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:

Write_Mikey_Write!

about 14 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 16 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.

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.

April 21, 2020 at 12:02am

#981659

Framed

I'm sharing this link, not to get embroiled in a political argument, but because this has implications for all writing.

https://www.fastcompany.com/90208548/this-is-how-tiny-changes-in-words-impacts-t...

This is how tiny changes in words you hear impacts your thinking
In a fascinating look at language, a Professor George Lakoff lays out how political parties can sway supporters with tiny tweaks in word choice.

It's not just political parties, of course; it's generally applicable to writing (including writing for video).

In 1973, America watched as then President Richard Nixon vehemently declared on national television, “I am not a crook” in regards to the Watergate scandal.

Not many people believed him.

Partly it's because of his word choices. Partly it was because everyone knew he was a crook and voted him in anyway.

The major mistake Nixon made was in his framing.

While I know what this author is trying to say here, I think we can all identify what Nixon's "major mistake" was, even if you're too young to remember Tricky Dick.

George Lakoff, a professor in cognitive science and linguistics at University of California, Berkeley, makes the point in his book Don’t Think of an Elephant! that when trying to get your point across, refrain from using the other side’s language.

Buy my book, and this guy's book too! Also, for the eleventeenth time, life isn't binary; there's more than two sides, always.

So, exactly what is a frame?

“A frame is a mental structure that is represented in the brain by neural circuitry,” Lakoff explains. Frames shape the way people see the world, and consequently, the goals we seek and the choices that we make.

I'd like to point out that "neural circuitry" is a metaphor. The brain isn't a computer. Also, that's a shitty definition, because it requires drilling deeper with more definitions.

Not that I believe that could define "frame" any better, but I'll take a stab at it, at least from a writing point of view: a frame is a mental metaphorical structure built on word associations, their emotional content, and their implications.

How'd I do?

Think about it this way: Something that has a “95% effective rate” will sell better than something with a “5% failure rate.” It’s all in how you frame it.

I've gotten into the habit of always turning statistics around in my head, anyway. It's a glass-half-full or half-empty kind of thing. Well, as an engineer, I see a glass overdesigned in capacity for its contents, but that's irrelevant. And as a drinker, I see a need for more beer.

Because subtle differences in languages might shape our thoughts and change how we experience reality, learning a new language can activate new frames. Quite simply, speaking differently requires thinking differently.

I stuck this article in my queue months ago and I'm just now getting to it, but I have a dim recollection of wanting to make a point based on this paragraph, something related to me learning French. But obviously, learning a new language hasn't helped my general memory all that much, as far as I can tell. Probably had something to do with this "thinking differently" thing being one of the reasons I decided to try learning a new language.

The rest of the article deals more directly with the political implications, specifically American politics, and that's not something I want to wade into here. It's the general ideas about framing as they relate to writing that I want to emphasize: word choice is important, and we need to consider not only the words but their possible emotional baggage.

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