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 29, 2024 at 10:51am

#1073318

Je Voyagerai

I thought I'd provide a travel plan update today, after discussing my general idea here a week ago: "Discomfort Zone"

On Monday, I looked at nearby travel agents. Just their websites and Google reviews.

I'm an avid Google reviewer myself, mostly of breweries (writing ability does have some practical use), so I know to take any review with a grain of salt, much like the item reviews here: each is (usually) just one person's opinion and experience, and everyone, reviewer and reviewee alike, has their good days and bad days. Also, one needs to be aware that some 1- and 2-star reviews are competitors trying to steer people away, while some 4- and 5-star reviews are salted by the company itself.

Still, it can be helpful to take a look at reviews all across the spectrum. If there are enough of them, you can be pretty sure there are some genuine experiences in there.

Anyway, one travel agent specifically bills itself as "luxury travel." I'm not elitist enough for that. Another had mediocre reviews at best. I settled on someone who's been active in town for almost as long as I've been here.

Tuesday, I drifted on over there to try to get a feel for what to expect. Sure, I could have called, but it's not far away.

Now, I know some people scoff at the idea of going with a travel agent. Why not just go and experience stuff? Make your own travel plans, your own schedule? Normally, I'd agree. But then I realized: part of the reason I travel is to experience stuff I haven't experienced before. And I've never used a travel agent, so why not experience that for the first time? Besides, I'm entirely too lazy to work out lodging and transportation details myself for a trip to an unfamiliar place.

They gave me a really good idea as to what to expect in terms of process and fees, and promised to get back to me.

On Wednesday, they sent me a few emails, but I had to visit the dentist to get some drilling and crown work done. That shot my whole afternoon (and most of my medical budget), and also the next day while I dealt with the soreness.

Still, I found the energy on Thursday to take a look at the emails, and immediately questioned my decisions when I saw what-all would be required. Passport stuff. Credit card info. Read these T&Cs. Sign here. Sign there. Initials everywhere. (I'm going on a trip, not buying a fucking house.) What kind of trip? What sort of lodging amenities do you want? What's your budget?

"Your vacation is an investment..."

No. No, it is not. An investment is something that's expected to increase in value over time (whether it actually does or not is another issue). Stocks can be investments. A house can be an investment. A car is not, unless we're talking about the classics. A vacation (or, in my case, a voyage, since retirement means permanent vacation to me) is purely spending; the only returns I expect are experiential and emotional.

Words mean things, you know.

On Friday, I called the contact person I'd spoken to and who'd sent me the emails, and she (her name is, amusingly, Karen, and she's a manager, having become the very thing she swore to destroy) assured me that most of these things can be worked out when I meet with the actual travel agent a week from Wednesday.

However, I have no idea what to expect for "budget."

When road-tripping here in the US, I have a pretty good handle on it: I know, to a fair degree of accuracy, what I can expect to spend on gas, food, and lodging each day, and maybe throw in a little extra to try beers at breweries. I'm also not entirely ignorant of what airfare costs, though that varies wildly depending on season and how much discomfort I'm willing to put up with (not a lot, at my age).

I don't want to cheap out, because contrary to popular belief, money does, in fact, buy happiness. Or, well, at least it buys comfort and booze, which is close enough. I also don't want to spend my life savings; I kind of need that for, well, life. Somewhere in the middle would be nice.

What's important to me is that I get value for my money. Like, not spending $100 (or the equivalent in euros) for something that's worth $50.

I figure maybe I can explain that to the agent when I meet with her, and we can work out the details from there. Budget will, of course, help to determine how long the trip lasts and how many different places I can go to. And right now, I'm flexible on those things, as well as on actual dates of travel (I'm hoping for this fall, because I'm definitely not going in the winter, which means if it's not this fall it'll be next spring, and I just don't want to wait that long).

In conclusion, I'm definitely doing this (barring, of course, emergencies). I just don't know any of the details, yet.

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