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:

Brandiwyn🎶

a day ago
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.

Aiva Raine

Nov 5, 2024 at 3:03pm
In response to "It's Hard Not To Scare"

Shaking head. Aside from the inevitable Simpson's spider goat song playing in my head which I will now share with you below, whenever I come across a story that has all these scientists doing ridiculous experiments for no logical reason, I must go- ah- scientists- CANCER? You do remember that problem right? Can we maybe cure that before we make cockroaches magnetic and elephants see pink elephants?

spider pig clip- and if you all now are singing the spider goat song- you're welcome! grin
https://www.youtube.com/watch?v=BARjPuUN36Y

1 Like

Aiva Raine

Nov 5, 2024 at 2:55pm
In response to "Anomalies"

One of your more humorous entries- I especially enjoyed- "Lofoten Vortex can be the name of my Bjork cover band."

Thanks for the laughs. Much appreciated.

2 Like

June 28, 2023 at 9:21am

#1051699

The Second Mouse

Here's another opportunity for me to rail against the smug supposed superiority of wormcatchers.

Why Waking Up Earlier Isn't Necessarily Better

Not that there's anything inherently wrong with waking up early, as I've noted before. It's the whole "turning it into a moral mandate" thing that bugs me, and the utter disregard for those of us with less common sleep schedules.

Still, this article (from, ironically, Time) makes a few good points I hadn't considered

A favorite trope of sleep research is to divide the entire human population into two cute, feathered categories: early birds (also called larks) and night owls. Often, these studies link people’s natural sleep patterns—called their chronotype—with some waking behavior or personality trait.

Kind of like astrology.

Research says that early birds are happier, more punctual, do better in school, and share more conservative morals. Night owls are more impulsive, angry, and likely to become cyberbullies; they have shoddier diets and, most critically, are worse at kicking soccer balls.

"Research" done by larks. This is like those studies that conclude cats are psychopaths, which, when you look into the authors, you find they're all dogs. Well, dog people, at least.

But can the population really be categorized so neatly? Or is the research painting an incomplete and overly moralistic picture?

No. Yes.

A study published May 24 in PLOS ONE by a group of Polish researchers takes a fresh look at the long-established link between being an early riser and being conscientious by examining a separate but potentially important variable that might underlie the link: being religious. The team found that people who woke up earlier tended to score higher on all dimensions of religiosity, leading them to conclude that being religious could help explain why early risers are more conscientious and more satisfied overall.

In other words: correlation, not causation. Not that religion is any kind of predictor of a person's morality, either.

“I think most people would recognize that, in reality, [chronotype is] more of a continuous type of variable,” says Brian Gunia, a sleep researcher, professor, and associate dean at Johns Hopkins’ Carey Business School. It exists on a spectrum: not everyone is always one or the other. But so much research uses this binary classification because people are usually able to self-identify that way, Gunia says.

And there's that. Life is rarely binary. Is someone who wakes up at 6:59 am a lark, while someone who wakes up at 7 is an owl? I have similar issues with how "generations" are labeled.

The bias that people who rise early are morally superior to evening people doesn’t just loom large in scientific research. It’s at the very heart of the U.S.’s founding principles of industry and hard work, says Declan Gilmer, a PhD student at the University of Connecticut who studies workplace psychology. “If someone gets up at 6 a.m., and they show up at work early, they’re viewed potentially as more committed,” he says.

Founding principles, my ass. Also, one can also show one's commitment to helping one's boss buy a second yacht by staying late.

“Some of the better work in the topic area has been trying to identify the genes that are most tightly linked to morningness and eveningness,” he says—genes that, if understood, could open the door to a more nuanced view of the topic.

Especially since genes alone can't give you the full picture. There are other factors, including environmental.

Very few chronotype studies include information about the time of day during which the research was conducted, but Gunia’s research has found that this seemingly simple factor can change data a fair bit. In a 2014 study of chronotype and ethical behavior, for example, “we found that morning people are most ethical in the morning, and evening people are most ethical in the evening, so maybe it’s more of a fit between chronotype and time [of day] than it is this idea that morning people are better or worse,” Gunia says. Studies that don’t take time of day into account “are missing half the equation.”

That seems so blindingly obvious to me that I should be surprised no one's thought of it, and yet, I'm not, really. I know if I had to wake up at 6 am to go be a guinea pig in a chronotype study, I'd be grumpy as fuck.

Humans don’t always fit neatly into one of two categories, even when it comes to their sleep preferences.

Duh. For instance, I'm biphasic. While I tend to go to sleep late, I also sleep most late afternoons. One of the great things about being me is that I'm usually able to go to sleep when I'm tired, and wake up when I'm not. This is the pattern I fell into, but even when I had to simulate being a wormcatcher, my gas tank would run out in the afternoons.

I'm not saying this makes me superior. It also doesn't make me inferior.

You don’t have to be a morning lark or a night owl. You can be any kind of bird you like—there are plenty of worms to go around.

The whole wormcatcher thing always bugged me anyway. Worms? Eugh. That's why I latched onto the alternative phrase: The second mouse gets the cheese.

Mmm. Cheese.

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