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 9 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🎁

a day ago
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

a day ago
In response to "Trash Pandas"

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

4 Like

Jayngle Bells

a day ago
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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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

September 16, 2024 at 9:42am

#1076872

Two Much

We all know I like Cracked. This is partly because, like me, they try to be amusing and informational at the same time.

Sometimes, that doesn't work out on one front or the other.

Companies That Make Only Two Wildly Different Products
It’s usually a bad idea for, say, a coffee shop to start selling auto parts

As a business, diversifying your product line is a surefire way to grow your customer base...

Not always.

Some companies, however, decide that they do two things. Two things only. Two wildly different things only.

5. Guinness

It’s not just a coincidence of nomenclature — the Guinness Book of World Records is the same Guinness that produces the world’s carbiest beer.

No.

I mean, yes, originally. Guinness reportedly started the World Records book as a way for pub-goers to settle bets, as the article notes. Later, the publication (which I'll call GWR) got sold to another company, with rights to the name.

That company then eliminated every world record related to drinking, on safety (read: liability) grounds.

While I understand their reasons, this ripped them from their roots. Would you call a piece of paper and a tree the same thing? No. Then GWR has nothing to do with the Irish manufacturers of dark and delicious malt beverage.

Guinness, the brewer, also sells merchandise, and even has showrooms for it. I've been in one in Las Vegas. Well, at least they did pre-pandemic, and I can't be arsed to see if they're still doing that. Point is, though, not only is GWR no longer affiliated with Guinness, but even if they weren't, Guinness would be selling more than two products.

4 Coors

Yep, the company that makes beer as cold as the Rockies is also involved in the production of high-tech ceramics.

I would argue that they don't make beer, but I'm admittedly a snob on the subject.

Also, it's Molson Coors now, and they suck even harder than AB/InBev, which at least has the decency to (mostly) leave craft brewers to their own devices when they Hoover them up.

3 Yamaha

The customers for Yamaha keyboards and motorcycles couldn’t be more different, and technically, they’re not the same company.

As noted, they're not the same company, so this shouldn't count, either. Also, why can't bikers be keyboardists, or vice-versa?

2 Bridgestone

Bridgestone is another Japanese company that successfully divided itself, not out of prudent business sense but for the love of the game — literally.

Tires and golf balls? Come on, they're both round, dimpled, and made of synthetic rubber.

1 Michelin

Speaking of tires, it’s often said that the Michelin tire company created their restaurant rating system to get people to drive more and wear down their tires faster, but it’s a little more complicated than that.

Okay, I'm going to grant them this one.

In 1900, people had to be convinced to buy and drive cars at all, so Michelin began publishing car-specific travel guides that included where to get gas, what time the sun set at different times of year (because there were no street lights), and yes, the best hotels and restaurants.

In the US, at least, there was another, way worse reason to have to know sunset times.

Now you know, once our car-centric society has destroyed the planet and global warming kills us all, you can blame the French.

Technically, the Industrial Revolution only kicked into high gear after Nicolas Léonard Sadi Carnot figured out the whole "thermodynamics" thing, so we might as well blame the French.

1 Like • Comment

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