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:

Write_Mikey_Write!

about 16 hours ago
In response to "Global Domination"

A little girl wearing a Hello Kitty shirt in Middle-of-Nowhere, Siberia will certainly make you wonder how it got there. My grandson has a souvenir t-shirt with a Terex 33-19 "Titan" on it. When it was built in 1973, the Titan was the largest, highest payload capacity truck ever built, a title it held for 25 years. The thing is, the truck on that t-shirt is on display in Sparwood, BC, Canada - over 1000 miles from here, and his mom got the shirt at a local thrift store. Makes me wonder.

Oh, yeah - I thought 'Schrödinger's Hello Kitty' was funny, too.

1 Like

Adherennium Plotting something

about 18 hours ago
In response to "Global Domination"

For whatever warped reason, 'Schrödinger's Hello Kitty' has tickled my funny bone.

3 Like

Brandiwyn🎶

Nov 12, 2024 at 10:50am
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.

3 Like

Turkey DrumStik

Nov 6, 2024 at 10:22am
In response to "Lo!"

Not gonna lie I need the holiday season chatter.

March 30, 2020 at 12:16am

#979659

Cunning Linguists

One left to do after this... entry #7 of 8 for

Journalistic Intentions

[18+]
This is for the journal keeping types that come to PLAY! New round starts July 1!
by Turkey DrumStik

How language shapes the way we think

This harks back to a post I made in the Before Times, way back on March 1: "It Doesn't Count"

. It's something that's long been fascinating to me, so I'm glad this one came up in my random walk through the prompts.

Now, I can't be arsed to look it up, but the idea that language shapes thought has its detractors -- or, at least, that it's as pronounced (pun intended) as this speaker asserts. Me, I'm no expert; I'm just drinking from the fountain here.

Part of my own cognitive problem is that I've never bothered to dive deep into any language except my own. My knowledge of idioms and grammatical constructions is very English-oriented, and American English in particular. I can play with it, twist it, and hopefully get my points across even if I don't hew to standard grammar and phrasing -- but it's definitely English.

What little I've learned of French in this regard has been enlightening. For example, we Yanks might say something like, "I have a headache" or "My head hurts." Well, the equivalent French phrase, strictly translated, would be something akin to "I have bad in head" or "I have wrong in head" (the bad/wrong word being mal). If I told someone I had a wrong in my head in English, they'd assume I need to be committed somewhere. I should note that I'm far from fluent in French, so I could have mal about all of this, but that's what it comes across as at this point in my learning.

Point is, that's a language that, as these things go, isn't all that dissimilar from English. I mean, obviously most words are different, English is less inflected, and the French syntax is often scrambled from an English point of view (ours is scrambled from their point of view), but for the most part, there are parallel parts of speech; things are countable; there's a left and right, up and down; colors are recognizable; and so on.

But that makes me pose a different question: why are there different languages in the first place? Language, obviously, evolves in communities that are separated. In that sense, it's similar to, but much faster than, biological evolution. But language also has a feature not usually found in (eukaryotic) evolution, which is cross-pollination. In biology it's known, as far as I understand it, as "horizontal gene transfer," and apart from genetic engineering, you just don't get a lot of that in nature (for the purists out there, it happens a lot in bacteria and other prokaryotes, but I'm not talking about them). But with language, horizontal meme transfer (using "meme" here in the sense that Dawkins originally coined the word) happens all the time.

Answering that question is above my pay grade, though the one thing I can say with a high degree of certainty is that it wasn't God's punishment for building a fucking ziggurat. (If it were, then why did language keep on evolving long after Babel?)

I can only hypothesize that it was groups of people who separated themselves and started thinking independently of each other. In which case, which is more true: that language shapes thought, or that thought shapes language?

I suspect there's a bit of both going on. Our ability to think in new ways

when learning a new language is, well, very human.

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