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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10 Most Recent Blog Comments:

tj-Merry Mischief Maker

about 21 hours ago
In response to "Hot Shoulder"

Maybe if we say, "Betelgeuse" three times in a row we can witness it.

1 Like

tj-Merry Mischief Maker

a day ago
In response to "Trash Pandas"

Being an avid camper, keeping items out of the reach of raccoons can be challenging. We've had them figure out how to open coolers, open chests, and even gnaw through rope used to suspend a cooler out of their reach.

2 Like

🐕GeminiGem🎁

Dec 23, 2024 at 12:56pm
In response to "Trash Pandas"

I wrote one of my Danger Dog stories based on a childhood camping trip that involved a raccoon stealing grapes from our cooler "Danger Dog's Camping Capers"

I had always heard that cockroaches will be the only thing that survives a nuclear war/apocolypse, but maybe it will actually be the raccoons.

4 Like

🌖 HuntersMoon

Dec 23, 2024 at 10:04am
In response to "Trash Pandas"

Just wait until we team up with the cats...

4 Like

Jayngle Bells

Dec 23, 2024 at 10:03am
In response to "Trash Pandas"

The back of my property is sheltered by buildings and cedar trees, so I live on a "racoon highway." Tinfoil works until one takes it for the team and moves it. The only scent that somewhat deters then is ammonia - but they alter their path within the already preferred walkway, not avoid it completely. They generally leave the garbage bin alone, but that's because of the bears, not because of anything we do 😂

I've had more wildlife problems in the city than I ever did in the country. The first time I saw a deer up close? In the middle of my street directly in front of my house. Probably looking for the trash panda party. Lol

4 Like

Annette

Dec 21, 2024 at 11:59am
In response to "Neutral Zone"

It made me happy that I was able to like it. I take the small victories.

1 Like

tj-Merry Mischief Maker

Dec 20, 2024 at 6:06pm
In response to "To Err Is Human"

Right from the beginning I find myself asking, "How many times can anything go extinct?" On a side note, the other day while shopping I saw a neanderthal in the produce section. He was trying to fit in with modern clothes, but it was pretty obvious he was a neanderthal.

1 Like

Annette

Dec 20, 2024 at 1:00pm
In response to "To Err Is Human"

Comments work.

"Like" button sends me this message:

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Annette

Dec 20, 2024 at 12:58pm
In response to "To Err Is Human"

Can I leave this comment?

Wordsmitty ✍️

Dec 19, 2024 at 9:03pm
In response to "Five Seconds"

Way, way back when I was little, it was a three second rule that I remember. Sometime since it was upped to five seconds because of all the chemical preservatives that are pumped into everything now.

1 Like

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.

2 Like • Comment • View 2 comments

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