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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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.

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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.

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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').

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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.

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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

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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

October 7, 2021 at 12:01am

#1018859

Personality

No, I don't get all my information from Cracked. But it's enough so the random numbers occasionally give me two or more in a row.

Why The Myers-Briggs Personality Test Is Nonsense

I see M-B stuff fairly often, though not as often as I used to. I haven't been a fan, but I also lack the background to criticize it in any meaningful way. Fortunately, others have stepped up.

If you use social media or dating apps, you’ve seen Myers-Briggs Type Indicators in profiles. INTJ, ENFP, STFU, and so on.

I see what you did there. But I don't use social media or dating apps.

The 16 different personalities are shorthand for how you view the world. Essentially horoscopes for people who think they’re too smart for horoscopes, they’re often found alongside words like “Believer in Science” or “I’d let Nikola Tesla touch my spheres.”

Basically, this.

It won’t shock you to learn that they’re about as predictive as a quiz on the Denny’s children’s menu, but why? And how did we reach the point where they’re so popular?

Of course, the article goes on to answer its own questions. I'll let you take a look for yourself.

But Myers-Briggs still has plenty of issues, starting with the fact that basing a test on Jungian archetypes is like basing your surgery on bloodletting. Jung himself warned that his archetypes were only rough concepts and, like the MBTI, they’ve since been discredited almost entirely.

I will say this, though: as with archetypes, or horoscopes, the test might have no validity in the real world, but I feel like if people want to use it for writing? Go for it. It's like... I have no belief in the supernatural, and yet I enjoy reading/viewing supernatural fiction. I know it's fiction; I don't try to apply it to The Real World^TM.

And no one is just an extrovert or an introvert like a stock TV character; not only are there too many factors to declare such stark differences, but introversion and extroversion are largely constructs that lack context (you can, for example, be introverted at work around colleagues but extroverted at a party among your friends).

I've described myself as an introvert before, though I've never been diagnosed as such (as if it were some sort of disease, which obviously it isn't). What I mean by being an introvert is that I need my downtime away from other people. Generally, when I am around other people, I can be as open and outgoing as anyone. Some of you have met me, and have probably noticed this. I just have a limited amount of energy for such things. If I had to be around other people for too long -- say a day or so -- it would be utterly exhausting.

But I don't get my self-diagnosis from the MBTI; it's just that after a few years of trying, I got to understand myself better.

Plus, if people take the test twice within a few weeks they often get different results, which doesn’t really make it an ironclad concept to build your personality around.

Confession: I've taken it a few times. What can I say? I'm a curious sort (I think that's on the test too). My results change almost every time. Also, some of the responses can be easily gamed. If you want to come across as an extrovert, for example, you can give a positive response to "I love being around other people." Or something. It's been a long time, so I don't remember how the questions are actually worded.

People say, "Well, you're only supposed to take it once." But if a test purports to measure a person's basic personality, and that personality doesn't change over time, it should give consistent results. If someone's personality does change over time, why bother testing it? It becomes a mood ring instead of a tattoo.

Free idea: use mood ring ink in tattoos. You're welcome.

None of this were a problem if people weren't using MBTI to determine a person's actual fate. It would be like if you went to a job interview and everything was going swimmingly. You're really clicking with the interviewer, and you're nailing the hard parts, like "What do you see as your greatest weakness?" and "If you were a bear, what kind of tree would you scratch your ass on?" Then the interviewer asks, "So, what's your zodiac sign?"

"Pisces," you say with great confidence, as even non-astrologers have their sun signs memorized.

The interviewer gasps through her teeth. "Oh, sorry, we don't hire Pisceses."

And you're done. I mean, okay, such a scenario isn't as bad as racism, sexism, ageism, anti-Semitism, or any of those other isms, but you're still getting banged on out of there for something you can't control.

And yet, even getting passed over for Piscesism can't be so bad; after all, you wouldn't want to work for a company that's that committed to pseudoscience, would you? Such a company would be fishy. But the MBTI has the patina of science, unlike astrology. You've just been kicked out for a completely bullshit reason.

The article goes into this, too; I just wanted to put my own spin on it.

If someone can find guidance from Myers-Briggs without building their whole life around it, well, we all need guidance from somewhere. But there’s something bleak about carving humanity up into immutable categories based on fundamental misconceptions like we’re in a YA dystopia.

Sometimes I get guidance from a die roll or a coin toss. I don't pretend it's fate working in mysterious ways; it's just that sometimes, any decision is better than waffling. I don't think MBTI is any better or worse than a 20-sided die, except that my D&D character never had to try to dodge a fireball using the MBTI.

So anyway, I hope I (and Cracked} didn't burst anyone's bubble here, but basically, MBTI might as well stand for "More Bullshit Testing Ideas." Or something. I suck at acronyms; where's that on the test?

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