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 an hour 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 4 hours ago
In response to "Gray Matters"

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

1 Like

tj-Merry Mischief Maker

about 23 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 20, 2024 at 10:05am

#1072938

A Midsummer Daydream

Solstice today.

At least, it's today in my time zone. Australians have to be different, so it's tomorrow if you're in Oz, and winter instead of summer.

I mean, the event occurs at the same moment for everyone, but at different times depending on your location. It can all be very confusing, I know.

I know a guy who likes summer, but gets sad at the summer solstice, because it marks the start of shortening daylight hours. I can understand that. I always mark the winter solstice by noting that daylight will increase, but for some reason, I ignore the complementary effect six months later.

Probably an example of willful ignorance on my part. It happens.

Some people have a favorite season. I used to, I guess, but I came to the realization that you can't have a favorite season without having to experience the other seasons, too. I do have a most hated season (winter) and a most hated month (February), but that's about it.

Which is kind of odd, because, by some definitions, February is mostly in spring.

("Okay, Waltz, you need to get your head examined.")

No. Well, maybe, but not for that reason.

Astronomical seasons indeed go from solstice to equinox, or vice-versa. Summer (in the only hemisphere that matters) starts today and will end on September 22. It also features four, not the usual three, full moons this year; the one in August will thus be a Blue Moon.

Meteorological seasons go by calendar month: June 1 to August 31 for summer, September 1 to November 30 for autumn, etc. They do this because temperature and weather cycles tend to lag solar quarters.

And by folklore, in some cultures, the solstices and equinoxes mark the middle of their respective seasons, not the end points. Midsummer is today (again, being a hemisphere snob). Midwinter is at the winter solstice. I've never heard of midspring or midautumn, but they follow logically.

By the folkloric definition, spring starts around Groundhog Day, which is one reason why there's a Groundhog Day: will winter weather linger, or start acting like spring? (That rodent is wrong more than he's right, so it's wise to bet against him over the long term, Bill Murray movies notwithstanding.)

Similarly, Halloween (or Samhain) marks the beginning of winter by that definition. The actual date is closer to Guy Fawkes Day, but whatever.

Point being that as of today, per some definitions, summer's already half over.

Reason enough to be depressed, as if we needed another one what with... *gestures helplessly at everything* ...and all.

In short, the events—solstices and equinoxes—are objectively real transition points, but the meaning we put on them can vary.

Yes, I've written about this sort of thing before, notably at the last winter solstice, and made many of the same points. But I can't expect everyone to read every entry. Hell, I don't read every entry. Apologies to those who do, for the repetition.

But hey, this (astronomical) summer, we might get treated to a nova.

I wrote about it back in March: "Celestial Spheres"

So that's something for us astronerds to look forward to.

If, that is, we live in the northern hemisphere.

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