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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:

Kåre Enga in Montana

about 21 hours ago
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

Simply Me

a day ago
In response to "It's a Crime"

I get it. The fear spreading jerks wants us to have an "us vs. them" attitude. The division gives them leverage. (they think)

1 Like

🌕 HuntersMoon

May 2, 2024 at 11:37am
In response to "Butt Of Course"

My wife tried waterskiing once... called it the 100-yard douche.

Talk about retaining water...

2 Like

Elisa the Bunny Stik

May 1, 2024 at 10:53am
In response to "Mad as Hell"

"But if you want to really see how anger is useful, you need to consider how it's used as a tool for manipulation."

This is a line that must be regarded with care. After all, there are people that get angry about the way they are being treated on a large scale and harness that anger to change their circumstances. There is some level of manipulation in this angry but still legitimate flavor of activism. We do have to consider the possibility that this angel's usefulness can have net benefits (such as environmental cleanup and protection; see the Live Canal as an example).

GeminiGem survived GoT

Apr 29, 2024 at 1:53pm
In response to "Supplementary"

Working in health care for a zillion years, I've heard it all. The whole thing about something being natural, therefore it would be good for a person is very, very, very...very prevalent. An internist I worked with had a fantastic comeback to that line of thinking. She would say, "Arsenic is natural, are you going to start taking that?"

1 Like

Lazy Writer est 4/24/2008

Apr 28, 2024 at 2:14pm
In response to "Revisited: "Exercise""

I will be 75 in July. My stepdaughter gave me a bike because I wanted to ride it again. Well, I started off shaky and stayed shaky...then I fell off it...A passing motorist stopped to help me get up...I have NOT been back on that bike and will not ever again.

🌕 HuntersMoon

Apr 28, 2024 at 10:43am
In response to "Revisited: "Exercise""

So, you're saying that now you pedal your ass differently?

Elisa the Bunny Stik

Apr 27, 2024 at 10:33am
In response to "Pleasure and Pain"

it illustrates why it's important to have people from different backgrounds, with different worldviews, working on scientific research.

This right here is why I get rather irritated by people poo-pooing college these days. I feel like that attitude effectively shits on minority populations that may not have strong finances but do possess persistent curiosity and want to learn about various subjects in-depth. In the 90s, these groups were exclusively told to go to trade schools even if they had the aptitude for college education. Nowadays the blanket "Everyone should consider a trade" mantra employs a more subtle (but still abundant) form of discrimination. I'm so over it.

1 Like

Kåre Enga in Montana

Apr 25, 2024 at 3:16am
In response to "Going to the Dogs"

I have ants in Thailand. Will I miss them in Montana? No.

Would it be nice to have a cat and a dog, here or there? Yes, but I have struggles keeping my neglected plants alive.

Not every situation is suitable for everything or everybody.

Kåre Enga in Montana

Apr 25, 2024 at 3:09am
In response to "Tipped Out"

I don't tip. In my travels I rarely find tipping the norm.

I also don't haggle.

When you add the tip... the true cost of America becomes evident. But what would I know? I travel while others who have double my income can't pay rent.

August 3, 2021 at 12:02am

#1014973

Tales from the Cryptid

Confession: I kinda like cryptids.

Why We Can’t Rule Out Bigfoot
How the null hypothesis keeps the hairy hominid alive.

It's not that I "believe in" Bigfoot (or Nessie, or the Jersey Devil, or chupacabra, or whatever). It's that I'm fascinated by what the stories say about us and about the way we approach knowledge.

This article is almost exactly seven years old now, but it's probably not like the science has advanced...

The international collaboration of scientists, led by University of Oxford geneticist Bryan Sykes, found no evidence that the DNA from the hairs belonged to a mysterious primate. Instead, for the most part, it belonged to decidedly unmysterious mammals such as porcupines, raccoons, and cows.

All animals (well, all things, really) are mysterious until they're not, and then a lot of them keep on being mysterious even as we study them. For instance, the opossum. If I didn't have a few climbing up on my deck to say hi every once in a while, I might not believe they existed. Hell, the first report of the Virginia opossum comes from John Smith

(yes, that John Smith, not one of his many namesakes). From that link: Strachey's notes describe the opossum as a "beast in bigness of a pig and in taste alike," while Smith recorded it "hath an head like a swine ... tail like a rat ... of the bigness of a cat." Okay, well, I think Strachey was hitting the hooch there; Smith's notes are a bit closer to the opossums I've seen.

Anyway, Bigfoot.

People often think that the job of scientists is to prove a hypothesis is true—the existence of electrons, for example, or the ability of a drug to cure cancer. But very often, scientists do the reverse: They set out to disprove a hypothesis.

Which is what I've been saying.

Bigfoot advocates have repeatedly claimed that professional scientists are willfully ignoring compelling evidence. The problem, in fact, is that the advocates haven’t been approaching the question of Bigfoot in a scientific fashion. So two years ago Sykes and his colleagues decided to run a scientific study of those hairs from an “anomalous primate.” And that involved creating a null hypothesis to try to reject.

Unlike some people, I don't have a problem with science being science even when the hypothesis is, at first glance, absurd, as in the case of most cryptids. At the very least, negative results provide skeptics with more argumentative ammunition. I'm not what you'd call a hard skeptic: show me compelling evidence and I'll change my tune. This applies to cryptids as well as psychic abilities, faeries, space aliens or whatever.

Thing is, everyone's level of "compelling" may be different.

The results were clear: The scientists found precise matches for all 30 samples in previously known mammals.

Does this mean Sykes and his colleagues have proved that Bigfoot does not exist? No... The question remains open, and—if Bigfoot doesn’t exist—always will.

Which is a long way to go about showing once again that it's very, very difficult, if not impossible, to prove a negative. Russel's Teapot

comes to mind.

That’s not to say Sykes’ study didn’t offer its own surprises. Two hair samples from the Himalayas matched a DNA sequence that was extracted from a 40,000-year-old fossil of a polar bear. Stranger still, their DNA was not a match to living polar bears.

And this is why I like to keep an open mind about these things. Even if we never find Bigfoot (and given that the classic Bigfoot sightings were revealed to be guys in ape suits) (well, hairy ape suits; we already wear ape suits), who knows what strange and cool things we might discover along the way?

Besides, it makes excellent story fodder.

Just to be complete, I'll save you a search: here's the Wiki article on Bigfoot.

6 Like • Comment • View 1 comment

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