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Thursday, May. 9, 2024

5:23am EDT

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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:

Lazy Writer est 4/24/2008

about 14 hours ago
In response to "Be Positive"

Mine always tell me to eat more fruits and veggies.

Annette

about 19 hours ago
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

Simply Me

May 3, 2024 at 11:58am
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.

February 20, 2024 at 10:48am

#1064499

Dahlia Dreaming

And now here's another one for "Journalistic Intentions"

[18+]...

Dahlia Dreaming

A good number of my "why the hell is it called that?" questions popped up before there was an internet. As a kid on a farm, I got to know lots of different cultivated plants—whether fruit, vegetable, or flower—whilst perusing a seed catalog. A paper one. That came through the postal mail.

This led to a lot of "why the hell is it called that?" moments that my parents, who were what passed for Wikipedia for me in those days (they'd eventually buy me an actual encyclopedia volume set, which I actually read and then promptly forgot most of), had no answers for. "Go look it up." Where? We live on a farm.

But I do remember that one of these moments was for the flower known as a dahlia. It seemed an even odder moniker than most plants' names, but the seed catalog didn't have much to say about it. It probably listed the botanical binomial, but I don't remember that. It's dahlia pinnata, according to Wikipedia, and while it's native to Mexico and Central America, it's not to be confused with a piñata. But as with many other cultivated plants, there are several subspecies. It gets confusing and beyond the point of this entry.

Nor is Wikipedia

much help with the etymology of the name. It seems it might have been named after a botanist named Anders Dahl. Which seems bogus to me, a typical European appropriation of an American species. The least they could have done is mangle one of the native names for the thing, like they did with, say, the raccoon. Except for the French, who call them washing rats, which is unfair to rats, who are often quite fastidious.

Also, I can't be arsed to find out if Anders Dahl was ancestral to the far more famous Roald Dahl.

None of which was what I set out to write about; I just did my usual assuming that if I don't know something, then no one else does, either. I have my parents to blame for that, too.

No, what I wanted to note, apart from the excellent use of depth-of-field in the photograph the title links to, is the petal pattern.

Those aren't true petals, incidentally. Each one of those petal-like pieces is a flower unto itself. But that's a bit like arguing whether a tomato is a fruit or a vegetable. No, what struck me is that the arrangement of the florets is similar to other petal and/or leaf arrangements found in nature, such as in sunflowers or artichokes: an instantiation of the Fibonacci sequence and the Golden Ratio.

Once you notice that particular spiral arrangement, you can't ever miss it. You can even find it all over the Mandelbrot set (which involves complex numbers), if you know what you're looking for. There are solid reasons for plants taking that general form (some animals, such as mollusks, do it, too), and none of them is that plants can do math. No, it's a bit complicated, but, basically, it's because it's easy and efficient.

I can appreciate that.

5 Like • Comment • View 2 comments

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