Complex Numbers (Book)

Blog Calendar

◄ August ►
S	M	T	W	T	F	S
				1	2	3
4	5	6	7	8	9	10
11	12	13	14	15	16	17
18	19	20	21	22	23	24
25	26	27	28	29	30	31
Archive				RSS

About This Author

Contact Me

Come closer.

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/

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 20 hours ago
In response to "Hot Shoulder"

Maybe if we say, "Betelgeuse" three times in a row we can witness it.

1 Like

tj-Merry Mischief Maker

a day 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🎁

Dec 23, 2024 at 12:56pm
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

Dec 23, 2024 at 10:04am
In response to "Trash Pandas"

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

4 Like

Jayngle Bells

Dec 23, 2024 at 10:03am
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:

Access Restricted

This is a private area. Access has been restricted and you may not view it under its current settings. If you feel you should have access, please contact the area's owner and request access.

Note: To contact a member via email, click the small envelope icon to the right of their handle. You must be logged into your account to send an email.

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

August 5, 2024 at 10:28am

#1074799

Mach It So

There are complex questions with simple answers. I can't think of any right now, but I'm sure they exist. At the very least, the answer is "a large quantity of plastic explosive." But today's article, from Thrillist, is a simple question with a complex answer.

How Fast Can Airplanes Go, Exactly?
An aviation expert breaks down how fast the answer beyond "really, really fast."

Science and human ingenuity have given an innumerable number of gifts to society: indoor plumbing, medicine, electricity.

And most importantly, beer.

Sure, it's also given us drone warfare, AI, and those weird Amazon Go stores—but let's focus on the positive.

Those are positive. They generate positive cash flow for a small number of individuals, which is positive for them.

One of the most exciting advancements of the modern era is the commercial airplane, which has made traveling around our massive planet in less that 180 days a true possibility.

And you'll experience maximum discomfort while doing so.

Also, editing is dead. Not only did they use "that" instead of "than," but the famous Verne novel was 80 days, not 180. Now, ISS astronauts circle the globe (for it is, indeed, a globe) twice in a bit over 180 minutes.

Now that I've called that out, I'll probably commit a massive blunder in this entry, if I haven't already.

The first flyers traveled at max speeds of 31 miles per hour, back in 1903. Now, 104 years later, planes have gotten a little faster. But just how fast? It turns out the answer isn’t so simple—it's a lot more complicated than getting a basic miles per hour answer.

Even more complicated are the various units used to describe speed. Miles per hour, sure, but there's also km/s, knots, millimeters per day, etc.

Thrillist spoke with Luciano Stanzione, who leads AviationExperts.org, an aviation consulting agency, to better understand how fast the modern commercial airplane is capable of flying.

I'm of the considered opinion that everything on the internet now is either paywalled, or an ad. I didn't hit a paywall here, so I wonder what the ad is for. Certainly not an "aviation consulting agency," right? Possibly Thrillist itself, based on the blurb at the bottom of the page.

“The airspeed is measured in KIAS (indicated air speed in knots) and in Mach speed (reference value to the speed of sound, used in higher altitudes)," Stanzione tells Thrillist.

Oh, yeah, the Mach number, too. I imagine it'd be fractional for subsonic flights, but I don't recall any instance where it's like "this airplane has a top speed of Mach 0.8." At least, not until I read this article. But all these different units make for opaque comparisons, like if your car's efficiency is in miles per gallon and you're looking to compare it with one whose efficiency is listed in kilometers per liter. Or whatever.

Speaking of complicated, the speed of sound changes with air density, which as I understand it is generally correlated with altitude.

The speed that airplanes can reach in the air can’t be as easily determined by the same metrics in which we measure the speed of objects on the ground. So bear with Stanzione for this scientific breakdown for a second.

There follows said "scientific breakdown," which I think is easy enough to follow, but I won't quote it here. Essentially, there's airspeed and groundspeed (no mention of whether the plane's carrying coconuts, though).

How fast will airliners be able to go in the future?

"That is impossible to determine but manufacturers are working again on high-speed jets, above sonic speeds," Stanzione says. "For a few decades now, improvements on speed have not been major. Let's see if new supersonic developments change that."

All the science fiction I've read indicates that it won't matter, because we'll have suborbital ballistic flights, or teleportation. Given the constraints of physics and capitalism, I don't see either of those things happening anytime soon, if at all.

Whatever the actual future of air travel is, though, at whatever speeds, you can bet they'll continue to find ways to make the process more degrading, uncomfortable, humiliating, and dehumanizing.

1 Like • Comment • View 1 comment

© Copyright 2024 Waltz Invictus (UN: cathartes02 at Writing.Com). All rights reserved.
Waltz Invictus has granted InkSpot.Com, its affiliates and its syndicates non-exclusive rights to display this work.