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 14, 2020 at 12:03am

#993295

The Inner Fright

Waking up to find it was all just a dream is the cheapest ever way to end a story.

PROMPT September 14th

You just woke up to discover you had dozed off in your 3rd grade class... how would you react realizing your entire life since then was just a dream?

I mean, I guess it worked for Lewis Carroll. But it's not the kind of thing the rest of us can get away with.

I have trouble sleeping when I know I have an alarm set. I've been known to sleep right through alarms. I'm not on a regular work schedule, so I no longer wake up at the same time every weekday, and that compounds the issue. If I have, say, a doctor appointment at 9:30 then I have the alarm set for 8:00 or whatever, even if I go to bed at midnight (way too early for me), when it goes off, I get up and slog to the shower. Along the way, I notice something: the light is dim, or some of the clutter on my nightstand is missing. That's when I realize I'm dreaming. To prove to myself that it's a dream, I will myself to rise up through the ceiling. Once I do that, I wake up for real and check the time. Whew - only 6 am. Still plenty of time to catch some more Zs before I have to wake up. I wake up again, and it's 9:45. Oh shit, I've missed the appointment! Wait... no. It's dark. I'm still dreaming. Levitate. Yep, dreaming. Then the alarm goes off, and there's someone in the room with me who's not a cat. Sigh. Dreaming.

Finally I wake up and it's 7:55, and I haven't gotten any decent sleep and I can't levitate so I know I'm not dreaming. Five minutes isn't anywhere near enough, so I get up and take a shower -- during which the alarm goes off because like an idiot I forgot to turn the blasted thing off.

I say "alarm" but I just use my phone these days.

The point of all of this is to relate that the whole "butterfly dreaming I'm a man" conundrum never made sense to me. Even if my mind starts out thinking a dream is reality, eventually there's some discontinuity that makes me realize I'm dreaming. Rarely, I can use that to my advantage and have a lucid dream, like my own personal holodeck. Usually, I just wake up, or at least I think I do.

Reality is persistent. That's how we know it's reality.

So if I woke up one day from a nap in a classroom and I'm eight years old, my first reaction would probably be that this is just another nightmare, and pretty soon I'll wake up for real, safely NOT in Miss Martin's classroom.

Miss Martin was one of those bachelor teachers of the old mold, with the pointy glasses and seventeen cats at home. Think Dolores Umbridge, but in green instead of pink. Possibly she was gay, but more likely she spent her adult life teaching assholes like Kid Me and decided that having children was not for her. Her like will never be seen again in our time, and that's a shame, but I have no good memories of that school year.

This whole thing kind of reminds me of the great ST:TNG episode "The Inner Light." In it, Picard falls into a deep coma on the bridge, and wakes up to find he's living a life on a pastoral planet. He lives the rest of a very long life on that planet, with a wife and kids, and the show expertly compresses something like fifty Earth years into less than an hour of runtime. I mean, really, if you watch no other Trek, that one's worth it by itself. Point is, spoiler alert, he finally wakes up back on the Enterprise and he's in his whatever decade Picard was supposed to be on TNG. Forties, I'd guess. But he's retained all the memories of this other life, even though it was, for all intents and purposes, a dream. The exact mechanics of this involves hand-waving some alien technology, which is kind of a specialty of Trek, but the salient character point here is that to him, it was not a dream.

And so I would hope it would be for me in the purely hypothetical situation outlined in the prompt above: that I would have memories of this life, know what was coming. I mean, in the episode it turned out that everything he experienced in his mind were based on things that happened long ago, but the parallels remain. That is, assuming that this new reality turned out to be persistent and not, in fact, a nightmare.

Consequently, as soon as Apple became publicly traded, I'd dump every last meager dollar fourteen-year-old me had into it.

And then with my luck, it would go bankrupt in that timeline.

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