#
Much Ado About the Square Root of Less Than Nothing

Seminar - UHON 301

## Instructor(s): Christopher Holden

## Course Description

**The version ChatGPT summarized:**

Discover the fascinating world of ℂ, the complex numbers, in this engaging course. No math background required! Learn how these numbers transform seemingly random formulas into simple and useful tools. Explore the human side of math and unravel the mysteries behind these innovative concepts. Join us on this journey of discovery and find out why ℂ is a game-changer in the world of numbers. Let's make math fun and accessible together!

**And my original:**

ℂ

One of the things people remember most about math class is briefly memorizing seemingly arbitrary formulas written in alphabet soup. A good example is the angle addition identity from trigonometry. Since you may not have held onto it this long (good for you if you have!), here is one of those formulas:

sin(a+b) = sin(a)sin(b)+ cos(a)cos(b)

This is the kind of thing that has people describing math as arbitrary, random symbolism. No wonder people forget it as soon as they have no more tests to take.

But what if I told you there's another world, a world of numbers so good that it turns that formula into something simple, meaningful and useful? It is known to mathematicians simply as ℂ, the complex numbers. It's what you get when you add a vertical dimension to the number line using i, the square root of -1, a number that seems impossible at first but opens many doors if you let it.

ℂ is a great unifier. For those who pass through its doors, the world can be said to hang together a bit better. But so few people ever get to see this. I think ℂ could be much more widely known and appreciated. In this course you'll learn how the complex numbers came to be, how to use them yourself, and see a bit of what they can do, in math and applications of numbers to reality. It's fine to come in as a nonbeliever, or even having no clue what i is even supposed to be. I think if you spend some time with these numbers though, you'll change your mind as I once did.

This class is also a chance to find the human side to math. Instead of racing through methods, we'll see how these ideas developed, in fits and starts, over centuries. You'll have a chance to dig into questions about why, not just what. And we'll especially explore the mystery of why ℂ seems so obvious, basic and useful to some while others still find it imaginary, not real. We'll open the door that ℂ helped mathematicians discover a couple centuries back: you're not stuck with the numbers you start with; you can invent as many kinds of numbers as you need.

I know this sounds like a lot of tough stuff. But be assured, this journey is for everyone, regardless of background. Seeing past geniuses wrestle with these ideas may help us feel better about needing some time to get used to them. There are no prerequisites, and no need to call yourself a math person to be here. We will be doing math, getting our hands dirty, but you will always have as much help as you need. We'll be there for each other. There are no tests.

The enigma of i does not take us out into the flights of fancy we expect from pure mathematics, but instead is a catechism that brings sense to chaos, unites diversity, and just makes you happy. Once you know ℂ, even the angle addition formula will make clear sense.

## Texts

We will use whatever sources we can find to learn the algebra and geometry of ℂ. There’s tons of tutorials online, and none are likely to teach you wrong math. This wealth of tutorial content though is balanced by a real difficulty in finding something deeper. If you can even find history, it is likely total bunk hero worship and weasel words. Prying into what math means, where it comes from, those kinds of bigger questions, means needing to be careful about the sources we choose and what we’re asked to believe. There’s too many people who have no idea what they’re talking about and end up falling back on fairy tale tropes and lying to or misleading us. Luckily most people don’t know much math history because it’s likely all wrong.

Besides what we can find online that’s been made recently, we’ll also look to some of the primary sources on the complex numbers, works by people like Cardano, Bombelli, Euler, and Gauss. Euler’s algebra textbook from 1765 is all we really need to learn the algebra of ℂ (he’s the one who got everyone calling √-1 i). Our other use for the primary sources will be to see how ideas about ℂ and its uses changed over time..

We will also read some of the secondary historical literature, both because its saves us from needing to read everything written by primary authors in several languages, and because its in these historical accounts that we’ll find some discussion of what the participants were thinking and talking to each other about, even if it can’t always be trusted. The math texts themselves are often rather barren of discussion thanks largely to the bad example set by Euclid way back when.

Note that this choice of subject also has the benefit of putting us into non-copyrighted, public domain works most of the time. These texts can often be free, and our library privileges should help with most of the rest.

Here are some examples of the texts we’ll be using:

*3 Blue 1 Brown*by Grant Sanderson (YouTube channel)*Do complex numbers exist?*by Sabine Hossenfelder (Youtube)*Imaginary Numbers Explained Bob Ross Style*by Tibees (Youtube)*Imaginary Numbers Are Just Regular Numbers*by Up and Atom (Youtube)*Elements of Algebra*(1765), 2015 English edition and*Introductio in analysin infinitorum*(1748) by Leonard Euler (translated by Ian Bruce, ND)*Disquisitiones Arithmeticae*(1801) and*Theoria residuorum biquadraticorum, Commentatio secunda*(1831) by Carl Gauss ,*Gauss and the early development of algebraic numbers*by ET Bell (1944)*The Origins of Complex Geometry in the 19th Century*by Raymond Wells, 2015*Complex numbers and geometry*by Liang-shin Han (former UNM professor)*An Imaginary Tale: The Story of i*by Paul Nahin*A Short History of Complex Numbers*by Orlando Merino

## Requirements

Like I said above, there are no math prereqs here. What you do need, and this is serious, not a platitude, is curiosity—a willingness to try, and to be responsible for that curiosity, to follow it, nurture it, and find ways to move forward when you're feeling stuck or confused, not by being a genius but by asking questions, reading, taking notes, and actually writing things down that may not end up being right. Taking those kinds of risks will maybe be hard to do but extremely valuable here, and very good practice for elsewhere.More specifically, each of you will help teach a couple sessions of our class, contribute to lively discussions, and complete many short writing assignments centered on math problems. By the end, I'd like to you make something designed to spread the gospel of ℂ, or to tell mathematicians why it's not all it's cracked up to be in their minds if that's still the way you feel.

Here's the list:

1. Read actively: take notes, work mathematical details with pen and paper, contribute to conversations.

2. Be a real part of what we're doing together as a group, and take responsibility for it.

3. \~10 Quizzes on basic material from readings and math calisthenics. Not a large grade component but an important check-in to prep for.

4. \~6 writing assignments based on math problems, \~2 questions each

5. Twice this semester teach us a small bit of math or related material: proofs of major theorems, calculation methods, historical context.

6. Make one piece of media for an outside audience that helps spread the good word about ℂ or takes it down a peg.

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