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.

3 Like

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.

2 Like

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.

1 Like

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.

1 Like

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.

2 Like

October 20, 2024 at 9:02am

#1078619

Revisited: "Unlucky"

Going back over 5 years, today's peek into the past takes us to an entry from July of 2019: "Unlucky"

I've written about luck several times in here, but this entry isn't directly about that; it's a play on the word "potluck." The entry is a response to a prompt, most likely from 30DBC: "What is your go-to dish to bring to a potluck? Does your family have any traditional recipes? (In Hawaii, these appetizers are called 'pupus'"

What I apparently didn't catch at the time was the reference to appetizers; in mainland North America, a potluck is when you bring a dish of any sort to a communal gathering. The dish could be an appetizer, but it could also be a main course (my entry goes on about casseroles for a good bit) or a dessert.

Casseroles, however, mystify me. I'm not sure why. They can't be that hard to do, or millions of Southern Baptists wouldn't do them, in all their various incarnations. You'd think that would teach them something about variety and the wonders of diversity, but... no. They just plop down their armies of Corningware, oblivious to the obvious metaphor.

Since then, I've tried my hand at various casseroles, with a good bit of success. They're less work than other dishes, but still harder than, say, popping a frozen thing into a microwave.

But what stumps me is that I don't know what the essential thing is that makes a casserole a casserole and not something else. Baking? You also bake lasagna; is that a casserole? Meat? Nonsense; there are plenty of vegetarian casseroles.

I've also had several entries about classification problems, like my perennial assertion that a hot dog is not a sandwich, but a taco.

So, yeah, no traditional family recipes here. Just stuff I've picked up along the road.

As I noted in that entry, if there were traditional recipes in my family, they'd be borderline inedible, and I'd have no problem letting them fade away.

Fortunately, one of the comments helped me to clarify the "what exactly makes a casserole a casserole" conundrum. I occasionally get comments here that thank me for helping the commenter learn something new, but I assure you, that works both ways.

Just lucky, I guess.

4 Like • Comment • View 3 comments

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