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

September 29, 2022 at 12:01am

#1038303

Be Happy!

I'm not really happy with this article.

A Nobel Prize-winning psychologist says most people don’t really want to be happy

It's not that I disagree. Anyone who's been following along should know that I think happiness is overrated (and fleeting at best). It's just that there are a few red flags here.

We think we want to be happy.

Assumes facts not in evidence. I'm not part of that "we," and I suspect I'm not alone.

Yet many of us are actually working toward some other end, according to cognitive psychologist Daniel Kahneman, winner of the 2002 Nobel Prize in economics.

Possible argument from authority. While I admit that economics and psychology have some overlap, a Nobel Prize is not an indicator of general intelligence, but of a successful breakthrough in a certain field (exception: the Peace Prize, which hasn't meant shit since at least the early 70s).

Kahneman contends that happiness and satisfaction are distinct.

Assertion without evidence. I mean, sure, I agree, but I need to be critical of articles that agree with me. Helps me avoid confirmation bias.

For example, in Kahneman’s research measuring everyday happiness—the experiences that leave people feeling good—he found that spending time with friends was highly effective. Yet those focused on long-term goals that yield satisfaction don’t necessarily prioritize socializing, as they’re busy with the bigger picture.

Fails to account for differences between extroverts and introverts; also nothing is said about those on the autism spectrum.

In an October interview with Ha’aretz (paywall), Kahneman argues that satisfaction is based mostly on comparisons. “Life satisfaction is connected to a large degree to social yardsticks–achieving goals, meeting expectations.”

How much of that is actual personal achievement, and how much of that is due to comparing one's outward measures of success to others'? Like, are you really happy just because you've hit your mark, or do you see that the neighbor has a better car or a greener lawn and decide you have to do better? (I'm the former.)

He notes that money has a significant influence on life satisfaction, whereas happiness is affected by money only when funds are lacking. Poverty creates suffering, but above a certain level of income that satisfies our basic needs, wealth doesn’t increase happiness.

Pretty sure I've noted before that the study he's referencing there is highly suspect. Ah, here it is, and it was only a couple of weeks ago: "Gut Wrenching"

The key here is memory. Satisfaction is retrospective. Happiness occurs in real time. In Kahneman’s work, he found that people tell themselves a story about their lives, which may or may not add up to a pleasing tale. Yet, our day-to-day experiences yield positive feelings that may not advance that longer story, necessarily. Memory is enduring. Feelings pass. Many of our happiest moments aren’t preserved—they’re not all caught on camera but just happen. And then they’re gone.

Again, doesn't jive with my personal experience. My memory is shit. For example, I couldn't remember how long ago I'd written that entry I just referred to. When I saw it was just two weeks, I was surprised. I don't remember many details of events or conversations—but the way that event made me feel is indelibly burned into my synapses. So, no, I have a much better memory of emotions than I do people, places, and happenings.

This theory helps to explain our current social media-driven culture. To some extent, we care less about enjoying ourselves than presenting the appearance of an enviable existence. We’re preoccupied with quantifying friends and followers rather than spending time with people we like. And ultimately, this makes us miserable.

Ah. Now I begin to understand why I'm so different. That quote doesn't describe me at all. I despise social media and its pretensions.

We feel happiness primarily in the company of others, Kahneman argues.

I feel happiness primarily in the company of beer.

However, the positive psychology movement that has arisen in part as a result of his work doesn’t emphasize spontaneity and relationships.

Wait, is this guy the one responsible for "Toxic Positivity"

? If so, no wonder I'm squicked out at this whole article.

Kahneman counts himself lucky and “fairly happy.” He says he’s led “an interesting life” because he’s spent much of his time working with people whose company he enjoyed. But he notes that there have been periods when he worked alone on writing that were “terrible,” when he felt “miserable.”

People are different. I know people who are happy working "alone on writing." Hell, that might describe some of my regular readers.

Still, it’s worth asking if we want to be happy, to experience positive feelings, or simply wish to construct narratives that seems worth telling ourselves and others, but doesn’t necessarily yield pleasure.

On that point, I conditionally agree: it's worth thinking about. For me, I'm content to be happy on occasion, and coast through everything else. I'd prefer to avoid misery, but to me that's not the same thing as seeking happiness; it's just pain avoidance.

2 Like • Comment

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