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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Brandiwyn🎶

a day ago
In response to "Blond Man's Bluff"

That's interesting about the two spellings of blonde. I always thought blonde referred to the person and blond referred to the hair. But I never bothered to look it up, So you've catered to my lazy today.

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Kåre เลียม Enga

Nov 9, 2024 at 1:02pm
In response to "It Was a Dark and Starry Night"

Ah... but why did he see those swirls when many of us don't? A totally different discussion... out of thin air... which is a frozen fog today when I gaze out the window.

Humans have great skills of observation; some of us extraordinary skills. Is The Scream depictive or evocative? 😱

Turkey DrumStik

Nov 9, 2024 at 11:04am
In response to "It Was a Dark and Starry Night"

Frankly, I'm more surprised to see reasonably good reporting from CNN. I'd given up on their ability to rein in their tendency toward loaded speech.

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JayNaNoOhNo

Nov 8, 2024 at 12:00pm
In response to "Bungalow Bash"

For the love of God, what did I just read? That is by far the worst use case of a bungalow I've ever seen, and a fundamental misunderstanding of the value of front porches—and the size of bungalows.

Was this his point? Nope, I know what his point was. For a guy so worried about presentation, he certainly undercut his own arguement.

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Turkey DrumStik

Nov 8, 2024 at 9:47am
In response to "Bungalow Bash"

"People don't change their minds on the internet. They usually do that in books, battlefields, or not at all.

Yeah, that's a little bleak."

Bleak and also inaccurate.

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Kåre เลียม Enga

Nov 7, 2024 at 10:19pm
In response to "Past and Future Intents"

I must live in the present. In the past I daydreamed a lot... probably to escape the bruises of reality; however, I need to be more aware crossing streets or even how I get out of bed (no rapid movements without thinking of consequences).

I can envision a future more peaceful than the present because this time-and-place isn't the only "present" that's ever existed, and I do try to see the larger view (emphasis on 'try').

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Lilli 🧿 ☕

Nov 7, 2024 at 10:58am
In response to "Past and Future Intents"

If the aforementioned exercises don't help; there's still liquor.

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Turkey DrumStik

Nov 6, 2024 at 10:22am
In response to "Lo!"

Not gonna lie I need the holiday season chatter.

Aiva Raine

Nov 5, 2024 at 3:03pm
In response to "It's Hard Not To Scare"

Shaking head. Aside from the inevitable Simpson's spider goat song playing in my head which I will now share with you below, whenever I come across a story that has all these scientists doing ridiculous experiments for no logical reason, I must go- ah- scientists- CANCER? You do remember that problem right? Can we maybe cure that before we make cockroaches magnetic and elephants see pink elephants?

spider pig clip- and if you all now are singing the spider goat song- you're welcome! grin
https://www.youtube.com/watch?v=BARjPuUN36Y

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Aiva Raine

Nov 5, 2024 at 2:55pm
In response to "Anomalies"

One of your more humorous entries- I especially enjoyed- "Lofoten Vortex can be the name of my Bjork cover band."

Thanks for the laughs. Much appreciated.

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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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