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:

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

September 15, 2021 at 12:03am

#1017439

Uh huh. Sure.

Finally, vindication! This isn't confirmation bias talking at all.

Sarcasm Spurs Creative Thinking
Although snarky comments can cause conflict, a little verbal irony also stimulates new ideas

Yeah, sure it does.

“Sarcasm is the lowest form of wit but the highest form of intelligence,” that connoisseur of witticisms, Oscar Wilde, is said to have remarked.

No, fart jokes are the lowest form of wit.

The reason is simple: sarcasm carries the poisonous sting of contempt, which can hurt others and harm relationships. By its very nature, it invites conflict.

That's only if people detect it. Or, in my case, when I do detect it, I play it straight, like I didn't. That sometimes throws them off balance.

Sarcasm involves constructing or exposing contradictions between intended meanings.

Because no other forms of expression do that.

And yet behavioral scientists Li Huang of INSEAD business school, Adam D. Galinsky of Columbia University and I have found that sarcasm may also offer an unexpected psychological payoff: greater creativity.

Sure, yep. That's why I'm so damn creative.

And sarcasm can be easily misinterpreted, particularly when it is communicated electronically...

All of my attempts at turning Comic Sans into the Official Sarcasm Font have failed miserably.

Anyway, the article goes on to describe a psychology experiment -- like most psychology experiments, the guinea pigs were college students -- to test the author's claim in the headline.

Given the risks, your best bet is to keep conversational zingers limited to those you know well, lest you cause offense.

Nope. I'd rather piss off strangers on the internet. Not here, usually - I have other sites for that.

Anyway, it's an interesting result though, to be serious about it, I have doubts concerning the sample size and lack of diversity in experimental subjects. But it's probably worth pursuing further. Not by me, though. I'll just have to read about it (this article is five years old and for all I know, they've already followed up).

But I'm reminded of an old story -- could be a joke, could be real; I can't tell -- about a college English class where the professor was talking about the double negative. He said something like, "You can string two negatives together to make a positive, but there's no instance of stringing two positives together to make a negative."

To which one student muttered, "Yeah, right."

Mini-Contest Results!

Great comments! Thanks for all of them. Hard to pick just one that stood out, because everyone had some real gems, but I'm going with Kåre เลียม Enga's five rules (though I must note that Lilli's point about tequila is absolutely true). For everyone else, I'll do this again soon! I really want to do MB entries more often; it's just that usually I get caught up in the post and then I forget. I'll try to be better at it, at least 2-3 per month (spaced out so I don't have to worry as much about CR eligibility).

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