Complex Numbers (Book)

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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/

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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:

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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 28, 2024 at 12:20pm

#1075828

Fact-Ory

I promise I have more articles in the queue from not-Cracked than from Cracked, but sometimes, the RNG likes to cluster things. I should probably mention that there are images in the article that may get you called in to HR at work, but that'll probably happen anyway, just from wasting your time reading this blog and a dick joke site.

5 Simple Science Questions We Bet You’ll Get Wrong
What is a day? What is the Earth made of? You don’t really know

Yeah, I get it. There's a lot I don't know. There's probably even some stuff I get wrong, though I try to fix that when I become aware.

Phrasing aside, though, as usual, the article has an interesting take on things.

Between time travel, fusion reactors and the brain transplants we hear are happening any day now, science can sound like an intimidating set of disciplines.

I don't know about brain transplants, but I did see something recently about a brain implant for thought-to-text.

No, not Muskmelon's crazy stunt; this one seems legit.

Of course, I immediately saw the potential to reverse this so that advertisers can pay to have their product beamed directly into our brains. And you know damn well that's going to happen, along with people using it for porn.

Today, however, let’s set aside the most complicated theories and applications.

Porn isn't that complicated, unless you're doing it in space.

5. How Many Minutes Does Earth Take to Rotate Once?

Oh, I know this trick question. A solar day is not the same as a sidereal day.

The Earth takes 1,436 minutes for each rotation. Or, it takes 23 hours, 56 minutes and 4 seconds.

The article explains this in good-enough detail, but the simple explanation is: that's how long it takes for the same distant star (that is not the one we're orbiting) to reappear in the same place in the sky, relative to any point on the earth's surface.

But also, that sort of knowledge makes people like me fun at parties, but doesn't much matter for our 24-hour-everyday lives.

4. Would It Be Sexy to Have X-Ray Vision?

No.

3. What Color Is the Sun?

Another trick question, and also involving the accursed daystar.

The Sun is yellow, say most people with eyes.

Except when it's red or orange.

People who know more about space might offer a different answer. The Sun is actually white, they’d say.

Again, details in the article, but basically, white is what happens when you mix a full spectrum of light colors together. And of course the sun has a full spectrum, else we wouldn't sometimes see rainbows. (I may be mixing up cause and effect; our eyes evolved to see that range for reasons).

But to confuse us even further, physicists refer to the kind of photon emissions from the sun as "black-body radiation," for reasons I won't go into but you're free to look up. So maybe the sun is "actually" black.

2. What Is the World’s Most Common Substance?

Oh, that's easy: Stupidity.

What is the most abundant material on Earth — or rather, what is the most abundant material on or in Earth?

Um... advertisements?

Is it rock? Sure, but you’re going to have to be more specific than that. Anything can be considered rock.

Well, maybe not smooth jazz.

The most common substance in the world is in fact... bridgmanite. We’ll forgive you for never having heard that word before since scientists only got around to naming it within the last decade.

Yep, okay, you got me. Never heard of it before this. My knowledge continues to expand, thanks to a juvenile humor website.

1. Should You Shoot C-4 Explosives, for Fun?

I'm a big fan of doing lots of things for fun, so... yes?

On the other hand, if you answered “yes” because explosions are cool, we have bad news. The C-4 will not explode, so shooting it might be quite boring.

Awww.

Yeah, my understanding was the whole point of C-4 was to have a stable explosive. Which sounds weird, but you want it to blow up when you want it to blow up, not while you're transporting it to the thing you want to blow up. So it requires a detonator or whatever.

Shooting Tannerite,

now... yeehaw!

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