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Complex Numbers

by Robert Waltz

Rated: 18+ · Book · Personal · #1196512

Not for the faint of art.

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/

Signature for Honorable Mentions in 2018 Quill Awards

A signature for exclusive use of winners at the 2019 Quill Awards

Signature for those who have won a Quill Award at the 2020 Quill Awards

Show Recent Comments

10 Most Recent Blog Comments:

Aiva Raine

a day ago
In response to "Knead Some Dough?"

I grew up on mostly Wonder Bread despite our family having a large dry farm (dry farm being a farm that grows wheat and barley and safflower without irrigation- it relies solely upon mother nature to rain). I was never fond of it and looked forward to the times my mom would laboriously mill and grind the grain we brought back during harvest into flour and make home-made wheat bread. There was and still is nothing better than fresh whole wheat bread with a little butter and honey.

I think we did the Wonder Bread because my mom wasn't a fan of cooking and making bread is a time-consuming process.

I quit doing the white bread decades ago. There are so many kinds of healthy, no HFCS, etc whole grain breads out there that are delicious that I have no idea why any adult would choose the crappy Wonder Bread.

1 Like

Wordsmitty ✍️

May 13, 2024 at 1:18pm
In response to "The Breakfast and the Breakfurious"

And for those who rather not read the Smithsonian article, the History Channel episode that covers Pop Tarts (if you have a cable TV account):

https://www.history.com/shows/the-food-that-built-america/season-4/episode-1

Annette

May 10, 2024 at 12:05pm
In response to "Meditation Clause"

Everytime a stranger asks me to close my eyes, I keep my eyes open to see what they are up to.
Meditation is not for me.

But I think most people would have 45 minutes every day if they took that time away from social media or TV watching.

2 Like

Wordsmitty ✍️

May 9, 2024 at 1:15pm
In response to "Super Stew?"

But . . .

What's the difference between one being in the soup or in a stew? Is one thinner or thicker? How do writers use soup and stew?

1 Like

Elisa the Bunny Stik

May 9, 2024 at 10:11am
In response to "Super Stew?"

"Served in a pot? Maybe in your country."

Honestly, in the US I'd think soup would be served in a pot due to the quantity that ends up being consumed.

1 Like

Lazy Writer est 4/24/2008

May 8, 2024 at 3:02pm
In response to "Be Positive"

Mine always tell me to eat more fruits and veggies.

Annette

May 8, 2024 at 10:52am
In response to "Be Positive"

Asking doctors for advice has its own pitfalls. Because there is no money in recommending a sensible nutrition pattern, they will say things like: Look it up online.

2 Like

🌕 HuntersMoon

May 5, 2024 at 8:50am
In response to "Revisited: "Fail""

Shame on you for making fun of the Hoggs. Being a Virginian, Bob, you should know the reverence that the name Hog inspires recalling the glory days of the Washington Redskins (now the Commanders). The Hogs

1 Like

Charity Marie - <3

May 4, 2024 at 5:42pm
In response to "Overdraft Dodgers"

I have four different accounts and each one has a purpose. Savings, personal debit card, auto draft debit card, and a b8lls account. The bills account is the only one that can overdraft as it's for manual big bills like rent, cell, electric, etc. Stuff that mist be paid. The rest are blocked from overdraft and the savings account has no debit card attached. My bank accounts have been hacked too many times so I keep it this way to protect myself. It is very rare I ever pay an overdraft. I have a money market account for money greater than $10K so it gains interest and isn't touched except emergencies. That is a secure account too.

1 Like

Kåre Enga in Montana

May 3, 2024 at 4:23pm
In response to "It's a Crime"

Thailand versus Montana versus New York versus Costa Rica.

Theft? Not in Thailand! Not a huge concern in Missoula, perhaps in NYC, definitely an issue in Costa Rica.

Violence? Not a huge concern in Thailand. More so in MT, NY and CR (where it wasn't a concern years ago), definitely a problem in Saint Louis and the US in general. El Salvador is known for violence.

Death by vehicle. Not a huge concern in either NY or MT. A rising problem in Costa Rica. Thailand is one of the worst places in the world in this respect.

You may be shot and killed in parts of the US but you're more likely to be run-over in Thailand.

1 Like

April 15, 2023 at 10:36am

#1048245

The Less You Know...

I've been wondering what to say about this one.

Cognitive neuroscientist explains why stupidity is an existential threat to America

While the article emphasizes the US, it's been clear to me, thanks to the international reach of the internet, that this is a human problem, not just an American one. But as I live in the US, I'll run with what it says.

It may sound like an insensitive statement, but the cold hard truth is that there are a lot of stupid people in the world, and their stupidity presents a constant danger to others.

That depends on how you define "stupid," I suppose. I think they mean it in the sense of "stubbornly and willfully ignorant," because I fail to see how someone with a legitimate learning disability (Down's Syndrome, e.g.) presents a constant danger to others.

It would not be a stretch to say that at this point in time, stupidity presents an existential threat to America because, in some circles, it is being celebrated.

Which fits "stubbornly and willfully ignorant." Also any politician (of any party) who runs on a "common sense" platform.

Although the term "stupidity" may seem derogatory or insulting, it is actually a scientific concept that refers to a specific type of cognitive failure. It is important to realize that stupidity is not simply a lack of intelligence or knowledge, but rather a failure to use one's cognitive abilities effectively.

Okay, so a better definition than I had.

To demonstrate that stupidity does not mean having a low IQ, consider the case of Richard Branson, the billionaire CEO of Virgin Airlines, who is one of the world’s most successful businessmen.

Clearly, this article was written before Virgin Galactic fell into a black hole.

Still, business failure can't always be attributed to stupidity (as defined here).

Branson has said that he was seen as the dumbest person in school, and has admitted to having dyslexia, a learning disability that affects one’s ability to read and correctly interpret written language.

I've known several people with dyslexia, and this may be some sort of selection bias on my part, but as a whole, they seem to have an above-average intelligence. Mostly, they just have trouble with words and spelling. While I pride myself on my spelling, it's not proof of my intelligence.

Branson’s smarts come from his ability to recognize his own limitations, and to know when to defer to others on topics or tasks where he lacks sufficient knowledge or skill.

I hope his rocket science competition here in the US is taking notes.

Stupidity is a consequence of a failure to be aware of one’s own limitations, and this type of cognitive failure has a scientific name: the Dunning-Kruger effect.

Ironically, people throw around "Dunning-Kruger effect" like they know what they're talking about, even if they don't. Shit, I'm guilty of that, myself.

It is easy to think of examples in which failing to recognize one’s own ignorance can become dangerous. Take for example when people with no medical training try to provide medical advice. It doesn’t take much Internet searching to find some nutritionist from the “alternative medicine” world who is claiming that some herbal ingredient has the power to cure cancer. Some of these people are scam artists, but many of them truly believe that they have a superior understanding of health and physiology.

I'd reverse that. Many of these people are scam artists, but some of them truly believe they have a superior understanding.

There are many people who trust these self-proclaimed experts, and there is no doubt that some have paid their lives for it.

My Platonic ideal of the successful businessperson is not Richard Branson, but Steve Jobs. No, I'm not an Apple fanboi, but consider: he started a company with a couple of other geeks in his garage, and lived just long enough to see it become the most valuable company in the world (by market capitalization), exceeding far older corporations such as ExxonMobil or IBM. Sure, he was, by all accounts, a massive dick, but that's irrelevant to being a successful businessman.

And yet, he might have lived longer if he hadn't fallen for the alternative medicine scams.

What’s particularly disturbing about the Dunning-Kruger effect is that people are attracted to confident leaders, so politicians are incentivized to be overconfident in their beliefs and opinions, and to overstate their expertise.

"The best lack all conviction, while the worst / Are full of passionate intensity." -W.B. Yeats

It is only when we try to become an expert on some complex topic that we truly realize how complicated it is, and how much more there is to learn about it.

As a dedicated dilettante, interested in a broad range of topics yet not willing to delve too deeply into any of them, this is something I have to guard against, myself.

What we are dealing with here is an epidemic of stupidity that will only get worse as divisions continue to increase. This should motivate all of us to do what we can to ease the political division.

Sure, it should. But it won't.

We are all victims of the Dunning-Kruger effect to some degree. An inability to accurately assess our own competency and wisdom is something we see in both liberals and conservatives. While being more educated typically decreases our Dunning-Kruger tendencies, it does not eliminate them entirely. That takes constant cognitive effort in the form of self-awareness, continual curiosity, and a healthy amount of skepticism.

Clearly, I'm better at this than anyone else.

That's a joke. I'm deliberately overstating my competence to demonstrate the D-K effect.

The article is optimistic about being able to fix this. But there's only one thing more dangerous than willful stupidity, and that's optimism.

4 Like • Comment • View 4 comments

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