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!

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

a day 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').

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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 13, 2019 at 12:13am

#958848

Just Right

Discuss the “Goldilocks Rule of Motivation” as described on the website below. “Human beings love challenges, but only if they are within the optimal zone of difficulty. Tasks that are significantly below your current abilities are boring. Tasks that are significantly beyond your current abilities are discouraging. But tasks that are right on the border of success and failure are incredibly motivating to our human brains.”

https://jamesclear.com/motivation#How%20to%20Stay%20Motivated

The argument presented is logical, but as with many logical arguments, I question the basic premise.

Human beings love challenges, but only if they are within the optimal zone of difficulty. Tasks that are significantly below your current abilities are boring. Tasks that are significantly beyond your current abilities are discouraging. But tasks that are right on the border of success and failure are incredibly motivating to our human brains. We want nothing more than to master a skill just beyond our current horizon.

There are things this theory fails to take into account. For one, the very human aversion to failure. For another, the also very human fear of success.

And it doesn't provide room for the other thing that's always held me back: depression.

As I've said before, some of my favorite art has been motivated by depression. It can be a profound driving force in creating emotional work. First and foremost among these, for me, is the music of people like Leonard Cohen and Bruce Springsteen, both of whom have publicly chronicled their battles with severe depression.

And yet, there's a difference. Springsteen achieved fame and fortune. Cohen - well, maybe not as famous, but you've heard his stuff whether you realize it or not. They're both pervasive in Western culture, and they achieved this in spite of their mental health issues, or perhaps because of them. Could you imagine a song like "Hallelujah" coming from a balanced, healthy and happy mind?

Most people are more familiar with some of the covers of this song, which is fine, but it was Cohen who wrote it.

Point is, these people have accomplished something very public and long-lasting, where for most of us, depression just keeps us from doing stuff in the first place.

Perhaps it came easily for them. Maybe the act of songwriting (or creating whatever art), recording, and performing something like that was right on the edge of success and failure for them.

For me, a good example of a task that's right on the edge of success and failure is cooking a halfway healthy meal. I mean, shit, I have to buy prewashed broccoli because I can't be arsed to wash the little buggers. And yet I still screw that up more often than not. Most days, even the thought of feeding myself can be overwhelming, no matter how simple. So it usually goes like this: I'm not hungry, so I'm not going to cook. Then, I start getting a little hungry, but I think about all the work it's going to take to open a fucking bag of broccoli, pour olive oil into a pan, mixing in other ingredients, etc., and I balk. Then I get really damn hungry, so ordering a pizza is looking more and more likely.

Well, okay, I haven't actually ordered a pizza since I decided to lose weight, but the principle remains. What motivates me to lose weight, I still don't know. Just to see if I can do it, I guess; my doctor told me I'd die if I didn't, and I just shrugged. I mean, I'm going to die anyway, right? We all are, at some point, and I might as well go out on my own terms.

So the idea of using my depression to fuel great art is laughable to me. So is the idea of motivation of any sort. I consider myself fortunate if I drag my ass out of bed in the morning.

So ring the bells that still can ring
Forget your perfect offering
There is a crack in everything
That's how the light gets in.

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