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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Turkey DrumStik

about 3 hours ago
In response to "Barley There"

And approximately 15 seconds later, someone's going to screech about genetically engineered crops.

That would be me...screeching for more genetically engineered crops.

Write_Mikey_Write!

a day 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

Nov 14, 2024 at 12:10pm
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

January 30, 2019 at 1:59am

#950746

Challenge Complete

It's the last day of the official competition, and the last War Chest Wednesday, so it's only fitting I repeat this prompt from November. Write a list of at least 5 blogging prompts to add to the Challenge War Chest to be used for future rounds of the 30DBC. Then, use one of your own prompts to write your entry.

This one was inordinately hard for me. I suppose that's only fair, as it is the last prompt of the month, and the others flowed easily. So yeah, I don't think these prompts are all that groundbreaking, but here they are anyway:

1) Write about one thing you want to do before you bite the dust.
2) If you had to go back and change one decision you made, what would it be and why?
3) Which movie and/or TV franchise would you most like to write for?
4) What job do you wish you had?
5) Abraham Lincoln is credited with saying "Better to remain silent and be thought a fool then to speak out and remove all doubt." Prove him right.

So, the prompt I'm going to use is the first one, but I'm pretty sure it's going to address #5 while I'm at it.

I don't know who coined the term "bucket list," and I can't be arsed to find out. I do know that it became the title of a moderately interesting movie with Jack Nicholson and God.

While I've been known to use the phrase from time to time, I usually call it my "fuck-it list." This is because I've already done most of what I set out to do, and have given up on the rest.

It's come to my attention recently that life is much less frustrating if you can separate "things that I cannot control" from "things that I can control," and to focus on the latter while letting go of the former. This is something akin to the infamous Serenity Prayer, which I resent because a) it's a prayer and b) it's associated with AA, and AA is for quitters.

And yet, far be it from me to reject an idea outright, so I started thinking about it.

Let's imagine that I wanted to win a marathon race. That is, to run 42.195km faster than anyone else in the race. This would depend on who else was in the race, and that is something beyond my control; if anyone has two working legs (or prostheses), or is over the age of 18 months or under the age of 105 years, I would not win. By this theory, the only thing I could control would be my own level of fitness.

But the assumption there is that I can control my own level of fitness. I've been walking every day for the past 5 weeks, which is, like, a new record for me for exercise. My walk is about 2.4km and takes approximately 30 minutes, giving me a walking speed of about 5km/hr, maybe a bit less. And when I finish, I'm out of breath, my calves are spasming, and my shins are on fire. Consequently, I cannot imagine walking 42km, let alone running that distance. It's hard to imagine any amount of training that would allow me to even finish a marathon.

Or take another example. Say I want to take a trip to space. How can I make that happen? Well, probably by paying Virgin Galactic a very large sum of money to fire me into a suborbital trajectory that technically takes me 100km up for a fraction of a minute, thus putting me, by definition, in space. Honestly, even if I were a billionaire, this would strike me as a colossal waste of money. There is nothing about this that I can control, except to decide that I don't really want to take a trip to space after all.

So here's the real example: I want to be published. Now, I'm not talking about self-publishing. Anyone can do that. Some can even be successful at it. What I want is for an editor to tell me, "This story is good and it meets our needs so here's some money." (Actually, it's not about the money, but that's part of the package.) And then have at least one person that I don't know purchase the story.

I have no control over publishers' or readers' decisions. The only thing I have control over is my own process: writing, editing, submitting.

But since I'm utter crap at selling myself and lousy at editing, I can't even control that much.

This is why I've given up on a some of my earlier goals.

But hey, at least I'm walking every day.

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