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').

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.

June 30, 2019 at 12:35am

#961783

Treadmill

https://www.theatlantic.com/family/archive/2019/06/big-houses-american-happy/591...

Are McMansions Making People Any Happier?
Homes have gotten bigger, but Americans aren’t any more pleased with the extra space.

Let's get this out of the way first: When the headline is a question, chances are good that the answer is "no."

And second: What the hell is this obsession that article writers have with happiness? Is it a uniquely American thing, perhaps having something to do with the infamous phrase in the Declaration of Independence?

But according to a recent paper, Americans aren’t getting any happier with their ever bigger homes. “Despite a major upscaling of single-family houses since 1980,” writes Clément Bellet, a postdoctoral fellow at the European business school INSEAD, “house satisfaction has remained steady in American suburbs.”

As a corollary to that first sentence, people aren't getting any less happy, either. Could it be that there are other issues involved besides that ever-elusive, undefinable "happiness?"

Why, yes.

To be clear, having more space does generally lead to people saying they’re more pleased with their home. The problem is that the satisfaction often doesn’t last if even bigger homes pop up nearby. “If I bought a house to feel like I'm ‘the king of my neighborhood,’ but a new king arises, it makes me feel very bad about my house,” Bellet wrote to me in an email.

So, it's not all that important what you have, as long as your neighbors (literally, in this case) don't have more.

And there we have one of the biggest problems with America (and, presumably, other countries) today.

I once met the guy who was, at the time, the richest motherfucker in the world. His estimated net worth was something like $5 billion, which seems laughable now - but, even accounting for inflation since then, $5 billion is a lot of money. We happened to be taking the same elevator, and I recognized him. It was certainly not mutual - I was, and remain, nobody special - but we nodded amicably at each other. That was the extent of our interaction.

I didn't compare myself to him.

Some people have been dealing with this by going, in my opinion, too far in the other direction, purchasing and living in what's euphemistically known as "tiny houses," but I'll continue to call "trailers," because that's what they are. Then the race is about seeing who can live in the smallest trailer, of course. That becomes the new status symbol. I understand there was even a "reality" show about it, which played up the concept, because it was probably funded by trailer manufacturers. I also remember hearing that a lot of the people who ended up in those closets got fed up with the situation really quickly. Presumably, it works for some people, but I wouldn't be one of them. I'm... happy... with my perfectly average-sized (probably more on the small side nowadays) home. Which, I'd like to add smugly, is paid for - that's what's important to me.

Bellet sketches out an unfulfilling cycle of one-upmanship, in which the owners of the biggest homes are most satisfied if their home remains among the biggest, and those who rank right below them grow less satisfied as their dwelling looks ever more measly by comparison. He estimates that from 1980 to 2009, the size of the largest 10 percent of houses increased 1.4 times as fast as did the size of the median house. This means that the reference point many people have for what constitutes a big home has shifted further out of reach, just as many other lifestyle reference points have shifted in an age of pronounced wealth inequality.

"Unfulfilling," indeed. You know, there two basic ways to deal with something that doesn't make you happy: a) change the situation or b) get used to it. Of the two, (b) sounds more... well, fulfilling... in the long run.

This vicious cycle used to be called "keeping up with the Joneses," for obscure historical reasons having something to do with the ubiquity of the name "Jones" and how the saying wouldn't work as nicely with the other two high-frequency names in the US, Smith and Brown.

You're perfectly happy with your used Chevy until the neighbor gets a new Cadillac, so you run out and buy a BMW on credit. The neighbor responds by leasing a Mercedes. This goes on until one of you either gives up or declares bankruptcy.

Screw that.

Find happiness in other places. Or give up on the concept entirely, because happiness is, by nature, fleeting, and requires constant effort to capture once again, much as you keep having to set the treadmill higher to get the same exercise value out of it.

Paradoxically enough, giving up might be what finally brings you happiness.

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