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 8 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 10 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.

July 4, 2020 at 12:14am

#987175

Forty

Happy boom-boom day, Americans! New rule: if I smell your barbecue, I'm coming over.

PROMPT July 4th

What could you give a 40-minute presentation on with absolutely no preparation?

One of my favorite scenes from TV is from a show I've never watched (apart from this scene). The only background you really need is that it aired shortly after Disney acquired Lucasfilm, and with it, the rights to Star Wars. So it's a bit dated now that the Skywalker Saga has officially ended, but taken in historical context, it's hilarious. Here it is:

Patton Oswalt is one of my favorite comedians, and he supposedly ad-libbed the whole monologue.

And yet... even though this guy is a professional actor and comic, and despite his impeccable geek cred, this cut clocks in at under nine minutes, part of which is intro. More, there are points at which he stumbles, backtracks, and even stutters. Now, I'm willing to admit that this might well have been all in character, but it doesn't come across that way. And still at the end of his roughly eight-minute filibuster, he practically collapses from dehydration and exhaustion (though I'm pretty sure that was a slice of ham).

As a bonus, you'll note Chris Pratt is in there, which adds to the awesomeness of this because he went on to join the Marvel Cinematic Universe, as I'm sure everyone knows, as Star-Lord ("Who?") in Guardians of the Galaxy.

I like to think Oswalt inspired at least some of Pratt's epic performance in those movies.

I, on the other hand, am not a professional actor, nor do I get paid for comedy (that's my free gift to the world; you're welcome). Okay, yes, I'm fairly well-versed in both Star Wars and the Marvel universe, but could I have done that?

No. Not without some preparation.

Now, I have done a bit of stand-up, but not for 40 minutes, and definitely not without rehearsal.

I used to know enough jokes to maybe fill 40 minutes, if I spoke slowly, but it wouldn't be original material. Besides, the internet has atrophied my formerly near-eidetic memory for jokes; why keep them banging around in my skull when I can google the punchlines and have the jokes right there in front of me?

Not to mention I'd have to make sure there were no kids in the room.

Hell, I once saw Lewis Black (another stand-up comedian and actor) when he performed in my town, and I had the honor of watching him utterly flub a joke on stage.

Of course, if you want the audience to fall asleep, I could probably have, at one point, given an ad-lib presentation on drainage design and erosion control practices in the Commonwealth of Virginia. But not these days.

On the other hand, I don't usually get stage fright, so that's something, I suppose. That trick you've heard about, the one where you imagine the audience naked? Doesn't work for me. It's supposed to make the speaker think that the audience is in an even more embarrassing situation than he or she is, I suppose. But all the blood flow to my brain ends up getting redirected.

I just realized that at least one of my teachers in high school must have used that trick. Sicko.

It's far more likely that knowing that I have to give a presentation will lead to those ubiquitous dreams where you're the one who has to perform naked.

As for Oswalt, after that show, he suffered some personal tragedy, but bounced back to work. And he eventually got to work for Disney/Marvel, as a side character -- well, several side characters -- in Marvel's Agents of S.H.I.E.L.D. We even got to see one of his characters sleeping in Star Wars bedsheets.

Good for him.

8 Like • Comment • View 7 comments

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