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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 22 hours ago
In response to "Butt Of Course"

My wife tried waterskiing once... called it the 100-yard douche.

Talk about retaining water...

1 Like

Elisa the Bunny Stik

a day ago
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

Kåre Enga in Montana

Apr 25, 2024 at 3:16am
In response to "Going to the Dogs"

I have ants in Thailand. Will I miss them in Montana? No.

Would it be nice to have a cat and a dog, here or there? Yes, but I have struggles keeping my neglected plants alive.

Not every situation is suitable for everything or everybody.

Kåre Enga in Montana

Apr 25, 2024 at 3:09am
In response to "Tipped Out"

I don't tip. In my travels I rarely find tipping the norm.

I also don't haggle.

When you add the tip... the true cost of America becomes evident. But what would I know? I travel while others who have double my income can't pay rent.

Kåre Enga in Montana

Apr 25, 2024 at 2:58am
In response to "Reality Cubed"

My dreams may be different from your dreams but if we share them with others then some form of the dream may live on. Regardless, if we don't eat we die. "Joy gives us wings." "Reality bites."

Aiva Raine

Apr 24, 2024 at 4:26pm
In response to "Tipped Out"

Agreed. I'll tip and well, for actual waitstaff at restaurants. I'll tip appropriately for bell-staff who assists with luggage at hotels, the more luggage, the bigger the tip. I'll give a tip to the valet as well. Beyond that- I don't tip. Certainly no tip for the fast food workers, coffee baristas, grocery store clerks, etc.

It annoys me to no end that we're being asked to tip for all kinds of jobs where they are being paid an hourly wage. I've worked as a waitress when I was young and my daughter currently does. In those jobs- she absolutely depends upon proper tipping to make a decent wage. And I hate the fact that she is dependent upon tips to do so.

It leaves women especially at the mercy of arsh**() on whether she wants to accept hostile, negative, sexist, harassing, demeaning, and negative behavior from her customers in order to make a living. It condones people treating their waitstaff like shite in order to get paid. While there is much improvement from the 30 years ago when I was working as a waitress in the skimpy outfits we were forced to wear- it's still an issue at non-chain, independent owned restaurants and diners.

Women are left to the mercies of jerk customers and their only recourse is the manager who they have to hope supports their waitstaff rather than the jerk customer. Tipping should be eliminated. All employees should be paid a livable wage and waitstaff especially should not be dependent upon tolerating crappy to obscene behavior from customers in order to make a living.

But, until that happens- I always tip at least 20% and often times more. Very rarely have I tipped lower and the service in those situations was abysmal and they still got 10%.

1 Like

December 20, 2018 at 1:14am

#947845

December

Another turn of the wheel.

https://www.vox.com/science-and-health/2018/12/18/18144477/winter-solstice-2018

As I wrote last month, this is one of the things that, were we so inclined, we could all agree on. It affects the entire human race (at least for now; I'm still hoping for space travel), even if its effects depend on your current latitude.

I should add a couple of observations to the article I linked above.

First, while the vast majority of humans live in the Northern Hemisphere, that's no excuse for hemispherism. Most of us see it as the Winter Solstice but, as noted, it's the Summer Solstice in the south. To avoid being exclusionary, I've seen it referred to as the December Solstice. A better name - one that's not tied to our arbitrary calendar - might be the Capricorn Solstice. However, that would make the other solstice the Cancer Solstice, and I can see people objecting to that.

Second, unless I missed it (I tend to skip boring parts, which is one reason I don't like internet videos), it doesn't define what "solstice" really means. Yes, it says "Technically, the solstice occurs when the sun is directly over the Tropic of Capricorn, or 23.5° south latitude." But the word is from Latin: sol (sun) + stice (from a root that means 'stationary'). So literally it means like, "the time the sun stands still." Which is pretty cool. Of course, the sun doesn't actually stand still - figuratively or literally - it's a bit more complicated than that. From the point of view of any observer on Earth, each day when the sun reaches its highest point in the sky (known as local noon, which is usually not when the clock says noon), it appears slightly more to the south between late June and late December. As the calendar approaches the December solstice, the sun's apparent location at local noon drifts a bit further south each day. On the solstice itself, it's at its southernmost point. Technically, the solstice occurs at local noon at whatever longitude the sun happens to be crossing when it's at zenith at the Tropic of Capricorn.

Yes, I know. If I'd been reading this, I'd have skipped the last paragraph.

Third, did you see the gif in the linked article with the time-lapse of Earth from space? It's pretty cool. With a little digging, I found that the satellite that shot these pictures is a weather satellite in geostationary orbit. When I saw the gif, my first thought was "that satellite is directly over 0/0." 0/0 being the point where the Prime Meridian crosses the equator. It's in the Atlantic, off the coast of Nigeria. I'm not so jaded that I find photos taken from geostationary satellites to be a mundane thing. Really, I'm not sure whether it's over the actual origin or not, but it's pretty close.

Anyway, science in general is cool, astronomy in particular is fascinating, and culturally, we have a long history of marking the solstices and equinoxes. As I've mentioned before, I don't observe Christmas, so the day I care about is the solstice. It's a time to consider rebirth and renewal; the moment when, just as everything seems to be about to plunge into eternal darkness, there's the promise of more light to come. It's more interesting to me than the very arbitrary New Year's Day (the proximity of which to the solstice isn't accidental), and more meaningful.

In the past, I've marked it for myself by staying awake through the longest night, maintaining a vigil for the returning, invincible sun. I think I might revive that tradition for myself on Friday.

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