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:

Turkey DrumStik

about 9 hours ago
In response to "Barley There"

And approximately 15 seconds later, someone's going to screech about genetically engineered crops.

That would be me...screeching for more genetically engineered crops.

Write_Mikey_Write!

Nov 14, 2024 at 2:04pm
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

Nov 14, 2024 at 12:10pm
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

October 19, 2018 at 12:50am

#943731

First Man

I went to see First Man today, that movie about Neil Armstrong and the space race.

Saw it at Alamo, of course, the theater with the beer taps, so I was able to get good and buzzed first. (See what I did there? I'm way too proud of that one.)

Spoiler alert: They land on the Moon.

I should note that Ryan Gosling really bugs the hell out of me. I don't know why. Maybe it's because he was in Blade Runner 2040. He looks like someone who gets his lunch money extorted on a daily basis.

Still, I think he nailed this one.

Okay, look, I'm just going to say it, even though I've said it before: I think the moon landing is not only the most significant achievement in human history, it's the most significant achievement that can ever be in human history. It's the moment that we first set boots on another world, satisfying the human desire for exploration and horizon expansion. Hopefully there will be more such moments, but that was the first. Yes, the discovery of extraterrestrial life (as per my last entry) will probably be the greatest discovery, but I'm making a distinction between achievements and discoveries.

That's arguable, I know. But it's why I went to see the movie. It didn't disappoint on that front.

It irks me that no one seems to be able to talk about the movie, or the events it depicts, without someone bringing up the "moon landing hoax" nonsense. (In this case, it's me bringing it up; it still irks me.) I'm not sure I've ever been able to articulate exactly what it is about that garbage that pisses me off so thoroughly, but I think the movie helped me figure it out - mostly because it focused on the human elements involved.

Using little more than a slide rule and a straightedge, we sent dudes to the moon in a rocket with the name of a god of the land on a mission named for the god of the sun. But these names were symbolic - this was a purely, epically human adventure. No gods transported us to another world. No demons, no monsters chased us into the heavens. No angels, no devils; only people.

And that's what bugs me about these conspiracy nuts. They offend me on a basic, visceral level. To deny this achievement is to deny humanity itself, to condemn us forever to mediocrity. It's as if to say, "Look at Earth. Look how we can't get anything right. It's impossible to believe that we can ever achieve greatness; therefore, we didn't, but we could fake it." It's the same sort of contempt for our potential that leads people to believe that aliens built the Pyramids, or fairies built Stonehenge. Because sometimes skyscrapers crumble and bridges buckle, because we still bicker among ourselves and kill and enslave, because we pollute the oceans with plastic and poke holes in the ozone layer, obviously we can't do anything right, according to these loudmouths.

But we contradict ourselves. We are large; we contain multitudes. We are destroyers and creators alike. We are thieves and we are philanthropists. We can bring darkness, and we can bring light.

We are the monsters, and we are the gods.

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