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🎶

about 22 hours 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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July 13, 2024 at 10:34am

#1073898

Surprise Me

A brief travel planning update:

I met with the travel agent on Wednesday. As a reminder, I'm using a travel agent this time because a) I'm entirely too lazy to coordinate my own itinerary; b) I don't know exactly what I want; and c) I've never done it before, so it'll be a new experience for me.

I told her basically what I wanted to do (wine and beer tourism in France and Belgium), so she suggested three locations in France: Loire, Burgundy, and Paris.

"So you'll fly into Paris and spend... how long there? Five days?" (I'd told her maybe three weeks for the entire trip.)

I'm like, "I'm not exceptionally interested in Paris. The city is dirty, and the people are rude and will be tired of foreign tourists after the Olympics."

I mean, I get flying into Paris. There's a big international airport there, and the other places I want to visit are generally in the northern part of the country. But thinking I want to spend time there is a big assumption.

Then I remembered that I'd heard the Paris craft beer scene had developed nicely in the past few years. Plus, she dangled "reservations at a Michelin-starred restaurant" in my hungry face. (I think they make most of their money from kickbacks, which is fine.) So okay, sure, Paris. Just not a lot of the touristy places. And maybe three days, not five. I know it's an enormous city and you can spend years there without running out of new things to do, but I'd rather focus on other locations.

The rest of the trip is countryside and small towns. Well, mostly. Nothing's detailed yet, but Belgium might involve staying in Brussels, for which I don't have any objection.

The problem with France, I've found, is that there's pretty much nowhere in the entire country that I don't want to spend time in. Well, maybe some of the freezing cold snow-peaked mountain parts. I wouldn't mind visiting them briefly, or seeing them from a warm, comfortable distance, but, as with every other high mountain range, the combination of colder temperatures and lack of air pressure is anathema to me. Plus, I might be expected to "ski," or "hike," or do any number of disgusting outdoor activities.

Yes, I've done it, like when I spent a week at a science camp in the Colorado Rockies. That's how I know it's not for me.

So, now I'm waiting for them to stitch together a draft of a travel plan. I'll get to look at it and change things if I want. I told her, "Don't worry; I like surprises... but only if they're pleasant ones."

She didn't find that nearly as amusing as I did.

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