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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Show Recent Comments

10 Most Recent Blog Comments:

Write_Mikey_Write!

about 6 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 8 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.

August 5, 2020 at 12:11am

#989926

Only the (not so) Lonely

I was an "only child," so this article caught my interest. So I'm sharing it. See? We can share.

https://qz.com/980226/neuroscience-shows-that-only-children-are-more-creative-mo...

Neuroscience shows that our gut instincts about only children are right

That headline can bite me. That's not the thrust of the article at all, as we shall see.

Conventional wisdom has it that only children are smarter and less sociable.

"Conventional wisdom" is almost always wrong. But flattery will get you everywhere with me.

Conversely, since those only children never have to share a toy, a bedroom, or a parent’s attention, it is assumed they miss out on that critical life skill of forever-having-to-get-along.

"Critical," my solitary ass.

But are their actual brains different?

OH COME ON.

Jiang Qiu, a professor of psychology at Southwest University in Chongqing, China and director of the Key Laboratory of Cognition and Personality in the ministry of education, led a team of Chinese researchers that sought to answer this question with more than 250 college-aged Chinese students.

FWEEEEET! Flag on the play. Minuscule sample size, representative of just one culture. 10 yard penalty, fourth down.

Hey, I just made a sportsball joke.

They used standard tests of intelligence, creativity, and personality type to measure their creativity, IQ, and agreeableness.

Fuck agreeableness.

On the behavioral tests, only children displayed no differences in terms of IQ, but higher levels of flexibility—one measure of creativity—and lower levels of agreeableness than kids with siblings.

You're also going to have to explain to me how to actually quantify creativity. I mean, I know there are issues with IQ tests, but at least some measures of intelligence can be quantified within a particular cultural and socioeconomic context.

Having worked on the 1896 study “Of Peculiar and Exceptional Children,” Hall cast only children as “oddballs” as “permanent misfits,” descriptions that have stuck over the years with remarkable persistence. “Being an only child is a disease in itself,” he claimed.

As developed countries transitioned from agrarian to industrial economies, family sizes tended to get smaller. I mean, I can't be arsed to find the data on that or anything, but I'm pretty sure it's the case. So, naturally, in the 19th century, growing up with a bunch of siblings would be considered "normal" and not doing so would be considered "abnormal." I assert that had the opposite been the case, it would be having siblings that would be considered "a disease in itself."

There is ample evidence suggesting the stereotypes of the “lonely only” are wrong.

As I'm pretty sure I've mentioned in here before, being sibling-free taught me a lot about self-reliance. I wouldn't want it any other way.

They found that only children, along with firstborns and people who have only one sibling, score higher IQ marks and achieve more, but aren’t markedly different personality-wise

Which right there contradicts the Chinese study, throwing both into question.

Jiang and his co-authors hypothesized a few reasons for their findings.

You can hypothesize all day. I mean, I do. Until you do the science, it remains opinion and educated musings.

Creativity —defined as having original ideas that have value—is strongly influenced by everything from family structure and parental views, to interactions and expectations

Okay, so you can define creativity, but again -- how do you measure it? If it can't be quantified, it's not science. And "original ideas that have value?" We share a world with seven billion other humans. I've had original ideas that, it turned out, someone else had already thought of.

The article goes on to talk about some measures of creativity: flexibility, originality, and fluency. Those sound more like general intelligence to me, but what the hell do I know?

Look, I'm not trying to argue from one datum (me), and if I hadn't been drinking wine I might seek out other sources to delve deeper into this, but so far it's not passing the bullshit test for me. Then again, I'm just not very creative, so maybe I'm just inflexible enough to provisionally reject this.

He also noted that just like IQ tests, creativity tests are not perfect measures of the thing they are measuring.

Duh.

Creativity involves spontaneity and intrinsic motivation—things which are a bit hard to assess on a test.

Pretty much by definition, you can't make a standardized test that measures creativity in a meaningful way. I mean, a truly creative person could find a way to bypass the strictures of the test, making it invalid. Right? Think about how James T. Kirk beat the Kobayashi Maru test. In either universe.

To summarize, this article panders to confirmation bias.

So back to my wine, and rewatching old episodes of Star Trek because it's been years since I actually watched them. Damn, some things were cringeworthy in the 1960s. But first...

Mini-contest results!

As is appropriate for a question about decisions, this was a tough decision for me. I appreciated all of the comments. From them I gathered that we all have different ways of making decisions, as I expected.

What clinched it for me, though, was the line: "I chart a course I want to take and weigh my decisions based on whether or not they move me along the course." I like that this implies a focus on making decisions based on achieving one's overall goals, and not just the pros and cons of an individual decision -- something that neither I nor the article's author seems to have considered, but in retrospect it seems important.

So this time, the Merit Badge goes to Turkey DrumStik

-- but again, I liked all the comments and everyone will get another chance soon.

I should also note that no, I'm not taking sides on which decision-making process is "best." Whatever works for you, works. But I hope that seeing how other people approach it helps us all with our own decision-making. As always, thanks for reading and commenting!

6 Like • Comment • View 5 comments

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