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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10 Most Recent Blog Comments:

Write_Mikey_Write!

about 21 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 23 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.

January 18, 2020 at 12:12am

#973629

Again Today

PROMPT January 18th

It’s Show and Tell Day here at the 30DBC! Think of an item that is special to you (it may help if you are able to see your chosen item while you write your entry). Describe it in as much detail as possible. Then, tell us its story. How did it come to be in your life and what makes it important to you?

My memory can be really lousy, sometimes. I can remember song lyrics and endless trivia about history, math, science, whatever, but I block out or forget the shadows of who I used to be. Sometimes I go back through this blog, or the one I kept before this one, and go, "I wrote that?" And I'm embarrassed, impressed, or both.

Point is, that's why I keep stuff around, and keep it for as long as possible: it's a reminder of what used to be, for good or ill. I'm not the kind of person to purge mementos of failed relationships or hard times. My last ex was that kind of person, on my behalf, which is why I don't have much stuff from my first marriage. Probably just as well. I really should let go of my resentment over that. It's really the only lingering negative emotion from that time; stuff is that important to me.

So trying to identify one object, one thing to talk about in response to this prompt initially gave me a kind of paradox of choice: so many things, and I'm supposed to pick just one?

Then I realized that the answer was right under my nose, and I mean that literally.

The detailed description? Well, it's a table. It's table-shaped, wooden, has four sturdy legs, and it can be pulled apart and extended. The finish is whatever you call that deep dark brown stain. Walnut, or whatever. Which is not to say that's the wood it's made of, just the color. I don't know what kind of wood it's made of. Tree wood, I'd imagine.

What else can you say about a table? It has matching chairs. It's high-quality, and probably wouldn't break even if I sat on it a bit over a year ago when I hadn't lost much weight. I usually don't work at a desk, but have my laptop set up on the table. So when I'm not on my back deck "enjoying" the weather, that table is where I sit.

But, like almost everything I possess, it's got a story. And it's not one I to which I can do full justice in a blog entry. So I'll try to condense.

I have - or had - a friend who, for the purposes of this entry and anonymity, I'll call W. We started out as co-workers, discovering a shared interest in science, SF/fantasy/horror books and entertainment, video games, snarky internet humor, drinking, and general geek stuff. Further, we have the same birthday (though not the same year; he's a bit younger).

That's about where the resemblance ended, because while I like to travel, W is way more restless than I ever was, and has been all over the world. He's also more interested in (shudder) the outdoors, hiking, hang-gliding, scuba diving, whatever. And, perhaps most importantly for this story, I'm a hoarding clutter-freak, and he's a minimalist light-traveler.

Nevertheless, it happened that my ex dumping me exactly coincided with W needing a place to stay. Since the ex had unilaterally gotten rid of a table of mine that I was quite fond of, replacing it with one she got from her mother, she took the replacement with her, leaving me table-less. Like I said, I should really let things like that go, but dammit, I liked that old table. It was perfect for playing D&D. Anyway, W moved into the spare bedroom with a little bit of stuff - it was during the recession and he'd just lost his job and consequently his house, but had some furniture - and he set up the table in the empty spot where my ex's had been.

It turned out that, despite our differences - the classic Odd Couple housemate situation, if anyone still groks that reference - we got along just fine, mostly because we generally kept to ourselves. Except, of course, when we'd sit at the table and drink excellent scotch or some new craft beer (W had gotten another job pretty quickly, but that didn't return his foreclosed-upon house to him).

After a couple of years or so of this, he had the opportunity to get a scuba instructor license, which involved him moving to the Philippines. Rather than schlep the table halfway across the world, he traded it to me in lieu of a month's rent. And that's how I became the official owner of the dark, sturdy table. Right after he moved out, I got another housemate, who I'll call A. She was a friend I'd met through my ex-wife. A moved into W's old room.

Not the end of the story, though. W and I kept in touch, and he also became friends with A. At some point, the life of an itinerant scuba instructor (like I said, all around the world) got old for him and he moved back to the States, living with his parents for a bit - until I had a heart attack, when he visited to provide moral support, and then moved in again because when you're middle-aged and restless, living with parents gets really old really fast. So for a while I had two housemates, both good friends, and the lack of friction between all of us still surprises me.

After a while, W got another job opportunity that required some education in a different state, so he left - but, again, we kept in touch and got together whenever we could. But then the actual job W got was on goddamn Maui. MAUI. So of course I took a month and visited him there. In a February. It was the first February of my life that I didn't hate, but then, any February that doesn't involve me having a heart attack the day before my birthday is a win in my book; being on a tropical island is a bonus.

W and I made plans to visit Scotland to sample all the excellent scotch they have, after he moved back to the mainland to take a different job.

And then, after fifteen years of being close friends, he disappeared. Ghosted me. Fell off the radar.

I don't make friends easily. There are few people I can get along with for more than a few hours or days. Losing one isn't fun. Not really knowing why just makes it worse. From what little I can tell, I think he'd lost his job, and maybe he couldn't face breaking our plans for Scotland. And he was developing health issues. I'd like to support him through those, as he did with me, but I don't even know where he is. Texts and phone contact just... stopped, on his part. And neither of us do social media. What little I know came from mutual acquaintances who do have social media.

Unlike with my ex, I don't harbor any resentment. I mean, I think I understand, even. If he couldn't do the Scotland trip, though, all he had to do was tell me.

So I have the table. I still drink at it sometimes, and when I do, I have a propensity for getting drunk and listening to Leonard Cohen and/or Brandi Carlile. Not something I'd recommend to anyone, but it works for me. It's one of the only ways I have to actually feel something.

I might do it again today.

Broken sticks and broken stones
Will turn to dust just like our bones
It's words that hurt the most now isn't it
Are you sad inside, are you home alone
If I could just pick up the phone
Maybe you could see a better day
And you won't waste away
Under my watchful eye
Because I'm your hero and you're my weakness

Who's gonna break my fall
When the spinning starts
The colors bleed together and fade
Was it ever there at all
Or have I lost my way
The path of least resistance
Is catching up with me again today

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