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Sunday, May. 26, 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:

Kåre Enga in Montana

a day ago
In response to "Plan Solo"

I'm more introverted than extroverted, but my only apprehension involves language and cultural barriers.

I'm bi- or ambi- but... always annoying. I waver between wanting to be left alone and not wanting to be alone.

I make my adjustments as necessary.

But LANGUAGE and CULTURE: true barriers that can cause problems for some and angst for me. I love strangers but find it frustrating when I can't connect.

For Belgium: make those plans soon. Today's connectivity means booking lodging in advance. In the old days it was show-up-and-ask-if-they-have-a-room. Still possible but not recommended unless cost is not an issue.

I prefer staying in hostels but rarely join in 'events'. I have gone out with fellow travelers however.

Kåre Enga in Montana

a day ago
In response to "Banned Camp"

I remember when the movie "Brokeback Mountain" received "F"s from folks who basically commented that they would never NEVER watch such filth. Even those who liked or enjoyed it missed the main theme of poverty (financial, cultural, individual...). Personally, I preferred the short story to the movie. Both made a point that America didn't/doesn't want to face: choice and lack of choice.

That said, if someone were to ban my poetry I could probably live off of the notoriety.

Burning at the stake though seems a more likely outcome.

2 Like

🌕 HuntersMoon

a day ago
In response to "Banned Camp"

"Any book worth banning is a book worth reading." Isaac Asimov

2 Like

Aiva Raine

May 21, 2024 at 11:34pm
In response to "More Science"

Giggles. Thank you for the laugh- especially this phrase, "I always picture Doc Brown's workshop in Back to the Future. Yes, of course I know it's wrong, but I'd rather be amused than right."

1 Like

Aiva Raine

May 17, 2024 at 6:57pm
In response to "Knead Some Dough?"

I grew up on mostly Wonder Bread despite our family having a large dry farm (dry farm being a farm that grows wheat and barley and safflower without irrigation- it relies solely upon mother nature to rain). I was never fond of it and looked forward to the times my mom would laboriously mill and grind the grain we brought back during harvest into flour and make home-made wheat bread. There was and still is nothing better than fresh whole wheat bread with a little butter and honey.

I think we did the Wonder Bread because my mom wasn't a fan of cooking and making bread is a time-consuming process.

I quit doing the white bread decades ago. There are so many kinds of healthy, no HFCS, etc whole grain breads out there that are delicious that I have no idea why any adult would choose the crappy Wonder Bread.

1 Like

Wordsmitty ✍️

May 13, 2024 at 1:18pm
In response to "The Breakfast and the Breakfurious"

And for those who rather not read the Smithsonian article, the History Channel episode that covers Pop Tarts (if you have a cable TV account):

https://www.history.com/shows/the-food-that-built-america/season-4/episode-1

Annette

May 10, 2024 at 12:05pm
In response to "Meditation Clause"

Everytime a stranger asks me to close my eyes, I keep my eyes open to see what they are up to.
Meditation is not for me.

But I think most people would have 45 minutes every day if they took that time away from social media or TV watching.

2 Like

Wordsmitty ✍️

May 9, 2024 at 1:15pm
In response to "Super Stew?"

But . . .

What's the difference between one being in the soup or in a stew? Is one thinner or thicker? How do writers use soup and stew?

1 Like

Elisa the Bunny Stik

May 9, 2024 at 10:11am
In response to "Super Stew?"

"Served in a pot? Maybe in your country."

Honestly, in the US I'd think soup would be served in a pot due to the quantity that ends up being consumed.

1 Like

Lazy Writer est 4/24/2008

May 8, 2024 at 3:02pm
In response to "Be Positive"

Mine always tell me to eat more fruits and veggies.

April 20, 2023 at 9:24am

#1048496

Heavy Size

More proof that we're utterly obsessed with size:

What is a black hole’s actual size?
What do we mean by a black hole's size? A photon sphere? The minimal stable orbit? The event horizon? The singularity? Which one is right?

Out there in the Universe, size definitely matters.

Always lead with a subtle dick joke.

Objects that are stable, both microscopically and macroscopically, are described by measurable properties such as mass, volume, electric charge, and spin/angular momentum.

And, lately, political affiliation.

But “size” is a bit of a tricky one, particularly if your object is extremely small.

Snort.

After all, if all the mass and energy that goes into making a black hole inevitably collapses to a central singularity, then what does the concept of “size” even mean?

Obvious jokes aside, I've long wondered how you can measure the size of something that warps spacetime, which makes the concept of "size" very slippery (as if covered with K-Y)

The first thing you have to know about a black hole is this: in terms of its gravitational effects, especially at large distances away from it, a black hole is no different from any other mass.

I've mentioned something like this before, I know, but can't be arsed to find it right now. Black holes have some legitimately scary properties, but if they were going to eat entire galaxies, they'd have done it by now.

After all, we’re taught that black holes have an irresistible gravitational pull, and that they suck any matter that comes too close to their vicinity irrevocably into them. But the truth is that black holes don’t “suck” matter in anymore than any other mass.

I'm not well-versed in the math involved, but there's a region around a black hole within which orbits decay in a predictable way. Outside that region, black holes act like any other mass.

This is complicated by the prevalence of gas, dust, and debris in the vicinity, all of which would cause any orbit to eventually decay.

When a black hole rotates, it no longer has just one meaningful surface that’s a boundary between what can escape and what can’t; instead, there are a number of important boundaries that arise, and many of them can make a claim to being the size of a black hole, depending on what you’re trying to do. From the outside in, let’s go through them.

The rest of the article does just that; it's very interesting, and only a little bit math-oriented, but there's no need for me to comment on it. After all, I just wanted to make the point (pun intended) that even science writers are not above making the occasional dick joke.

8 Like • Comment • View 1 comment

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