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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Show Recent Comments

10 Most Recent Blog Comments:

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').

1 Like

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

1 Like

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

October 23, 2022 at 12:05am

#1039581

No Geese Involved

Finally bought a car yesterday, after 15+ months without one. My ass is still sore from the reaming the dealer gave it, but at least my feet will be less sore and, hey, new car. Actually slightly used car, but whatever.

In a great example of the random number generator not giving a shit, today's article (from Cracked) has nothing to do with cars, loans, sodomy, or sales associates.

16 Portuguese Words (That American Slang Could Sure Use)

Always happy to learn new words. Or make them up. When I see words in a foreign language, I can never be sure they weren't just made up. I mean, all words are made up, but whatever.

Did you know Portuguese is the 6th most widely spoken language on the planet (suck it, French and all but six other languages)?

No, because it's not. And if it were, that should be "suck it, French and all but five other languages."

Or... well, maybe it is. What a language is can be fuzzy. I've mentioned before that some linguists consider Scots a dialect of English; others call it a separate language. I've heard that some regional dialects of Mandarin (one of the most widely-spoken languages in the world) are mutually unintelligible; I imagine this is kind of like how Cockney is all Greek to Americans. Other languages are considered different even if they're mutually intelligible. And how far along do you have to be in learning a language to be said to "speak" it? I know some French, but I don't consider myself a speaker of the language. I know maybe 30 words in Japanese (if you include tempura, sushi, and hentai), but I certainly don't speak it. For these and other reasons, any listing of language by popularity necessarily requires some amount of guesswork.

You could list languages by number of native speakers, in which case Wikipedia puts it at #7.

That list puts Mandarin at the top, followed by Spanish, then English. Or by total number of speakers including those for whom it's a second, third, etc. language; that list moves English to #1 (no surprise) and pushes Portuguese down to #10.

To be fair, the Wiki page

on Portuguese does call it #6, and that's likely where this author got the number.

I'm just throwing this information out there; I've emphasized before that Cracked is a humor site, not a source for scholarly research. Neither is Wikipedia, but that's still more reliable than Cracked. I think it's safe to say, though, that Portuguese is most likely in the top 10 no matter how you look at it.

That's quite a big feat, especially for a country most people can't find on a map.

Look, I'm sorry, but if you can't find Portugal on a map, you haven't looked at a map. And I'm not entirely sure, but I think the majority of Portuguese speakers (which, again according to Wiki, are called Lusophones for some arcane reason I can't be arsed to research) are in Brazil. Which you can also find on a map if you've ever looked at one.

I'll grant, however, that Portugal is a remarkably small country to have birthed such a widely-spoken tongue. The other biggies (Chinese, English, Spanish, etc.) come mostly from larger countries, whether measured by population or land area.

Anyway. The rest of the article goes into some actual Portuguese words, none of which I was familiar with because they don't describe the most important product of Portugal, to wit, port.

I won't list all of them; you can go to the original link for that. But a few comments are warranted.

16. Saudade

The most internationally famous of Portuguese words.

Nope, this is the first place I've seen or heard this word.

It's about missing someone but with an added dash of total doom.

But I like it. If I had more time, I'd discover if it's related to the Spanish soledad. But not tonight.

14. Achigã

This is the name the Portuguese give to the black bass, a type of fish the Portuguese got from Michigan to try and fight off other predators from the surprisingly widely contested rivers of Portugal. And why the hell are we talking about a fish in a list of Portuguese words Americans should know? Well, the “Achigã” quickly destroyed all presumed rivals and became the dominant freshwater species in the country – that's as American as a fish as it gets.

Buuurrrrn.

But true.

13. Feitiço

This one just translates directly to "sorcery" or "spell," but it has a pretty interesting story. The French were the first to co-opt and rebrand it, turning it into "fetiche," and obviously making it sexual because, duh, France, and that's how Americans got the word Fetish.

I didn't spend a lot of time fact-checking this one, but it does indeed seem to follow that etymology. I'm just leaving this here because I can't rag on Americans and not rag on the French too.

12. Otorrinolaringologista

This one translates to "otolaryngologist," a doctor specializing in nose health, or whatever. What's great about the Portuguese version is how much more needlessly complicated it is because someone decided to add the word "rino" -- also an abbreviation of "rhinoceros" for the Portuguese -- into the mix.

Okay, no, come on, dude. "rino" or "rhino" comes from the Greek for "nose," and an otorhinolaryngologist (English, sort of) is commonly called an ear, nose, and throat doctor. A rhinoceros is so named because "ceros" comes from the greek for "horn," and "nose-horn" perfectly describes a rhinoceros.

The animal is understandably shortened to "rhino," but then you have things called rhinoviruses, which have nothing to do with rhinoceroseses, and everything to do with your nose.

7. Pneumoultramicroscopicossilicovulcanoconiótico

I'm just emphasizing this one here because a) apart from the suffix, English has the same word for the same thing, and b) come on.

It refers to the disease one gets from inhaling fumes out of a volcano.

Just to save you the click if you're feeling lazy.

4. Peru

We sure hope the stereotypes about Americans aren't correct because we're about to add an extra level of geographical confusion you probably don't need in your life. If you're an English speaker, Peru is the name of a country. For Portuguese speakers, however, Peru is a country, sure, but also an animal. The animal in question is ... turkey (not the country, the animal you eat on Thanksgiving).

And I'm including this one because it's amusing in My sight.

2. Actualmente

While "actualmente" sounds just like "actually" and is also an adverb (English speakers use the "ly" suffix to make adverbs whereas Portuguese ones use the "mente" suffix), it's a completely different thing -- actually. "Actualmente" actually means "currently" or "current times."

French does the exact same thing. "Actuellement" means "currently" in that language. I imagine it's because they're both Romance languages so they share the same Latin root, but I'm running out of time here.

1. Excitado

While it originally began its word life sharing the exact same meaning as "Excited," things quickly and inexplicably got off the rails. While that's not its official meaning in Portugal, whenever a Portuguese speaker hears an English speaker say "I'm excited about x," it's inevitably awkward because it reads less as "being hyped for something" and more about "being sexually aroused by x."

And again, French is the same way. Never tell a French person "Je suis excité" unless you want to doink them. Or make them run away in terror because they don't want to doink you because you're an ugly American.

So, okay, I learned a little today. Well, actually yesterday now. Well, actually some months ago when I first saved this article. And I do mean actually, not actualmente.

Any day when I learn something is a good day. Even if my ass still hurts from getting reamed at the car dealership. Je ne suis pas excité.

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