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.

3 Like

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

June 22, 2024 at 8:42am

#1073015

Discomfort Zone

My passport expires in two years. I would like to use it at least once before I get it renewed. Plans I had to travel internationally got interrupted by you-know-what.

Yeah, I'm not doing an article today; I want to talk about this, instead.

The last time I left the country was over 10 years ago; I went to England and stayed with WDC friends there. It didn't require a lot of planning or schedule-keeping on my part, except for, obviously, the flights: no hotel itineraries, no figuring out how to drive on the wrong side of the road, no dining reservations, no navigating a wildly different language (just a moderately different one).

I had a great time, but like I said, I've been languishing in the US (and maybe a little bit in Canada, which hardly counts) since then.

My desire to visit Belgium still stands, though the person I was going to travel with has other priorities now. Which is fine; I'm used to traveling alone and actually enjoy it.

But I've gotten it into my head that, first, I want to go to France. No, not because of the Olympics. Fuck the Olympics and its crowds. If you ask me "what could possibly make Parisians more grumpy," that would be my first answer. Hell, it might even drive them in Seine.

(Not sorry.)

No, it's because un) je comprends un peu de français maintenant, deux) il y a de bon vin là, trois) j'ai entendu que les bières en France sont mieux qu'avant.

(Okay, fine; that's intended to mean "I understand a little French now," "There's good wine there," and "I heard that the beers in France are better than before." Chances are I screwed something up, but at least I tried.)

Of course, there's no reason not to do both countries in one trip, except for that pesky "planning" and "transportation" thing. Unlike the UK, I don't know anyone in France (yet). Or Belgium. And that's mentally blocking me from traveling there alone.

I know that's a "me" thing. Another "me" thing is that I despise flying, especially flying steerage. It's not fear; I've internalized that I'm safer on an airplane than I am sitting on my deck where a tree could fall on me or a mosquito could bite me with a dread disease. No, it's the discomfort, inconvenience, and security theater.

Either of those "me" things by themselves, I can deal with. I've flown across the US, including once to Hawai'i and back. I've also driven across the country (though obviously, not to Hawai'i) solo, without seeing anyone I knew for days at a time. Not to be a dick about it or anything, but I can even ameliorate the coach-class issue by upgrading; since I haven't traveled much in the last four years, my budget for that sort of thing is well-padded.

There are other considerations, too. For example, I know some people travel to get away from it all. I do not. I require guaranteed internet connectivity. Supposedly, my mobile carrier has options for this, but I still don't fully understand them. At a bare minimum, I'd want access to Google Maps. At a slightly less naked minimum, I have a Duolingo streak going that I am loathe to give up. There's also the daily blogging streak that I was hoping I could extend until the end of this year, but as I hate writing on a phone keyboard anyway, that can slip if it needs to (I've already decided I'm not bringing the laptop).

For another example, I'm still a little unclear about how to pay for stuff while out of the country. Even in England, sometimes they had trouble with taking my ass-backwards American credit card, even though it's supposedly specifically tailored for international travel. I imagine that's even worse in the Eurozone. And I don't know how to deal with actual currency exchange. Still, American tourists infest Europe like a disease; obviously, they've found a way to accommodate us. I just don't know what to expect or plan for.

Speaking of which, I also hate being a "tourist." I'd rather be seen as a visitor. Hawaiian shirt and camera notwithstanding.

I shouldn't let these things slow me down, but I do. I'll need to work on that.

On the plus side, I have an electric plug converter that should work in any country civilized enough to have electricity in the first place. You know, for charging up the mobile phone I'm not going to travel without.

In any case, if this happens, it'll be after the goddamn sports thing. I don't necessarily want to visit Paris, but I suspect that crowds everywhere will be worse during that event.

One option I'm considering is, unlike international travel itself, something I've never done: putting it all, or most of it, in the hands of a travel agent.

This appeals to my inherently lazy nature, but I expect it takes a lot of the spontaneity and control out of my hands (yes, I know that "spontaneity" and "control" are basically antonyms, but we've already established that I'm weird). I also have no idea how to go about it, but I'm pretty sure I can figure it out.

I know some readers are seasoned international travelers and might find my hesitation perplexing. But, as I said, I'm just not used to this stuff. My experience with it is limited.

So that's today's blog. I did it mostly to get my own thoughts in order, but advice and comments are always appreciated.

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