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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Show Recent Comments

10 Most Recent Blog Comments:

Wordsmitty ✍️

about 2 hours ago
In response to "Gray Matters"

"who were between age nine and 39," - so the old folks need the rejuvenation formula first to turn back the years.

I wonder if anyone has studied why hair goes gray in certain places and not everywhere. Maybe different kinds of stress?

1 Like

🌖 HuntersMoon

about 5 hours ago
In response to "Gray Matters"

I've always believed "better gray than gone away..."

1 Like

tj-Merry Mischief Maker

about 24 hours ago
In response to "A Mammoth Discovery"

This makes think of a book, "Evolution and Genomic Diversity" by Gene Poole.

1 Like

tj-Merry Mischief Maker

a day 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

Dec 23, 2024 at 4:55pm
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

June 30, 2024 at 7:40am

#1073352

Revisited: "Rock you like a..."

Today's retrospective takes us back to September of 2018, a brief entry about the weather: "Rock you like a..."

So now, six years less a season later, how have things changed?

It's hard to avoid people talking about the impending doom of Hurricane Florence, so I might as well add my voice to the babble.

I remember that one. While it hit the coast as a Cat 1, it was so slow-moving that what it lacked in wind damage it made up for in massive flooding.

No, I'm nowhere near the coast. Looks like the worst we'll get is a bunch of rain. Nothing new there; it's been raining more often than not all summer. I'm at a high spot and I have a sump pump and a backup generator.

I swear my sense of time is horribly distorted. If you asked me yesterday when I got my generator, I'd have probably answered, "I don't know. 2020, maybe?" Our power sometimes goes out in a light breeze, so it's not just for hurricanes. I justify the presence of the (very loud) generator by claiming I have a medical device that requires electricity, which is true, but the main reason is I don't want to be without internet for more than five minutes.

I also have a house surrounded by trees, though, so I might be boned anyway.

Since then, I've had at least one of the trees taken down, one that was leaning noticeably toward the house. Still boned when the next 'cane blows through.

Some good friends of mine live in Raleigh, NC. They're boned.

They got through it just fine. Couldn't go anywhere for a day or so, if I recall correctly, but safe and dry. They just bought a house in southwest Virginia this year, which means the next hurricane will head for them. And yes, hurricanes can still do significant damage that far inland. Not as often, but it happens.

I have a fondness for Virginia Beach and the Outer Banks. I'll miss them.

My fondness for Virginia Beach ended when I heard that my favorite watering hole there was closing. And miraculously, most of the OBX is still there. I just spent a night there last year.

It seems like a lot of the places I've been to have been hit by something.

Some sort of familiarity bias. Also, I travel more than a little, and every time I do, the chance of a place I've been getting smacked by some disaster goes up a little.

Well, as long as Las Vegas is still- wait, what? The Colorado River is at historic low flows, and Lake Mead is dangerously dry?

Since then, the water level there has fluctuated,

but it's still pretty low. I also haven't been to Vegas since the pandemic started.

I'd ship 'em some of our rain if I could. Let me tell you, if I ever hear another breath about "drought" around here, I'mma lose it at someone.

We've had a bit of a dry spell this year (I write this during a drizzle, however, and we're supposed to get thunderstorms this afternoon and evening). My grass is partly brown. Good thing I don't live in an HOA; I refuse to waste water on lawn irrigation.

They've been predicting a very active hurricane season this year, but those long-range predictions are questionable at best, so I don't pay much attention to them. Weather is notoriously hard to predict with any accuracy more than a week out, so I'll do what I do every year: wait and see.

And try to remember to get my generator serviced.

1 Like • Comment • View 1 comment

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