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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Brandiwyn🎶

a day ago
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').

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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.

Aiva Raine

Nov 5, 2024 at 3:03pm
In response to "It's Hard Not To Scare"

Shaking head. Aside from the inevitable Simpson's spider goat song playing in my head which I will now share with you below, whenever I come across a story that has all these scientists doing ridiculous experiments for no logical reason, I must go- ah- scientists- CANCER? You do remember that problem right? Can we maybe cure that before we make cockroaches magnetic and elephants see pink elephants?

spider pig clip- and if you all now are singing the spider goat song- you're welcome! grin
https://www.youtube.com/watch?v=BARjPuUN36Y

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Aiva Raine

Nov 5, 2024 at 2:55pm
In response to "Anomalies"

One of your more humorous entries- I especially enjoyed- "Lofoten Vortex can be the name of my Bjork cover band."

Thanks for the laughs. Much appreciated.

2 Like

November 14, 2021 at 12:02am

#1021569

Past Tense

PROMPT November 14^th

A Mystery Genre prompt tonight. You are researching your genealogy and find that ancestors from a different generations and different countries made visits to the same remote place. Tell us about this remote place. Why were your ancestors going there?

I don't much care about genealogy.

This is not to say that I think people who do are wrong. Everyone needs a hobby, and if that's yours, great. I'm only talking about my own proclivities here.

I know a bit about my adoptive family's ancestry. They shaped who I am just as surely as genetics must have (when it comes to the ancient nature vs. nurture debate, I'm firmly on the side of "both"). I know about it because my parents, aunts and uncles told me about it. I know, for instance, that my father's mother died when he was very young; later I put two and two together and realized that she was a victim of the Spanish Flu epidemic of the late 1910s / early twenties. I've even tried to find records of them, out of curiosity, but ran into brick walls after grandparents (only one of whom I ever knew, and she died the day after I turned seven). But I didn't care enough to dig deeper.

As for my biological family, I have no interest in finding out about them at all.

I am who I am, and I prefer to think of myself as unbound from any ancestral shackles. Go back far enough, and we all come from the same place anyway.

It always bugs me in shows and movies when a character is like, "Oh, I need to find out who my REAL parents are" when they've been raised by perfectly acceptable human beings, people who cared about them. And they're going to snub all that for the family who gave them away? Rude.

In one of them, the character was a law enforcement official; she discovered that her biological father was a criminal, so she started acting like a criminal because she thought that was a part of who she was. Turns out (plot twist) he wasn't her bio father after all, so that was all her, not genetically programmed.

I know that's just a story, but stories reflect attitudes, and that was an attitude I didn't appreciate.

So, to address the actual prompt, I'll write a story of my own, something I rarely do in this blog, but it fits. The only thing I'll preface this with is to keep in mind that every place on Earth is remote from somewhere.

"So, Dad, I was looking into family history." Ted spooned another morsel of chili into his face.

"Oh?" The older man dabbed his napkin at the corners of his mouth. "Find any skeletons?"

Ted swallowed. "Just one weird thing I don't understand. Seems like everyone on your side was born in Baltimore."

Ted's father leaned back in his chair. "Yep," he acknowledged.

"I was. You were. Your father, his father... not sure about great-grandpa, but even his father was born in Baltimore, and they were still living in England at the time. And we've always lived here in California."

"Well." Dad pressed his lips together. "That's all because of the prophecy."

Ted blinked. "No, really, what's the deal with that?"

"I'm not kidding, son. There was a prophecy. Let's see." His eyes searched the ceiling. "Nope, I don't remember the exact wording. Something about how a son born in Baltimore would one day save the family."

Ted snorted. "Right."

"My dad thought I might be the one, so he took your grandmother to Baltimore to have me. His dad thought he would be the one, so, same deal. And on back as far as I'm aware."

"And none of you fulfilled the prophecy?"

Dad shrugged. "I don't know. Maybe. What exactly is saving the family?"

"Maybe it's about me," said Ted.

It was Dad's turn to snort. "Son, I love you, but you're 45 years old, single, childless, working at Wal-Mart, and living above my garage. You're the last Thistlethwaite. You ain't savin' nothin'. Prophecy's bullshit."

"I'm 47," Ted corrected him.

"Whatever." Dad stood with a grunt and picked up his empty bowl. He headed toward the kitchen. "Doesn't change my point."

"Hey, at least I have a job. What do you do all night while I'm at work, anyway?"

Dad raised his voice from the kitchen. "Sleep, like a normal person. And I worked for thirty damn years to put food on the table for you and your no-good mother."

"You call this food?" Ted mumbled, pushing the chili around the half-empty bowl.

"What was that?"

Ted cleared his throat. "I said, at least Mom could cook."

"You're a disappointment, son."

"You're disappointed now? Wait 'til you see your nursing home."

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