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 5 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 7 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 24, 2022 at 12:01am

#1035616

Home on Deranged

For a dedicated indoorsman, I sure post a lot of articles about the (shudder) outdoors.

11 Common Survival Mistakes That Can Get You Killed
These simple mistakes can lead to catastrophic scenarios. Here’s how to avoid them

1. Going outside
2. Going outside
3. Going outside
...
11. Going outside.

But, you know, staying at home has its hazards too. Falling down stairs. Slipping in the bathtub. Owning a weasel. That sort of thing. So let's see what the actual article has to say.

We all make mistakes. It’s only human. But what happens when our mistakes are combined with treacherous conditions in the outdoors? What happens when our blunders combine into a series of unfortunate events?

Evolution in action.

1. Going Alone

Well, shit. As an introvert, I do this all the time.

After my divorce lo these many years ago, I got it in my head that I was going to travel across the continent, from the easternmost point to the westernmost point of the contiguous US. You can read all about it in my offsite blog (link in the sidebar on the left), which I haven't used lately but some of the earliest entries were about it. So I'm not going to rehash the whole thing here—just the endpoint. This involved parking at a ranger station in western Washington and hiking three miles through climax rainforest to the Pacific.

At that time, hiking three miles was easy for me. I made it to the ocean, everything was fine. No one was around, though campsites indicated that it was a fairly popular spot. But this was November, when the weather can be a bit iffy in the PNW. Still, that day, the temperature was perfect for hiking; the sun was shining just a few degrees above the western horizon...

...and that's when I realized I still had to hike three miles back and it was about to get very, very dark.

Long story short, I made it, but I got back to the car when it was full dark out. Obviously, I wasn't eaten by a bear, but that's more luck than planning. My whole body shook from exertion, and I could not get warm enough, even with the seat heater on and the vents blasting on full. Insult to injury, I had to spend that night in a motel in Forks. This was right after Twilight came out, so you couldn't walk down the street without tripping over a cardboard cutout of a lame character from the movie.

Point is, yeah, I made this mistake, and got lucky.

2. Getting Lost

Again, not something that you have to worry about if you stick to buildings and roads (and poorly maintained rainforest trails), and have some basic directional sense. Having a GPS doesn't hurt, either, but one shouldn't be over-reliant on technology.

Prevention: The best way to prevent getting lost is to get more familiar with land navigation. It’s not enough to bring a map and compass with you into the wild. You’ll need to know how to shoot an azimuth and back azimuth, account for declination, and determine distances (among many other skills). You’ll also need to know that steel or iron objects can pull your compass needle away from magnetic north, so keep your rifle barrel away from your compass as you take bearings.

Prevention: Don't go outside.

3. (Not-so) Edible Plants and Mushrooms

I had a whole entry about poisonous shrooms exactly one month ago: "Everyone Calls Me Mushroom Because I'm Such A Fun Guy"

Short version: Don't go outside.

4. Failure To Light

It’s ironic but true. The times when we need a fire the most are the times when fire can be the hardest to produce. Cold, windy, rainy conditions are prime time for hypothermia (the dangerous lowering of your body’s core temperature), and these are the toughest conditions for building a fire.

No, it's not "ironic." This, too, can be solved by utilizing our thousands of years of technological innovation and sticking to gas or electric heating inside a comfortable building. I mean, shit, our ancestors built for us a splendid mansion, and y'all wanna sleep in a mud hut.

Skipping a few here. If you refuse to heed my advice and want to go into the not-indoors, you can look at the link.

10. Ignoring Your Instincts

You mean like my instinct to find the nearest bar or craft beer dispensary and settle in there?

11. Underestimating The Risks

There are risks involved in enjoying the great outdoors. Every year, a handful of people run into real trouble in the wilderness – often because they didn’t understand the risks they were truly facing.

So, like I said, there are risks involved with staying home, too. Sure, I continue to make jokes about not venturing into the not-so-great outdoors, but—as my anecdote above indicates—I do like to go out from time to time. Perhaps I make the opposite mistake: overestimating the risks. But to me, that's preferable. A life of complete safety is no life at all (and also unattainable), but I'd rather take my chances with shitty motels than deal with tents, campfires, and nosy elephants (or whatever animals are around; I don't know).

It's not about not taking risks. It's about risk management, balanced with whatever enjoyment you might get out of the adventure. I will never again go camping, for several reasons, but if I did, I'd go with someone who knows what the hell they're doing.

Because that person is not me.

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