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 an hour 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 3 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.

May 21, 2023 at 9:03am

#1049920

Revisited: "Naturally"

Spinning the dials on the ol' time machine, today we land on "Naturally"

, from just over two years ago.

The entry was a 30DBC response, and the prompt was: Write about your favorite outdoor activities to do in the summer. Are there any activities you haven’t done that you want to try?

I have a distinct memory of shuddering when I first saw that prompt. I have a bio somewhere—not on a dating site, because I don't do dating sites—that describes me as a "dedicated indoorsman."

So, I opted for humor; specifically, hyperbole:

Well, my only favorite outdoor activity in the summer is: rushing from an air-conditioned car into an air-conditioned bar; if the accursed daystar is burning, add "while shading myself as much as possible."

And that's not entirely true. I rarely drive to bars, on the theory that I might have to leave my car there to Uber home, thus necessitating another trip, the next morning, while hung over. So the "car" in question is a rideshare.

I'm just as likely to walk to the bar. This is true no matter the outdoor temperature. That doesn't mean I'm enjoying the outdoors; it just means I'm a cheapskate.

Still, given a choice between being too hot and too cold, I will pick too hot every damn time, so if I'm going to do anything outside, it'll be in the summer. Of course, one is never given such a choice; either it's winter and too cold, summer and too hot, or a week or so in between when it's actually pleasant to be outside for five minutes when it's not raining.

This is only a slight exaggeration. We usually get more than a week of mid-range temperatures. Sometimes as many as two!

But this does give me the opportunity to talk about a phrase that triggers my grumpiness, to a possibly irrational degree: when someone says something like "I'd rather it be cold because I can always put on more clothes, but there's only so many I can take off."

It's not that those people are different from me. As with being a morning or night person, people prefer different things and it's hard to change that, even if I wanted to, which I don't. If this weren't the case, we wouldn't have people living in Alaska or, conversely, Arizona. So, okay, you prefer the cold; that's good to know. We can be friends, but we'll never take a Caribbean cruise together.

I think it's partly the implied naughtiness: tee hee, now you're picturing me naked. Thanks, I'm already doing that (if you're female) or repulsed by the idea (otherwise).

Worse, though, is that it's dead wrong.

Ever seen people adapted to extreme heat? I don't mean the scantily-clad jungle-dwellers of various tropical societies; they've usually got some shade to cool off in. I mean, like, the desert-dwellers, the real-life inspiration for the Tusken raiders of Star Wars fame. Do they run around mostly naked? No, they're completely covered in loose-fitting clothing. This serves as portable shade and takes advantage of the slightest breeze to provide evaporative air conditioning.

And for me, when it's cold, it doesn't matter how many layers of clothing I wear; I'm going to freeze anyway. Once my hands or feet are cold, that's it. I'm done. If I wear enough clothing to delay this freeze, I'm uncomfortable and can barely move.

So I prefer to do as nature intended, stay indoors and run my A/C in the summer and heating in the winter. Yeah, compared to the vast bulk of humanity, both now and in the past, I'm living in privileged luxury. So? Might as well take advantage of it.

No, I'm of the firm opinion that we evolved to build shelters for a reason, and that reason is that so we could use them. Don't get me wrong; I love nature. I'm a big fan of watching webcams of natural areas.

And that's an excuse to re-link the webcam I talked about in that entry, since we're coming up on the summer solstice once again: The Brooks Falls Brown Bears

from Alaska.

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