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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Adherennium Plotting something

about 2 hours ago
In response to "Global Domination"

For whatever warped reason, 'Schrödinger's Hello Kitty' has tickled my funny bone.

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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.

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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.

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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
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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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May 15, 2024 at 9:55am

#1071131

Absolutely

"Only a Sith deals in absolutes." —Obi-Wan Kenobi

The danger of absolute thinking is absolutely clear

The problem with that quote from one of the Star Wars prequels is that it is, itself, an example of absolute thinking. This has been pointed out by better minds than mine, but it seems relevant to today's article (from Aeon). Of course, the Jedi weren't as pure and noble as their PR tried to paint them to be; controlling others' minds is generally considered "evil," regardless of purpose.

But if there's anything Star Wars is known for, it's lightsabers. If there's anything else it's known for, it's bad dialogue.

Anyway, to the point:

Think of the most happy and well-adjusted person you know – what can you say about their thinking style?

Well, that would be me. (If that thought doesn't scare you, nothing will.)

Are they dogmatic, with an all-or-nothing outlook on the world? Do they place totally rigid demands on themselves and those around them? When confronted with stresses and misfortunes, are they apt to magnify and fixate on them? In short, do they have an absolutist thinking style?

I absolutely do not. Well, with some exceptions.

Absolutist thoughts are unqualified by nuance and overlook the complexity of a given subject.

I've railed on this before. One solution would be to drink more Absolut.

There are generally two forms of absolutism; ‘dichotomous thinking’ and ‘categorical imperatives’.

These $2 words are explained in the article, and I'll quote them, too:

Dichotomous thinking – also referred to as ‘black-and-white’ or ‘all-or-nothing’ thinking – describes a binary outlook, where things in life are either ‘this’ or ‘that’, and nothing in between.

Some rare things are binary, or effectively so. Flip a fair coin, for example, and the chance of it landing on edge is minuscule; the practical effective outcome is either heads or tails. But, agreed, most things aren't. Like "alive" and "dead." At first that seems binary, but there are nuances to life; if someone's in a coma, e.g.

Categorical imperatives are completely rigid demands that people place on themselves and others. The term is borrowed from Immanuel Kant’s deontological moral philosophy, which is grounded in an obligation- and rules-based ethical code.

Hey, great job explaining "dichotomous" and "categorical;" but you had to go and throw in "deontological,"

didn't you?

Yet we all, to varying extents, are disposed to it – why is this?

"We all?" Wow, that's some absolutist shit right there.

Primarily, because it’s much easier than dealing with the true complexities of life.

I'm all for "easy," but not if it's going to make things more complicated in the long run.

The article (which is from 2018, so things might have changed in the field since then) goes on to describe the author's work on the subject which, like all psychological science, should be read skeptically. I'm not saying it's wrong, but I'm not saying it's right, either. See? I'm not an absolutist.

These findings support the recent ‘third wave’ therapies that have entered clinical psychology. The most well-known of these is ‘mindfulness’, but they all advocate a flexible outlook, acceptance, and freedom from attachments.

Ugh. "Mindfulness" again. I'm in favor of flexible outlooks and whatnot, but I have yet to accept the usefulness of "mindfulness," and I might never do so. Then again, I might. Thus once again showing that I have some flexibility.

Many argue that the world is a harsh place, and that it is the stresses and misfortunes in life that make people depressed, not their thinking style. Wrong!

Once again, the irony of proclaiming "WRONG!" in an article like this doesn't escape me.

Countless people suffer misfortunes and do not get depressed or anxious, while others seemingly suffer no misfortune at all, and are blighted with depression and anxiety.

Or it could be, though I don't have studies and shit to pull out, that depression and anxiety are health issues, caused not by absolutist thinking but by chemical processes in the brain. This is like saying that lots of people slip and fall on ice, but not all of them break their legs doing so.

So, yeah, I'm not taking the words here to be absolute truth, but I found it to be an interesting enough take on things to share it.

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