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 21 hours ago
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🎁

a day ago
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

a day ago
In response to "Trash Pandas"

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

4 Like

Jayngle Bells

a day ago
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

tj-Merry Mischief Maker

Dec 20, 2024 at 6:06pm
In response to "To Err Is Human"

Right from the beginning I find myself asking, "How many times can anything go extinct?" On a side note, the other day while shopping I saw a neanderthal in the produce section. He was trying to fit in with modern clothes, but it was pretty obvious he was a neanderthal.

1 Like

Annette

Dec 20, 2024 at 1:00pm
In response to "To Err Is Human"

Comments work.

"Like" button sends me this message:

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2 Like

Annette

Dec 20, 2024 at 12:58pm
In response to "To Err Is Human"

Can I leave this comment?

Wordsmitty ✍️

Dec 19, 2024 at 9:03pm
In response to "Five Seconds"

Way, way back when I was little, it was a three second rule that I remember. Sometime since it was upped to five seconds because of all the chemical preservatives that are pumped into everything now.

1 Like

tj-Merry Mischief Maker

Dec 19, 2024 at 3:43pm
In response to "Five Seconds"

I also ate carrots right out of the garden growing up, I also picked and ate fruits and berries growing in the wild without washing. I do remember food dropped on the floor was either rinsed off and then eaten, or if it was something not easily rinsed, like buttered toast buttered side down, we gave it to the dog.

In our house, there's not a five second rule; unless it's something dogs can't eat, it's whichever one gets it first.

2 Like

October 15, 2024 at 7:42am

#1078334

Pushing the Button

This BBC article (whose headline resembles a Cracked headline) is a few years old now, but all that means is they might be missing a few recent ones.

The moments that could have accidentally ended humanity

In recent history, a few individuals have made decisions that could, in theory, have unleashed killer aliens or set Earth's atmosphere on fire. What can they tell us about attitudes to the existential risks we face today?

Already, I have a bad feeling about this. Sounds like an Agenda, which I don't expect from the BBC.

In the late 1960s, Nasa faced a decision that could have shaped the fate of our species. Following the Apollo 11 Moon landings, the three astronauts were waiting to be picked up inside their capsule floating in the Pacific Ocean – and they were hot and uncomfortable. Nasa officials decided to make things more pleasant for their three national heroes. The downside? There was a small possibility of unleashing deadly alien microbes on Earth.

No. No, there wasn't.

They thought there might be such a possibility, so I understand the caution, but in hindsight, there was never any danger from that.

A couple of decades beforehand, a group of scientists and military officials stood at a similar turning point. As they waited to watch the first atomic weapon test, they were aware of a potentially catastrophic outcome. There was a chance that their experiments might accidentally ignite the atmosphere and destroy all life on the planet.

Again: no, there wasn't. They thought there might be such a possibility, and they went on with the tests anyway, which says more about humans than I care to admit.

This sort of writing demonstrates some sort of fallacy that, well, I don't know what it's called, and it's hard to back-search these things.

The people in those cases were acting on imperfect knowledge. I mean, we all are, all the time, but not usually with imaginary stakes so high.

Let me maybe give you an example. You're walking through the woods one day, and you come across a pedestal with a single, big, red button on it. There's no writing, no pictograms, nothing to indicate what (if anything) pressing the button might do. From cartoons, you know that pressing the Big Red Button is generally a Bad Idea, but those are cartoons, not reality.

You're human, and you're faced with a mysterious button in a strange location to find a button. The urge to push it is real, palpable. How can you, curious ape, just... walk past it? You can't. Billions of years of evolution can't be overcome by the moral lessons in cartoons, not easily. Maybe it opens a secret cache with 10 million dollars in nonsequential, unmarked bills. Maybe a genie would come out to grant wishes, and you could wish for more than $10M. Okay, that's unlikely. Maybe nothing will happen. That seems most likely. Maybe a piano will drop onto your head when you press it, killing you. Or maybe it connects to the gigantic planet-exploding device hidden deep in the core. Wow, that's really unlikely.

What you're doing, whether you realize it or not, is risk assessment. Each possible event has a probability. Each possible event also has a desirability. Assigning probabilities is tough, and it's even tougher without Bayes' Theorem to help you assess the probability through the lens of prior knowledge. It's also very difficult if you have no way of knowing what every possible outcome is. But the point is, if you can do that, even roughly, and multiply the probability by the desirability, you can more easily rank your choices. Finding a stash of money? Low probability but desirable. Blowing up the planet? Really low probability, but undesirable.

In the end, billions of years of evolution win out, and you press the button.

Only through hindsight can you really know if you made the right choice or not.

Getting back to the examples from the article, yes, of course, I'd heard about them before. Take the astronauts thing. High probability: no deadly space microbes. Result: no disaster. Low probability: deadly microbes. Result: Astronauts die, Earth saved. Or, Result: Everyone dies. That last one would be Bad, so I don't blame them for hedging with the "deadly microbes" scenario.

It's kind of like the Trolley Problem, only your action shifts the trolley onto a random track where a random number of people are tied.

I don't know if I'm explaining all this very well, but my main point is, the article does a really bad job distinguishing "what we thought the probability was at the time" and "what the probability actually was, in hindsight."

The remainder of the article does a much better job explaining risk and our reaction to it, but the way it's described at the beginning stuck in my craw, so I could no more resist posting about it than you could resist pushing that Big Red Button.

3 Like • Comment • View 1 comment

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