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!

about 8 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 10 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.

March 6, 2024 at 8:16am

#1065695

Les Langues d'Amour

Speaking of backing things up with science... from Big Think:

Everyone is wrong about “Love Languages.” Here’s why.
Big Think spoke to the author of "The 5 Love Languages" about the popular relationship theory — and its lack of scientific support.

I'm not immune to knee-jerk reactions. Sure, I try to reason through them later, but, as I suspect most people are, sometimes I just hear about something and react with instant rage or disgust. I'd speculate that this was an evolutionary adaptation on the part of our ancestors, who had to make survival decisions in a dangerous environment, but one of the things that sets me right off is speculation based on evolutionary psychology. So I won't.

One of the things I immediately scoffed at when first hearing about was "love languages." Then, after taking some time to find out what was meant by that, my reaction softened somewhat. After all, the concept boils down to "we all appreciate different things," which I wouldn't deny.

If you're reading this, and somehow still wondering what the hell I'm talking about, well, the précis is right there in the bullet summary at the top of the linked article: The idea, created by evangelical pastor Gary Chapman, is that to make your partner feel truly loved, you must show them affection in their preferred "language."

Birthed in a perennially best-selling book over three decades ago, the idea remains ever-present in popular media, dating apps, and social media. The book’s resonant success might make you assume that the theory was born in the lab of a superstar relationship psychologist, but the notion has humbler origins.

Actually, I immediately suspect any book, especially purported nonfiction, that achieves huge success, because I distrust the zeitgeist. And it's exceedingly rare that a book written by an actual scientist climbs to that level. The only one I can think of offhand is Hawking's A Brief History of Time, and that one had pretty pictures. (It was also worthy of the success.)

The architect of the theory is evangelical pastor Gary Chapman.

Okay, here's where I feel compelled to make an embarrassing confession: In my view, evangelicals are all guilty until proven innocent. Some of them are responsible for those grifting megachurches. Others are doing their best to turn our secular country into a theocracy. Many outright reject science in favor of superstition. The ones you hear about are always trying to make life difficult for gays, trans people, and anyone else who doesn't conform to their narrow view of what people should be like. When your ranks include the likes of the thankfully-late Jerry Falwell and Pat Robertson (and those are just the famous ones from my home state), you have a hell of a lot to answer for.

This may seem to contradict my radical acceptance of diversity. And maybe it does. But to me, there's a big difference between being born into a demographic, and choosing to align oneself with a certain philosophy. Even that difference, though, contradicts my assertion that we never really "choose" anything. So I don't know. I guess that, in this way, I've been prejudiced (even though I already had a counterexample in my cousin's evangelical wife, who's one of the kindest people I know). I should work on that.

Everyone's a hypocrite in some way. I figure at least I recognize some of my hypocrisies.

Point being, when I heard the author was an evangelical pastor, I instantly dismissed anything it might have to say. And I was (probably) accidentally right to do so... but not because of the author's affiliation.

The popularity of his book, quiz (taken more than 133 million times), and theory confirm to Chapman that love languages work.

No. No, it doesn't confirm any such thing, any more than the popularity of books on astrology confirm that astrology works.

Still, popularity and anecdotes do not prove that love languages actually work, or even exist at all. For that, science must weigh in.

You knew that was coming, because I'm writing these comments.

In a paper recently published in the journal Current Directions in Psychological Science, a trio of social psychologists specializing in relationships reviewed the science conducted on love languages and found it wanting.

And yet, one paper does not a definitive takedown make.

First off, research contradicts the notion that people have a primary love language, they write. Chapman’s quiz is fundamentally flawed, they argue, because it forces takers into binary this-or-that choices.

I'm on board with that finding, myself. As I keep saying, life is rarely binary. Not everything has to be "cool" or "sucks." Maybe it's just me being Gen-X (another labeling system I question), but to me, there's always "meh, whatever," too. In various shades.

Second, the five love languages overly simplify forms of love in relationships and are based on a homogenous sample. For example, the love languages “do not include mention of support for a partner’s autonomy or personal goals outside of the relationship, factors that have been associated with relationship satisfaction,” the authors noted. They added that Chapman formulated his ideas after counseling couples “who are all married, religious, and mixed gender and likely share traditional values.”

That is, to me, a serious problem. Like with the study I mentioned yesterday, which was all Brits. Or a lot of postdoc papers, which draw from a sample core of "American college students who want a few extra bucks."

Third, and most glaringly, the limited research that has been conducted does not support the idea that speaking someone’s preferred love language yields greater relationship satisfaction or success.

As the article admits, though, the research is scant. So though I want to agree with the conclusion, I can't quite rise to the level of "science says it's bullshit."

Chapman had a chance to respond to these issues, and this is the part where my mind gets blown because it doesn't fit in with the prejudice I stated above:

Chapman, who throughout the interview expressed intellectual humility that any scientist would admire, wanted to make clear that he appreciated researchers’ work to study love languages. “I’m not against research,” he said. “Scientific research is wonderful.”

So there you have it. I'm not changing my opinion that "love languages" are airy nonsense, but maybe I came away from this with a little less animosity to evangelicals in general.

And the world just got a little, tiny bit more tolerant.

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