruel V. Churchill, and "Functions of One Complex Variable" by John B. Conway.

In terms of the branch of math that deals with these ideas, as you mentioned, it is a combination of algebra and geometry. Complex analysis falls under the broader category of analysis, which is a branch of mathematics that deals with functions, limits, and continuity. It also has connections to other branches such as number theory and differential equations.

I hope this helps guide you in your exploration of complex numbers and their applications. Keep up the curiosity and determination to understand these concepts, and I'm sure you will have a solid grasp of them in no time.