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

November 12, 2022 at 12:01am

#1040558

Bona Frieds

Hope you're not hungry.

15 Delicious Foods That Used to Be Considered Gross

"Delicious" is, of course, a matter of opinion. The headline would have probably been too long if Cracked had qualified that, though.

As usual with such a long list, I'm only going to share a few of these.

15. Lobster

Lobster was so plentiful in the areas colonized by early Americans that they stopped eating it as soon as they could. Only prisoners, the poor, and livestock -- which were pretty much considered the same things -- would deign to eat it, and it was even used as fish bait.

You know, I've been hearing this for so long and with such certainty of delivery that I started to question it. So I checked a source that's marginally more verifiable than Cracked, and discovered

that while this probably is the case, in Europe it was often associated with wealth, before a bunch of Europeans came over here.

So this is more of a case of changing popularity over time, which happens with many foods.

Also, keep in mind that sometimes the wealthy like to eat expensive things just because they can, and because the poor can't. This has little to do with the actual taste of the food. See also: caviar. That shit's disgusting.

14. Chicken Wings

Though now a staple of [sportsball game whose name is copyrighted] parties and other manly gatherings, chicken wings were considered the grossest part of the chicken, to be either unceremoniously thrown out or used for soup stock at the most.

Ah, one of my favorite things to rag on. "Let's take the chicken part that used to be made into dog food and turn it into sports food." Look, they're still the grossest part of the chicken (except maybe the beak and intestines) and are only popular because they're marketed to be. And because of the hot sauce, of course.

12. Foie Gras

There is so much wrong with this entry that I'm not even going to paste it here. Also, everything about foie gras is foul. Pun intended.

9. Peanuts

Peanuts came to America from Africa, and like most delicious African foods, they were immediately dismissed by colonizers as unfit for humans until three things happened. First, the Civil War reduced people to choking down whatever protein they could get their hands on, and peanuts were definitely preferable to rats.

I'm no fan of peanuts, but yes, if I had a choice between peanuts and rats, I'd eat the peanuts.

Then P.T. Barnum began selling peanuts as circus food.

Having been marketed by the Platonic ideal of "huckster" ("sucker born every minute" etc.) is not something that I'd use to recommend a product.

Finally, peanut butter happened. Even the most frothing bigot can’t resist a spoonful of peanut butter.

Admittedly, I'm okay with peanut butter. I still don't know why I like peanut butter (but only the real kind, not the candy kind like Skippy) and not peanuts, but I never claimed to be entirely consistent.

7. Mushrooms

The Western world shunned mushrooms on account of their tendency to make you see God and/or kill you until the French insisted they were the height of cuisine in the 18th century

I do like (commercially available) mushrooms. I know several people who can't stand them, mostly due to the texture (they say). I can understand that. When you really think about it, eating mushrooms is weird. It's a fungus, neither animal nor plant (but, oddly, closer to the former than to the latter), and thrives in shit. There aren't many fungi that we eat. Yeast, sure, but we were ingesting that (as part of bread or fine fermented beverages) long before we knew what yeast actually was.

On the other hand, when you really think about a lot of things that we eat, you start to question them. Eggs, for example. Or:

3. Oysters

The story of oysters is a very straightforward one of supply and demand. They were once so plentiful that Charles Dickens characters looked down on the patrons of oyster houses that lined the London streets one “to every half-dozen houses.” Then we filled the oceans with so much pollution that it was hard to get a good oyster, prices went up, and the rich just equated “expensive” with “good.”

Like I said, sometimes rich people do rich people things just because the poor can't.

Still, you have to wonder how people figured out oysters in the first place. "Let's crack open this rock and see if there's a tasty treat inside."

1. Burgers

There are people who, not unreasonably, would take a juicy burger over the finest steak any day, but back in the Upton Sinclair days, ground beef was seen as unclean at best and possibly containing dude at worst.

It's actually more complicated than that, and ground beef certainly predates fast food. I'm pretty sure, though, that the popularity of hamburgers didn't take off until there was enough supply to make them cheaply, and that required access to electricity to power the grinders.

When I was a kid it confused the hell out of me that hamburgers didn't contain ham. Just another linguistic weirdness of English, in this case with the word deriving from the city of Hamburg which may or may not have had nothing to do with the invention of the hamburger. Nowadays you can talk about beef burgers, veggie burgers, turkey burgers, even cheeseburgers (which aren't made out of cheese) but never ham burgers. And no one calls it a hamburger anymore, either.

Now you're hungry, aren't you?

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