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.

2 Like

August 27, 2024 at 11:01am

#1075789

They Invented It; We Perfected It

In a twist of cosmic coincidence, today's article is also from Cracked and is also about opposites.

5 Words That Mean the Exact Opposite If You Go to England
Jack Reacher makes no sense, and to understand why, you need to understand British English

Clearly meant for an American audience. If a British rag had written this article, it would have been "5 Words Yanks Get Backwards."

England is a strange land, where they speak a language known as English. You might have trouble understanding what anyone there is saying.

Strewth.

For example, you might hear someone described as “mean,” and you think that refers to how they say cruel things. You later realize it really refers to how they’re stingy with money.

In fairness, those traits often go hand in hand.

5. Tabling an Issue

In America, when someone says, “Let’s table that discussion,” they mean, “Let’s stop talking about that for now.”

Except these days, you'd say "let's put a pin in that and circle back later."

In Britain, when someone wants to table something, it instead means they want to discuss it right now.

And this nicely illustrates the hazards of verbing nouns. "Take it off the table" or "put it on the table" would be close equivalents, and less ambiguous.

4. Public School

If you went to public school in the U.S., that means you went for free, in a school set up by your local government.

Generally badly.

In England, however, the term “public schools” refers to the most elite schools in the country, places like Eton and Harrow... They’re run by private institutions and charge fees.

I've known about this difference for a while, and I gotta say, in this case, I'm going to side with the US.

But it's not like either side of the pond is going to change its usage soon, so it's just important to know there's a difference. The way I remember it is that a pub, originally public house, is also privately owned.

3. The Doughnut Effect

One of the few delicacies that doesn't have a different name in the two Englishes. We say cookie; they say biscuit. They say donut too; they just don't use the lazy spelling.

However, this item isn't about delicious treats, but about how cities grow rings around them... differently in each country. Oh, just read it..

2. Luck Out

If you luck out, that means you stumbled into some good luck.

That is, obviously, the US version. As one should expect by now, the UK version is the precise opposite.

Though the character Reacher is American, the author Lee Child is British. It appears here that the author is using “luck out” in the exact opposite way from how Americans do.

I'm mostly just including this because the subhead up there called out Jack Reacher.

And this is not an invitation to discuss the literary merits, or lack thereof, of the Reacher books. Which I've never read, but I've enjoyed the Amazon adaptation so far.

1. A Moot Point

When I was a kid, I read books in both American and British English, which is why I'm largely bilingual now. But I gotta admit, this one confused me for the longest time.

If a British person says a point is moot, they mean it’s up for debate, while when an American says it, they mean it’s closed for debate.

The article actually goes into some of the semantic reasons for that, which is one reason I read Cracked.

This one's similarity to the "table" one is, however, too obvious not to comment on. But the article did that for me, too.

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