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:

tj-Merry Mischief Maker

about 14 hours ago
In response to "Gray Matters"

I wonder how many nine year old people have gray hair? I also wonder how much stress most nine year olds have?

3 Like

Wordsmitty ✍️

about 17 hours ago
In response to "Gray Matters"

"who were between age nine and 39," - so the old folks need the rejuvenation formula first to turn back the years.

I wonder if anyone has studied why hair goes gray in certain places and not everywhere. Maybe different kinds of stress?

3 Like

🌖 HuntersMoon

about 20 hours ago
In response to "Gray Matters"

I've always believed "better gray than gone away..."

2 Like

tj-Merry Mischief Maker

a day ago
In response to "A Mammoth Discovery"

This makes think of a book, "Evolution and Genomic Diversity" by Gene Poole.

1 Like

tj-Merry Mischief Maker

Dec 24, 2024 at 5:16pm
In response to "Hot Shoulder"

Maybe if we say, "Betelgeuse" three times in a row we can witness it.

1 Like

tj-Merry Mischief Maker

Dec 23, 2024 at 4:55pm
In response to "Trash Pandas"

Being an avid camper, keeping items out of the reach of raccoons can be challenging. We've had them figure out how to open coolers, open chests, and even gnaw through rope used to suspend a cooler out of their reach.

2 Like

🐕GeminiGem🎁

Dec 23, 2024 at 12:56pm
In response to "Trash Pandas"

I wrote one of my Danger Dog stories based on a childhood camping trip that involved a raccoon stealing grapes from our cooler "Danger Dog's Camping Capers"

I had always heard that cockroaches will be the only thing that survives a nuclear war/apocolypse, but maybe it will actually be the raccoons.

4 Like

🌖 HuntersMoon

Dec 23, 2024 at 10:04am
In response to "Trash Pandas"

Just wait until we team up with the cats...

4 Like

Jayngle Bells

Dec 23, 2024 at 10:03am
In response to "Trash Pandas"

The back of my property is sheltered by buildings and cedar trees, so I live on a "racoon highway." Tinfoil works until one takes it for the team and moves it. The only scent that somewhat deters then is ammonia - but they alter their path within the already preferred walkway, not avoid it completely. They generally leave the garbage bin alone, but that's because of the bears, not because of anything we do 😂

I've had more wildlife problems in the city than I ever did in the country. The first time I saw a deer up close? In the middle of my street directly in front of my house. Probably looking for the trash panda party. Lol

4 Like

Annette

Dec 21, 2024 at 11:59am
In response to "Neutral Zone"

It made me happy that I was able to like it. I take the small victories.

1 Like

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.

3 Like • Comment

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