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Monday, May. 6, 2024

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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:

🌕 HuntersMoon

about 17 hours ago
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

a day ago
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.

🌕 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

April 18, 2021 at 12:01am

#1008685

Just the Facts

Today, just a gentle reminder that we don't always know what we think we know. From Cracked, so it's also amusing.

5 Everyday Science Facts (Nearly Everybody Gets Wrong)

Only it's not limited to science.

We all love science, sure. It's what makes things explode when we want them to, and keeps things from exploding the rest of the time.

Leave it to Cracked to succinctly explain the primary purpose of science.

5. "You Are What You Eat"

And if you eat a lot of a nutrient, you build a whole lot of it into your body. Right? That sounds reasonable enough. But that very last observation is kind of only true with fat, which is something your body likes to store away (and even fat storage is more complicated than that). With other stuff, you have to eat it, but if you eat extra, your body just kind of discards it.

This is demonstrably true, but the article does a pretty good explanation of why it's so.

Anyway, my mom always used that line on me: "You are what you eat." She was a lousy cook and I was a lousy kid, so one time, I said "You're saying I'm garbage?"

That was the last time I said that.

4. "Save The Rain Forest, It's The Earth's Lungs"

Most oxygen doesn't come from trees at all. It comes from algae in the ocean. And you don't get many trendy charities for looking after algae, because no one had fun climbing algae as a kid.

I gotta give 'em this one. Assuming it's as well-researched as the other points, it's something I never looked into, so I didn't know. In any case, there are plenty of other reasons to save the rainforest.

3. "Let's Keep Squeezing Solar Panels Everywhere!"

Having conceded that trees are great, though not in the way some people think, let's also reassure you that we think solar power is great. The tech works, it's advanced enormously recently, and we should build a bunch of new solar arrays. But many people seem to take this to mean that we need a solar panel, like, right there, right where they’re staring right now.

Minor quibble with this one: even deserts have biodiversity and what life exists there is dependent on sunlight. Build giant solar arrays out in Nevada and what happens? Shade on the ground. The ground that's harboring some really delicate life. So if we can put solar panels on things that are already blocking out the ground, like roofs and roads, I have no problem with it. But the article also goes into the economics of such.

2. We Have No Way Of Picturing Extreme Probabilities

This one's a pet annoyance of mine. Large and small numbers are incomprehensible. Hell, to most people, numbers are incomprehensible. It doesn't help that a lot of people don't see the world in terms of probabilities; only in black and white. Either I win a hand of poker, or I don't. Either I win the lottery, or I don't. If I get vaccinated, I could still catch the disease, so why bother? (The answer to that last one is because your chance of catching the disease drops by several orders of magnitude.)

1. None Of Us Can Picture Distances In Space

There's a Douglas Adams quote about how big space is, something about how it's considerably more than even the distance from here to CVS.

The actual quote is "“Space [...] is big. Really big. You just won't believe how vastly hugely mind-bogglingly big it is. I mean, you may think it's a long way down the road to the chemist, but that's just peanuts to space.” I first read it when I was fairly young, so I didn't know "chemist" was Brit for "pharmacist," and was confused. What's really confusing is that their word for "peanut" is the same as ours.

In any case, this one is kind of related to #2. Some things are just hard for our earthbound ape minds to comprehend.

4 Like • Comment • View 1 comment

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