Complex Numbers (Book)

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Complex Numbers

Complex Numbers

A complex number is expressed in the standard form a + bi, where a and b are real numbers and i is defined by i^2 = -1 (that is, i is the square root of -1). For example, 3 + 2i is a complex number.

The bi term is often referred to as an imaginary number (though this may be misleading, as it is no more "imaginary" than the symbolic abstractions we know as the "real" numbers). Thus, every complex number has a real part, a, and an imaginary part, bi.

Complex numbers are often represented on a graph known as the "complex plane," where the horizontal axis represents the infinity of real numbers, and the vertical axis represents the infinity of imaginary numbers. Thus, each complex number has a unique representation on the complex plane: some closer to real; others, more imaginary. If a = b, the number is equal parts real and imaginary.

Very simple transformations applied to numbers in the complex plane can lead to fractal structures of enormous intricacy and astonishing beauty.

http://www.ddewey.net/mandelbrot/

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Show Recent Comments

10 Most Recent Blog Comments:

Write_Mikey_Write!

about 8 hours ago
In response to "Global Domination"

A little girl wearing a Hello Kitty shirt in Middle-of-Nowhere, Siberia will certainly make you wonder how it got there. My grandson has a souvenir t-shirt with a Terex 33-19 "Titan" on it. When it was built in 1973, the Titan was the largest, highest payload capacity truck ever built, a title it held for 25 years. The thing is, the truck on that t-shirt is on display in Sparwood, BC, Canada - over 1000 miles from here, and his mom got the shirt at a local thrift store. Makes me wonder.

Oh, yeah - I thought 'Schrödinger's Hello Kitty' was funny, too.

1 Like

Adherennium Plotting something

about 10 hours ago
In response to "Global Domination"

For whatever warped reason, 'Schrödinger's Hello Kitty' has tickled my funny bone.

3 Like

Brandiwyn🎶

Nov 12, 2024 at 10:50am
In response to "Blond Man's Bluff"

That's interesting about the two spellings of blonde. I always thought blonde referred to the person and blond referred to the hair. But I never bothered to look it up, So you've catered to my lazy today.

3 Like

Kåre เลียม Enga

Nov 9, 2024 at 1:02pm
In response to "It Was a Dark and Starry Night"

Ah... but why did he see those swirls when many of us don't? A totally different discussion... out of thin air... which is a frozen fog today when I gaze out the window.

Humans have great skills of observation; some of us extraordinary skills. Is The Scream depictive or evocative? 😱

Turkey DrumStik

Nov 9, 2024 at 11:04am
In response to "It Was a Dark and Starry Night"

Frankly, I'm more surprised to see reasonably good reporting from CNN. I'd given up on their ability to rein in their tendency toward loaded speech.

2 Like

JayNaNoOhNo

Nov 8, 2024 at 12:00pm
In response to "Bungalow Bash"

For the love of God, what did I just read? That is by far the worst use case of a bungalow I've ever seen, and a fundamental misunderstanding of the value of front porches—and the size of bungalows.

Was this his point? Nope, I know what his point was. For a guy so worried about presentation, he certainly undercut his own arguement.

1 Like

Turkey DrumStik

Nov 8, 2024 at 9:47am
In response to "Bungalow Bash"

"People don't change their minds on the internet. They usually do that in books, battlefields, or not at all.

Yeah, that's a little bleak."

Bleak and also inaccurate.

1 Like

Kåre เลียม Enga

Nov 7, 2024 at 10:19pm
In response to "Past and Future Intents"

I must live in the present. In the past I daydreamed a lot... probably to escape the bruises of reality; however, I need to be more aware crossing streets or even how I get out of bed (no rapid movements without thinking of consequences).

I can envision a future more peaceful than the present because this time-and-place isn't the only "present" that's ever existed, and I do try to see the larger view (emphasis on 'try').

1 Like

Lilli 🧿 ☕

Nov 7, 2024 at 10:58am
In response to "Past and Future Intents"

If the aforementioned exercises don't help; there's still liquor.

3 Like

Turkey DrumStik

Nov 6, 2024 at 10:22am
In response to "Lo!"

Not gonna lie I need the holiday season chatter.

June 1, 2022 at 12:02am

#1033122

Ceci n'est pas une pipe

Another link courtesy of Turkey DrumStik

today. I promise I have way more of my own than ones I've been pointed to, but such are the vagaries of the random number generator, peace be upon it.

So we begin June with a rather depressing -- but fascinating -- article.

The Devastating Decline of a Brilliant Young Coder
Lee Holloway programmed internet security firm Cloudflare into being. Then he became apathetic, distant, and unpredictable—for a long time, no one could make sense of it.

I would never have heard of Cloudflare until just now, except that I have a script blocker installed to help prevent popup ads and some tracking. Usually, it's unobtrusive. Sometimes, some actual content gets blocked and I have to figure out which of the scripts are causing that, which requires research and experience. Point being that Cloudflare is a recurring one that I know it's safe to, at least temporarily, disable.

Ironically, per my script blocker, this source, Wired, doesn't seem to use Cloudflare.

Now. I'm not going to quote much from the article today. It's very long and, in the tradition of the rambling New Yorker style, it pops back and forth between anecdote and meat. Honestly, the story of Cloudflare, or even Lee Holloway, is background to me, mostly just the example used to anchor the information about the medical issues involved.

Because it's long, I realize it's unlikely most of my readers will take the time to actually delve into it. So, if you're going to read it, do it now, because I'm about to spoil the plot.

Ready?

Here we go.

Dude helps start a tech company that both does good things and makes money (rare). He shows signs of going off the rails. His first wife leaves him, taking the kid. After getting married again (which could also be taken as a sign of mental issues), and having another kid (ditto), and going further off the rails, he gets diagnosed with a rare (though not as rare as a company that both does good things and makes money) form of early-onset dementia for which there is no cure.

Having dealt with different forms of dementia in both my parents, in addition to my abiding interest in philosophy, I was interested mostly in the underlying discussion. It's not that I don't feel for the guy, or his family; it's just hard for me to work up too much sadness for people I don't know.

This central question is expressed in the article, after the first part of the anecdote I mentioned.

What makes you you? The question cuts to the core of who we are, the things that make us special in this universe. The converse of the question raises another kind of philosophical dilemma: If a person isn't himself, who is he?

And that's what I'm focusing on here.

I've had occasion to hang out with people, many of whom are very nice so this isn't a cut on them, who airily declared things like, "You are not your body."

I disagree.

I mean, okay, in the ultimate case, sure. I have, fortunately, rarely had occasion to look at a real live dead body (okay, you know what I mean), but when you see such things in movies, people are like, "That's Joe. Poor guy." Well, it's not Joe, is it? It's Joe's body. Joe's left the building. It's a bit like when you look at a photograph or painting of someone and say "That's Joe."

Ceci n'est pas une pipe.

But a living person has continuity, not just of mind as the article mentions, but also of body -- though even that is more of a pattern continuity than a matter continuity.

Still, the existence of dementia (or other brain issues) points, in my opinion, to the inescapable conclusion that insofar as you can say "I am" at all, you are your body, which perforce includes your brain.

The "You are not your body" crowd is expressing, as I understand it, a form of mind-body duality, the idea that the mind, or spirit, or soul, or something, exists independent of the physical form, perhaps even persisting beyond death.

But when my parents, first my mom followed by my dad, fell ill, I seriously began questioning such things. Not that I ever fully accepted an afterlife, but it's such a pervasive belief that I never really argued about it, either.

If the mind persists beyond death... which mind? By the time my father died, his brain had deteriorated to the point where he was largely nonfunctional. If I met his ghost, would I meet the one with Alzheimer's? His spirit from ten years earlier, before he had to go to a nursing home for it? From 40 years earlier, when I was a child? Or the one from when he was a teenager, running away from home to go to sea? Or something else entirely (at which point I couldn't really say he was "Dad").

The point being that the brain changes all the time, and these changes are reflected in the mind. We can call disorders of the brain "mental illness" or "dementia" or whatever, but that's just a way more obvious change. It's a label we put on so that people can understand that something the affected does is an illness with physical signs, not just them being weird.

Some might argue that it's not the mind, but the soul, whatever that is, that persists. Okay, I'm not going to try to shatter any closely-held beliefs here, but if "you" are the sum of your experiences, tethered to an ever-changing mind tethered to an ever-changing brain, which "you" survives? If you don't have your memories, you're not really "you." If you don't have a neural network to shape your thoughts, or a body to keep running and send signals back and forth to, who or what are you?

From my surface understanding of information theory, which, oddly, seems to be a good candidate for explaining Everything, I've heard that no information can truly be destroyed. So in one sense, we live on, in echoes, having affected people and things along the journey.

But personality? That's inextricably tied to the material. Might as well ask "what happens to a candle flame after it's snuffed out?"

Anyway. That's my philosophy for the day, inspired by the story of a guy whose work echoes in most every corner of the internet.

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