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 13 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 15 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.

April 2, 2022 at 12:02am

#1029896

Suspect Motivations

Today, I'll discuss motivation again. What motivated me to do this? Nothing. It came up at random from my queue.

Motivation: How to get it, keep it and use it, even during a pandemic when you may need it most

The funny thing about motivation -- well, one of the funny things -- is that it's not enough to want to do something. For instance, take that perennial want: weight loss. So you want to lose weight. Most people do. At least until you find out how much bloody work it is. And how much you have to give up of life's simple pleasures to do so. I did it for two years, until I decided... fuck it, this is nice but it's not worth giving up everything I love. I mean, come on, think about the things you love the most. If someone told you that you could maybe, no guarantee, live an extra five years if you never did anything you loved most again, would you take that monkey's paw?

Another funny thing is that it's not enough to want to do something. You also have to want to want to do it. And you have to want to want to want to do this. Take the derivative as many times as you need to; eventually, the resulting equation devolves into a constant, and that constant is Longing.

Non-math people may not grok what I just said, but whatever.

We've all done it. We've chosen a diet or workout plan that will get us in shape; said we're going to spend less time on our cellphones and more time reading; or committed to making the most out of our day by being as productive as possible.

And there it is again: the Holy Productivity. All hail Productivity. Productivity will save us all.

But now, as the coronavirus continues to spread across the globe, it can be difficult to find the motivation to set and achieve goals, especially when the future feels uncertain.

Yeah, look, the article is two years old, from the beginning of lockdowns and shit. Doesn't matter.

While it can be hard to find and maintain the motivation to make a habit stick, experts have some motivational secrets to propel you through the quicksand of inertia to achieve your goals.

My expert secret: Forget about it. If you really want to do something, you'll just do it. If you're not doing it, then you don't really want to do it; you just think you do.

I visited a shrink once when I was feeling unmotivated and paralyzed at work, in hopes that he'd help me figure out this mess in my head. Instead, he told me, basically, "You just don't wanna." "But I do want to," I said. "But you're not doing it."

Shrinks haven't helped me much.

The typical advice for finding motivation is to "keep your eyes on the prize," to remind yourself of your ultimate goal. However, a recent study found that the secret to accomplishment may be to focus on effort first and rewards last.

You know what else I figured out? It's that in order to do something new, you have to give up something, because there's only 24 hours in a day. And that's the opposite of my definition of happiness, which is not having what you want, but wanting what you have. I want to keep what I have, not start something new that might not work out.

Sure, there have been exceptions. I'm nearing 950 days straight doing lessons on Duolingo. But to do so, I had to give up video game time. Everything is a tradeoff.

"When deciding, we tend to concentrate on and base our decisions on rewards. When executing an action, we switch our attention to the effort required," Ludwiczak said. "If this effort is more than we anticipated, we might abandon the task, deciding it's not worth it."

And usually, it is not.

In addition to a routine and a healthy diet, working toward a goal with others can also help you to stay motivated, as it keeps you accountable to someone other than yourself.

Likely, this depends on the individual. I hate letting other people down. I have no problem letting myself down, as I can do the mental gymnastics required to convince myself I never really wanted whatever in the first place.

But for others, the opposite might be the case.

"Think about hiking," she said. "When you're setting out on an adventure, you need your backpack, supplies and equipment, but you also need a compass to tell you where you're heading. Without one or the other it is not likely to be a very successful trip."

If you're trying to get me motivated by using a hiking metaphor, you're wasting your time. As soon as I think about hiking, my brain switches to "how about a beer instead?"

But hey, who knows -- maybe the article will be useful to someone else. Me, I can't be arsed.

2 Like • Comment

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