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.

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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.

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.

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August 24, 2024 at 11:39am

#1075658

Finally!

This week, the agent got back to me with dates for travel. I still don't have tickets in hand (or, more likely, email), so things might still change, but right now my trip's scheduled for about 9/21 to 10/10. As a bonus, she came in under my budget, even with my refusal to fly steerage.

Finally, hallelujah, eureka, etc... wait. That's just four weeks away. Shit. Snuck up on me.

Fortunately, there's not much else to do. I still want a travel laptop or tablet, but that's easy enough; I just have to venture (shudder) into a local retailer. Staples, maybe. Or Breast Buy. I already have a plug adapter. Maybe visit another store to buy some other clothes, but that's easy enough, too.

It did occur to me that at one point, I'd heard the EU was going to start requiring visas (or visa-like applications) from US travelers, so I looked that up. {xlink:https://www.cntraveler.com/story/americans-will-need-visa-for-the-eu}Pushed back to 2025. Whew. Seems to me the travel agent would have told me if that was required, but one never knows.

I think I've figured out the best way to get cash there: ATM card. Problem: I haven't used an ATM in well over a decade, so I don't actually possess a debit card. It's rare that I actually need cash, as I pay for almost everything with a credit card.

The reaction to this usually surprises and disappoints me, and look, I'm hard to disappoint because I'm a pessimist. I usually get comments like "I hate credit cards" or "you must have a really low interest rate" or "wow, your debt must be crushing" or some other response about the inherent peril of using credit cards.

Thing is, no, I don't pay extra. I choose cards that feature cash-back promotions, so they pay me. And funny thing about credit cards: at least for now, if you pay off the balance in full when it's due, the amount of interest charged is zero. So I basically use it like a debit card.

But there's one drawback (besides needing to make an effort to stay on top of things): if you use the credit card to get a cash advance, that grace period of zero interest goes right out the window, at least for the portion of the balance that was the cash advance. I don't know exactly how it works, because I've never gotten a cash advance, but I can see that fucking up my budget for months, because it accumulates interest during the period between taking the cash advance and paying it back. Does that interest then garner further interest? I don't know, and I don't want to find out the hard way.

Point is, I called my bank and asked for a debit card. I also made them give me a PIN for my primary credit card (same bank) because, apparently, some European merchants use chip-and-PIN systems instead of tap-and-signature. Ask me, that's a more secure system anyway, and I don't know why we can't adopt it here. The bank rep didn't instill a lot of confidence in me, though, because she kept saying things like "PIN number" and "automatic ATM machine." I think I reached their Department of Redundancy Department.

The idea is that I expect to pay for most things in Europe with the CC, but there are some situations where cash is useful. I don't want to carry a lot of dollars, pay someone to convert them to euros, then pay someone again to convert them back to dollars as I return. At least I won't have to worry about tipping... right? Right?

Back in the old days, we used traveler's cheques, but I don't think that's a thing anymore. And most of the countries I've visited up until now were cool with taking US money (though at an unfavorable exchange rate).

Apparently, there's a 1% fee for using the debit card at a foreign ATM, but I think that's still cheaper (and easier) than currency exchange. Which I might still have to do on the return, but whatever.

That is, unless one of you more seasoned international travelers have any better ideas. I'm open to suggestions.

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