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 3 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 5 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.

September 18, 2020 at 12:01am

#993619

Is Time Ever Really Free? Are We?

Some prompts, I just have a hard time relating to.

PROMPT September 18th

When you have an unexpected hour of free time, how do you fill it?

Sleep.

At least, that would have been one of my answers back when free time was a precious commodity for me.

The other answer, on those ultra-rare occasions when I wasn't dead tired, was reading.

Seeing this prompt, though, reminded me of how great it feels to not have pressing obligations all the time. It was like my chest was constantly in a vise, and when I retired, those clamps finally loosened up.

I had so much going on, so many things I had to do and things I wanted to do but couldn't bear to go without, so something had to give, and that something was sleep. So when I got a chance, sleep was my preferred recreational and restorative activity.

Still is, really. It's magical: you need to make time pass, so you fall asleep and then suddenly, boom, it's time to do... whatever it is you needed to do.

I've heard some people say that they wish they didn't have to sleep. Think of all the things you could get done! More time to be productive! More opportunity to serve your masters! Don't you want to do more, more, more?!

No, thanks.

Anyway, this sort of thing doesn't apply to me now. I do exactly what I want to do most of the time, so as far as I'm concerned, it's all free time -- visits to dentists and whatnot notwithstanding. Though I could argue that while I, like most people, dislike trips to the dentist, I dislike tooth problems more, so I want to do that, too.

I've got one of those coming up in a couple of weeks. Root canal. Not my first root-io, either, so I know the drill. (Yes, that's a pun.) It's not painful like you hear about; it's more... well, they numb you thoroughly and you're sitting in a dentist chair for two hours while they do unspeakable things to the insides of your tooth, and the occasional tiny wisp of smoke drifts from your wide-open mouth across your field of vision. Even that wouldn't be so bad if you didn't occasionally smell it.

So even if you wanted to, you couldn't actually sleep through it. But it's incredibly, mind-twistingly boring. (You can consider that a pun, too, if you want.)

That's the worst part of it. You're sitting still for a couple of hours and you can't do anything, not sleep, not talk, not read, not watch a movie, not play a game... just stare at the ceiling and try not to think of what's really happening to the once-living tissue inside of your tooth. I mean, I'm sure it's no picnic for the dentist or her assistants, but at least they can think about the BMW they can buy with what you're paying them.

I expect they don't put you completely out because there's no real reason to, and there's always a risk with general anesthetic. But you'd think at least, by now, they'd have it so you could put on some kind of VR headset and watch Little Shop of Horrors during the whole ordeal. Okay, maybe that's a bad choice of movie. Still.

Afterward, I won't be able to eat or drink anything for a while, so I'll do the one thing that I know will make the time pass until the anesthetic wears off:

Sleep.

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