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 6 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 8 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 29, 2021 at 12:18am

#1010947

Promptly

PROMPT May 29th

We need your help filling the Challenge War Chest with new prompts for future rounds of the 30DBC! Write three of your own prompts and then use one of them to finish the rest of your entry.

Yeah... I've been dreading this day.

As I'm sure I've said before, I only have a limited supply of prompts in my head. I have to use one every week for the Cramp, and with that it's often the case that I can't come up with one until the last minute.

When you ask a computer to generate a random number, you don't really get a random number. It's what's called a pseudorandom number, based on a seed value. And that seed value will always generate the same sequence of pseudorandom numbers. This is a good thing for debugging programs. It is not such a good thing when you want an actual random number.

To get around this, sometimes the generator will pick some environmental value as the seed. I hear background noise level is a common one; it tends to vary unpredictably. As for the computations it uses once you give it a seed, well, that's above my pay grade, but from what I understand it involves chaos theory and... well... complex numbers.

The point is, humans -- even the one writing this -- are horrible at choosing random numbers. Unpredictable numbers, sure, but if you ask a bunch of human to give you a number between 1 and 10, for instance, you will get 1 and 10 relatively rarely. But if you roll a 10-sided die multiple times, or have a computer select the numbers pseudorandomly, you'll get them with approximately the same frequency as the other 8 numbers.

What I mean is... we all have our biases, and most of them are unconscious. So, for example, if I just saw my cat sniffing at the mint plant that I've somehow managed to keep alive for months (the average life expectancy of a plant in my care is about 3.4 seconds), I'll come up with a prompt like "Write about how a pet's curiosity gets her in trouble" or something like that. (Because mint is similar to catnip, and I'm quite good at killing the plant on my own, thanks, I don't need your help, kitty.)

How does this relate to prompts, you ask? Well, ideally, I'd pick them at random. But that requires something like a predetermined list that you can choose from at random, or at the very least, several sets of ideas that you can blend together. Both of these still require some thought, and thought may be unpredictable to an outsider, but it's not random.

So anyway, here are three prompts that were prompted (heh) by things in my current environment.

1) What's your least favorite kind of litter, and how would you eliminate it?
(There's a discarded paper plate on my deck because I was using it to feed the cats, and I can't be arsed to pick it up right now because it's wet.)

2) Write about a memorable rainstorm.
(The paper plate is wet because there's a soft rainfall going on right now.)

3) How do you avoid being rained on?
(I'm able to sit on the deck and type this on my laptop while it's raining because I have a nice patio umbrella that's useless when it's windy, but right now it's not windy.)

Now I'll use a random number generator app to pick one. This is how I pick things to talk about when I'm not doing this blog challenge. I could use WDC's Virtual Dice, but my phone is more convenient.

2) Write about a memorable rainstorm.

In August of 2017, a solar eclipse streaked across the United States.

I'd known about the eclipse for years, because, hello, nerd here. I figured I'd wait until a year or so before the eclipse, and find a brewpub along its path, and make reservations at a nearby hotel. There's a good chance of a brewpub being in the path of such an eclipse, because the path spanned the US, and we have a lot of brewpubs. I figured I'd sit outside to have a beer and watch it.

Well, that didn't happen; what with one thing and another, I procrastinated, as is my wont, and it was only like two weeks before when I thought, hey, I really would like to see a total eclipse. I figured, okay, I'll go to Wyoming and find a spot out in the middle of nowhere because there's a lot of nowhere in Wyoming and that state, of all the ones crossed by the eclipse, had the smallest chance of being overcast on an August day. Any more west, and you run into the notoriously cloudy PNW; east coast, and you get hurricanes sometimes in August.

But then my friends wanted to go, too, and neither of them could be arsed to go to Wyoming. We settled on Misery. I called it "the Missouri compromise," because at first they were thinking closer states like Kentucky or South Carolina, which, like I said... chance of rain.

My friend miraculously found a hotel room available the night before and after in central Misery, so we spent a lot of money reserving that, and off we went.

The day of the eclipse came, and... there were clouds. Fortunately, they were the thin, high ones, clouds that only very barely covered the face of the accursed daystar. We saw the eclipse. It was awesome. But that's obviously not relevant to the prompt.

What is relevant is that, the next day, we were driving back along I-64, and the skies opened up and poured down rain fit to sink Noah's ark. I couldn't see in front of my hood. The wipers became useless; I could see better without their constant slapping. That kind of rain.

Now, when you're driving down an interstate in conditions like that - and I've done it in rainstorms, fog, and at least one blizzard - you're fucked. If you slow down, you risk getting hit from behind. If you speed up, well, don't do that. If you keep going at your pre-deluge speed, you risk slamming into the idiot in front of you who slowed down. If you pull over to the shoulder or under an overpass to wait it out, you risk someone else having the same idea and, again, slamming into you.

Normally, such a storm lasts about five minutes, and then you get better, if wet, visibility. Well, this one must have also had the idea to take I-64 east, so we were in that thing for what felt like hours but was probably only half an hour. Long enough to get into many accidents. Spoiler: we didn't.

Driving down that interstate, hands clutched on the wheel, peering ahead for the telltale signs of brake lights (which would be the only way I could see a car ahead of me, and then probably too close to safely slow down or stop), all I could think of was:

"Gee, I'm glad it wasn't doing this while we were staring at the sun yesterday."

There's another eclipse in the US in April of 2024. I'll have to drive to see that one in totality, too. Assuming I'm still alive for it, I'm looking forward to that. Rain or no rain.

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