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:

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 1, 2020 at 12:08am

#992001

Cheese? What Kind of Cheese? I Want Brie

Hm. Another WDC birthday week. That means I'll be 16 soon. Can I have a car? I was promised a flying one.

PROMPT September 1st

Happy 20th Birthday, WDC!

I know this is cheesy, but I have to do it... In your entry today, write about what you love about Writing.Com. How have you grown as a writer? How are you (as a writer) different from how you were when you first joined this site? Thank the members who helped you get where you are today.

You know, my hesitation on prompts like this is not that it's cheesy. I can do cheesy. It's that I've been here for just short of 16 years, and no matter how much I try, there is no way I wouldn't forget someone significant if I tried to list all of the people who are important to me here. And, sadly, some of them have left the site and others... well, they've left far more than the site. Everyone that I've interacted with over the years -- occasionally unpleasantly, usually quite the opposite -- has helped to make me what I am today. That includes you, since you're reading this. But also everyone who enters the Writer's Cramp, organizers and participants in the October NaNo Prep Challenge, anyone whose work I've reviewed or has reviewed mine, all those who have given me awardicons and Merit Badges (and vice versa), those who take the time to give me feedback on my newsletters or comment in this blog, anyone whose blog or newsfeed I follow, the members I've exchanged IMs with, the many who I've met in person, those I occasionally talk to on Zoom... everyone.

That helps to address the earlier parts of the prompt, though. No matter what else, it's the people that make this site what it is. Oh, the setup is important, of course -- we'll get to that in a bit -- but it would be empty without you.

How have I grown as a writer? Well. I'm not sure I know how to answer that. I'm the worst judge of my own stuff. I mean, sure, sometimes I look back at stuff I wrote back in the noughties when I first joined, and I'd forgotten I wrote it, and it makes me laugh. Other times it makes me cringe (but I keep it around anyway). I expect if I'm still here in another 16 years, I'll say the same thing about this year's output.

I just don't know. I think I've improved -- I hope I've improved -- but I have no objective measure on that. Changed? Sure. I have more confidence now, but I also have a better idea of my limitations.

Aside from the people themselves, as I mentioned above, there are several things about this site that I think help anyone grow as a writer. The most obvious of these is reviewing. When done with intent, it can really help a writer determine what works and what needs work. And reviewing itself is an exercise in writing. Also, even the most cursory review at least lets you know someone's reading your stuff. I don't do enough of it, myself; I will work on that in the future.

I know I said I didn't want to thank anyone individually, but I'd be horribly remiss if I didn't call out three specific individuals for thanks. The first two are obvious: The StoryMistress

and The StoryMaster

. They've built a great community and a wonderful platform for all of us. I know very little about web coding and even less about graphic design, so their work might as well be sorcery for all I understand it.

The third to thank is maybe not so obvious to anyone else: Artemismad Scientist

, who, approximately sixteen years ago this week, first told me about the site. Without her recommendation, I might not be here at all. Yeah, I know, some of you don't see the downside to that.

Before I go, I need to announce the Merit Badge winner from yesterday.

Mini-Contest Results!

Great comments, all. I don't really have the time, tonight, to address all of them individually, but I did appreciate every one, which made it hard to pick just one as a favorite.

But I'm going to go with ⭐Princette♥PengthuluWrites

, who wrote:

I hope for Cthulhu to awaken in September and put us all out of our misery. I figure, considering how 2020 is going, that's not as farfetched as it might otherwise seem. If Cthulhu doesn't rise in September, I'm putting it down for December. Gotta end the year in style, right?

Because I can totally see that happening. Not in September though. Or December. But this is going to take a bit of an explanation.

One of my favorite writers was Roger Zelazny. He, sadly, died back in the 90s. His last full novel before he left us was titled A Night in the Lonesome October, and you can probably call it my favorite book. I don't want to spoil the thing too much, but basically, the premise is this: On certain rare occasions, it is possible for the door to the realm of the Elder Gods to be swung wide, releasing eldritch horrors to doom us all. Possible, that is, but not certain; it hasn't happened yet (in the story).

What is this rare occasion? It is a full moon falling on Halloween.

Guess what's happening for real on Halloween 2020. Go ahead. Take a wild guess.

Point is, ⭐Princette♥PengthuluWrites

gets the MB this time, but I'll do this again soon so there will be more chances. Yes, even with the 30DBC running.

10 Like • Comment • View 6 comments

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