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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:

Robert Waltz

about 16 hours ago
In response to "Ignorance is the Opposite of Bliss"

Kåre เลียม Enga

- While competition is often touted as the driving force of progress or evolution, it's actually cooperation that's most important. Even opposing sportsball teams, while seeming to be in competition, have agreed on a set of rules, the enforcers of these rules (referees or umpires or whatever), and the penalties for breaking the rules.

The Space Race is another example. On the surface, it was a dick-measuring contest between US and USSR. Each country relied on internal cooperation, obviously, but they also cooperated with each other—if only through things like weapons treaties and promises not to go to war over it.

Competition is a factor, of course. But survival depends on cooperation.

1 Like

Kåre เลียม Enga

about 19 hours ago
In response to "Ignorance is the Opposite of Bliss"

Denver Broncos beat the Cleveland Browns... just saying. No, I didn't know. I looked it up! And although I know a thing or two I rely on the experts behind google to find out for me.

Is this blog entry pertinent? As Our Dear Leader fires experts left and right (but mostly left, because he's always right), we're left with loyalists who may or may not be experts or have any knowledge of the field at all.

You delineate the weakness of competitiveness. One ring to rule them all negates the contribution of millions. In the case of reaching the Moon or Mars, the crew cannot be competitive and even the technicians have to be a competent expert in what may be a detail (like proper heat shielding for any shuttle). China, USA and India can compete on getting to Mars; but, it will be a cooperative crew of experts that gets there.

I loved reading my poetry to a group of writers who could pick it apart. I miss that. Since I am the boss, I don't have to change one comma and I can ignore their suggestions... but I'm better off doing that than submitting to an AI program that reduces everything to General American for an audience of 12 year old Vacation Bible School students (a.k.a. the education system of Southern Baptist Confederacy of Oklahoma).

2 Like

tj wanderlust-words-in-motion

a day ago
In response to "Steaming Ship"

Thanks for clearing that up. On a side note, if I have a 16 foot fishing boat and stick my kayak in it, is it now a ship?

Speaking of space "ships", I've often wondered why the termed the craft used to shuttle people/beings between the planet and ship, or even between ships, shuttle-crafts.

In sticking with terminology, if the larger vessel is a spaceship and carries smaller crafts, shouldn't it be a spaceboat?

1 Like

tj wanderlust-words-in-motion

Feb 13, 2025 at 3:21pm
In response to "Canyons on the Moon"

Or, maybe, they were just scratched out by Schrödinger’s cat

tj wanderlust-words-in-motion

Feb 12, 2025 at 3:44pm
In response to "Day of the Dollar"

I don't care what "special day" it is most of the time, but some are amusing and all can be celebrated with beer (or beverage of choice), so here's to Plum Pudding Day

1 Like

tj wanderlust-words-in-motion

Feb 11, 2025 at 12:35pm
In response to "Paint in the Neck"

I just hope they aren't still using lead based paint!

1 Like

tj wanderlust-words-in-motion

Feb 11, 2025 at 11:06am
In response to "Leaning Tower of Pizzarotti"

It seems when developing land and building on it, anything goes. In high school, I worked for a resort and marveled at the townhouses project they were partaking. The land they picked was cheap because it was all swampland. But "fill" was dumped, the townhouses where built on floating slabs, and the units were sold. It took a few years, but eventually the floating slabs began sinking and the problems started.

2 Like

🌖 HuntersMoon

Feb 11, 2025 at 11:00am
In response to "Leaning Tower of Pizzarotti"

OK. I know, I'm obsessed but why does this remind me of the current administration's approach to restructuring the government? Anything to save a buck. Ignore the foundation.

2 Like

🌖 HuntersMoon

Feb 10, 2025 at 9:03am
In response to "Paint in the Neck"

Given the current situation, maybe they'll update their logo... by changing the paint color to orange.

2 Like

tj wanderlust-words-in-motion

Feb 9, 2025 at 3:30pm
In response to "Revisited: "Bloomington""

I have been to Bloomington a few times myself, the one in Minnesota, not Illinois or Indiana.

2 Like

October 12, 2024 at 8:39am

#1078145

A Few Observations

Tomorrow, I intend to get back to my usual annoying self here by posting a Revisited entry. As for today, well, I'm still in recovery, so I'll just summarize some of my observations on the trip that I may or may not have already blogged about (because I can't remember and can't be arsed to re-read what I've already posted).

I gained a greater appreciation of the intricacies of French wine. Contrary to French beliefs, it's not necessarily superior to other countries' wine, but it does take a different approach. Not that I know a whole lot of details; that takes years of study and practice. I don't mind practicing.

It's popular in some circles to tout the general superiority of Europe over the US. In other circles, the opposite is the case. As it turns out, in my opinion at least, that they're just different. Europe does some things better, like public transportation and healthcare delivery. The US does other things better, like... um... well, I can't think of anything offhand, but I'm pretty sure I would if I weren't so tired. I guess we invented the Internet and GPS, so there's that.

There are whole sections of Brussels where there are more chocolatiers on each block than all other businesses combined.

There are others that are pretty much entirely restaurants.

I gather that it is easier and takes less time to become a medical doctor anywhere than it does to become a chocolatier in Belgium.

Real estate prices (both rent and buy) are amazingly cheap there, compared to the US. At least in the country and small towns. And that's with a denser population.

Unlike my above comment about French wine, Belgian chocolate really is the best. Sorry, Switzerland. At least you won't send assassins after me for that observation. ...Right?

The old joke about how Brits think 100 miles is a long way, while Yanks think 100 years is a long time? It applies to mainland Europe as well. You just have to convert to SI units.

Best quote of the trip, by a Belgian tour guide, uttered between bites of a Belgian waffle topped with whipped cream and chocolate: "You Americans eat like you have free health care."

So that's all I'm going to post about the trip... unless, of course, something else reminds me of a thing I did there. Like I said, tomorrow, we're back to our regularly scheduled programming.

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