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:

Annette in France

about 8 hours ago
In response to "Paris, I Guess"

Bienvenue en France.

1 Like

WakeUpAndLive️~Happiness

about 12 hours ago
In response to "Paris, I Guess"

last one today:https://parisjetaime.com/eng/event/olivier-giraud-how-to-become-a-parisian-in-one-ho ur-e047

1 Like

WakeUpAndLive️~Happiness

about 12 hours ago
In response to "Paris, I Guess"

Could not resist sending the link: https://www.atlasobscura.com/things-to-do/paris-france

1 Like

Elisa: Middle Aged Stik

about 17 hours ago
In response to "Paris, I Guess"

Your remarks on Paris looking like many cities reminds me of this video (which I promise is short). It's a sassy look at Paris, and I'm here for it.

https://youtu.be/ByUPFYxy5vw?si=lT57RpTDntAhFdBs

2 Like

WakeUpAndLive️~Happiness

about 18 hours ago
In response to "Paris, I Guess"

if you have two hours to spare tomorrow? A beer bike tour in Paris. Check it out, still available. https://parisbeerbike.com/

3 Like

WakeUpAndLive️~Happiness

about 19 hours ago
In response to "Paris, I Guess"

Je suis sure, vous avez aimez la ville!!!!

2 Like

Aiva Raine

a day ago
In response to "Get Baked"

FYI- basting them with olive oil before placing them directly on the oven rack- NO tinfoil- will allow the skin to crisp up a little as well and add a nice flavor to the skin. Recommend Idaho baking potatoes as well for best results.

Sum1's In Kalispell Montana

a day ago
In response to "Get Baked"

I have long advocated using only the oven for baked potatoes, and am guilty of punching holes in the tops of one. I don't use Aluminum Foil either. But my other half likes to nuke them first, or only nuke them. It's the speed thing for her. I prefer to take my time when I cook, and do things the way I do them, not her way. But, sometimes, ,ya gotta keep the peace in the house. She won't buy Russet potatoes, but those are the only ones to use for this. I may have to exert a little effort an buty Russets just for Baked Potatoes. She loves the English Chips (really enjoyed those in England when I was there, so maybe she'll like the link I sent her (from you!))

2 Like

Cubboo!

a day ago
In response to "Get Baked"

Yes! I discovered this a few years back (slitting the potato about 1/4 - 1/2" lengthwise, then one or two slits across the longer slit.) It came out wonderful! It seems like this was called an English jacket potato, or something like that.

2 Like

Annette in France

a day ago
In response to "Get Baked"

I'll try your oven baked potato in winter when it's sensible to run the oven for 90 minutes.

2 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.

3 Like • Comment • View 2 comments

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