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:

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 11, 2024 at 1:26pm

#1072479

No Cornea Jokes Today

One of the immeasurable joys of getting older is we get to go visit doctors and other medical professionals more often. For instance, me, this morning, so this is going to be a fairly brief personal update rather than deep philosophical musings about whether Chicago pizza is actually pizza (it's not) or scorn at the sorry state of science reporting.

Some long-time readers might remember that back in the dark ages of 2021, I had cataract surgery in both eyes. Today's visit to the eye doctor served as a general check-up and 2.5-year follow-up to that.

Now, I know I like to kvetch, whinge, moan, and complain in here. But this time, I have nothing to complain about. Everything, according to the eye doctor (don't make me spell optha-whatever), is fine. Sure, my vision is slightly worse than it was 30 years ago, but it's a lot better than it was 5 years ago.

What I mostly wanted to share, though, was that the doc, who I go to both because she's very thorough and because she's married to a former colleague of mine, is a massive nerd. I mean, you expect that from science-oriented professions, but she keeps up with the latest ocular-testing gadgets, and this time, she had a new toy: a virtual reality headset programmed to do part of the eye exam.

You put on the headset, and a little blue floating cartoon lady (you can tell the gender from the voice and the ponytail; I have a ponytail sometimes, too, but my voice and goatee give away the game) walks you through an exercise where they measure your visual focus distance. This involves staring at a target while a light show goes on all around you like you're moving through space in a game.

Turns out my focus isn't quite right, but not off enough to cause serious eye problems or require glasses. Which is still weird to me, not needing glasses for anything at my age. Well, sunglasses for protection from the harsh glare of the accursed daystar, sure, but not prescription ones, and I've always needed shades.

And the only reason why?

Science.

I'd be effectively blind right now without science. Well, not really; without science, I'd have died before I could go blind.

So if you ever wonder why I bang on about nerd stuff in here a lot... well, that's one reason why.

4 Like • Comment • View 1 comment

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