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Saturday, Dec. 21, 2024

6:37am EST

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Complex Numbers

by Waltz Invictus

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:

tj-Merry Mischief Maker

about 13 hours ago
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

about 18 hours ago
In response to "To Err Is Human"

Comments work.

"Like" button sends me this message:

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2 Like

Annette

about 18 hours ago
In response to "To Err Is Human"

Can I leave this comment?

Wordsmitty ✍️

a day ago
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

a day ago
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

🌖 HuntersMoon

a day ago
In response to "Five Seconds"

2 Like

Aiva Raine

Dec 17, 2024 at 7:39pm
In response to "Need for Speed"

Even with the actual explanation- it still left me going- sooooo what's the difference between the special expensive "speed tape" they use for "temporary fixes" and duct tape? LOL- sure sounds like duct tape to me and they are using it for exactly the same reasons you would use duct tape. I'm guessing that the only difference between it and duct tape is the "speed tape" is slightly stronger and handles temperature differences better. It still comes down holding the plane together with tape which doesn't exactly fill me with confidence. Especially if it's a Boeing plane.

1 Like

tj-Merry Mischief Maker

Dec 17, 2024 at 6:14pm
In response to "Need for Speed"

I suppose they call it speed tape to keep people calm. If they called it what it is, temporary repair tape, they may lose passengers.

2 Like

Waltz Invictus

Dec 17, 2024 at 9:43am
In response to "Need for Speed"

Ichabod Crane-- Mince Pie

, I've heard that speed tape is extremely expensive. I can only assume the maintenance guys at Delta got a five-finger discount. But if you're going to steal from work, might as well go big and not just lift a box of paper clips.

1 Like

Ichabod Crane-- Mince Pie

Dec 17, 2024 at 9:06am
In response to "Need for Speed"

One of the last jobs I had was with Delta. I did ask the maintanace guys about that tape once and they pointed out high speed will rip duct tape or any other tape right off. They used it personally for their boats sometimes because of its strength. But looking out a window at 400 miles an hour and 30 thousand feet up and see duct tape on the wing. Certainly made me nervous first time I saw it.

3 Like

August 23, 2024 at 10:40am

#1075620

Unsolved, Unresolved

Today's link isn't a podcast. I don't do podcasts. Shouldn't they call them something else now that the iPod is no more? Anyway, from Vox, a collection of stuff from a bunch of podcasts:

17 astounding scientific mysteries that researchers can’t yet solve
What is the universe made out of? How should we define death? Where did dogs come from? And more!

The key word in the headline there is "yet."

“Whatever we know is provisional,” Priya Natarajan, a Yale physicist, told us about research on dark matter. But the sentiment also applies to science overall. “It is apt to change. What motivates people like me to continue doing science is the fact that it keeps opening up more and more questions. Nothing is ultimately resolved.”

This is, of course, a good thing. But "nothing is ultimately resolved" can be misleading.

Unexplainable isn’t about how scientists don’t know anything.

There's a tendency in humans to indulge in binary thinking. That is, believing that since science can't tell us everything, then it can't tell us anything. This is, of course, arrant nonsense.

We’re drawn to questions because they are optimistic. They invite us to dream of a better world in which they are answered, where the gaps between questions and our capabilities to answer them are smaller.

Ugh. Optimistic? Maybe I should stop asking questions. Besides, finding out that the universe will definitely end is hardly optimistic. I mean, what's the point of doing anything if it's just going to wind down and stop one day? I mean, sure, trillions of years from now, but that's not so long, is it?

Here are some of the questions that astounded us the most.

I won't list all of them. That would take too long. Just ones I have comments on or can make jokes about.

1) What is the universe made out of?

String.

String theory: the Universe is a big ball of string, and God is a cat.

(Okay, this section is actually about dark matter, unsurprisingly, but my string theory pun is funnier.)

2) How did life start on Earth?

Lots of people think it started elsewhere and migrated to Earth. I have another article about that in the queue. Which is fine and may be a testable hypothesis, but, either way, it doesn't get at the real question, which is how did life start, query, full stop.

3) How did dogs evolve from wolves?

That's not the best-phrased question. It's not like wolves haven't been evolving for the past 20,000 years or whatever. Though that's hardly long enough for a lot of natural selection to take place. Artificial selection, sure, which is why we have both chihuahuas and Great Danes and call them both dogs. It's like the idea that we evolved from chimps: no, we and chimps (and bonobos) share a common ancestor.

5) What will animals look like in the future?

I'm guessing the answer is "animals."

10) Is there anything alive in the human poop left on the moon?

You know, I get that this question is academically important. It is also unique on this list because while we don't know the answer, we know precisely how to find the answer, and all it'll take will be another trip to the moon, this time carrying pooper-scoopers.

11) Was there an advanced civilization on Earth before humans?

I know it's kinda vogue-y to ask that question right now, but the simple truth is, if there were, we'd know. We might not find their artifacts or other archaeological remnants, but any "advanced" civilization worth that adjective is going to need energy and produce waste. That waste would show up all over the globe, just like our waste is doing now.

Besides, if there had been, all the oil would have been gone before we started digging for it.

No. There wasn't.

12) What is the definition of “life”?

Yeah, that's not a science question so much as a philosophy question. As I've noted before, though, science can inform philosophy.

14) What did dinosaurs sound like?

I hope they figure this one out, and it turns out they meowed.

17) How will everything end?

With some questions as yet unresolved, I'm sure.

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