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 9, 2019 at 12:53am

#960429

Snooze Button

https://aeon.co/essays/its-time-to-celebrate-the-humanity-of-the-communal-snooze...

American work culture, seeping around the globe, threatens to ruin the pleasures and benefits of public, communal sleep

The article opens by describing the culture shock some Americans felt in China when everyone else took a one-hour nap.

Thing is, communal sleep wouldn't work in the US, not just because we run screaming in terror whenever someone uses the adjective "communal," but because there will always be the one asshole who thinks it's funny to take a Sharpie and draw willies on the faces of the sleepers.

In fact, in the US, it is not only sleep that invites judgmental responses. There’s smoking (anything); food (how much, or how little, and its role in one’s health); alcohol (too much, of course, and who can abide the abstemious, the judges of judges?); and exercise and sex, with names for people who are seen as too active, and other names for people who aren’t active enough. Yet those judgments are founded on ideas about health and moderation. Part of what makes them obnoxious is that they’re not altogether wrong, and the intrusion touches a nerve.

I've been railing against this shit for years. The US was founded by people who were too uptight for British culture. Really think about that for a minute. These Puritans' attitudes shaped our social fabric - for good and ill. Mostly ill.

We’d all be better off dropping the moral judgment, and the policies that inevitably follow from it.

Much as I admire Ben Franklin, the whole "early to bed, early to rise" quote mentioned in the article has always annoyed me. And it's out of character for him. I'm pretty sure it was satire, trolling, or some proto-performance art.

There is an evolutionary argument for the staggered types of sleep across cultures. According to the psychologist Frederick Snyder’s sentinel hypothesis of 1966, for most of human history, sleep left people vulnerable to animals, other people and environmental dangers – not to mention the wrath of the spirit world. So, a community with varied chronotypes ensured someone was always alert to keep watch and sound an alarm.

Okay, look, I have an admission to make here. Whenever I see any sort of evolutionary hypothesis shoehorned into an article about any kind of human activity, my inner alarms go off (not the ones that wake me up, but the ones that warn me about bullshit). Most of these are just that: hypotheses. The first one I remember reading was an "explanation" for why boys carry their books swinging by their side, while girls carry them up against their torsos: "It's because the males were hunters, and that's how they carried spears, while the females carry babies like that." I was raised on a farm, so I know bullshit when I smell it. I suspect it's more to do with the well-known and noncontroversial observation that the average male has greater upper-body strength than the average female, followed by peer pressure to follow the crowd.

But even that is a hypothesis; the only thing that makes me like it better is the fact that I came up with it myself, which makes everything superior. My point is that we can't always know how our ancestors lived; and besides, such hypotheses ignore the 4 billion or so years of evolution that preceded the whole upright-ape tool-using food-cooking big-brain thing, which most of these stories focus on.

And they are stories; most of them lack evidence of any kind.

In this case, however, the evolutionary hypothesis doesn't detract from the basic thrust of the article, which I read as "stop fucking sleep-shaming," which I've been saying for a while now, so I don't need to say any more about it.

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