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

March 15, 2021 at 12:01am

#1006406

The Ideas of March

Oh, wait, that's supposed to be "Ides," not "Ideas."

Here's the thing: I've been up way longer than usual at the moment; I've had a couple of beers, and I've been busy doing stuff all day. So I'm a little burned out.

I pick articles at random to feature here. As luck would have it, just now, my usually friendly random number generator had a good laugh at my expense, and came up with this ponderous pondering:

The mathematics of mind-time
The special trick of consciousness is being able to project action and time into a range of possible futures

By the way: If you're put off by the word "mathematics," don't be; there's not a single equation in the article. As far as I can tell, it could just as well have been titled "The philosophy of mind-time."

This was fascinating when I first found it, for some reason, and it continued to stretch my brain today. The only problem is, today, my brain is already stretched as far as it can get right now.

So... I'm just going to leave this here. I hope someone else will be interested. I neither agree nor disagree with the article, incidentally; I just like the way the author presents their case. This stuff is seriously above my pay grade, but at the moment I can't even think of ways to riff off of it for humorous effect.

Yeah... I know... I'm cheating today. I'll give myself this one. Hopefully, I won't make a habit of it.

I'm going to go turn off my consciousness now. Tomorrow we'll see if I can make jokes, or at least be coherent.

2 Like • Comment • View 1 comment

March 14, 2021 at 12:01am

#1006352

Make It So

I don't know if I can do this one without offending someone. But I'm going to try.

TV’s first interracial kiss launched a lifelong career in activism

On Nov. 22, 1968, an episode of “Star Trek” titled “Plato’s Stepchildren” broadcast the first interracial kiss on American television.

So, that would be five years to the day after the JFK assassination, well into the civil rights / desegregation era. Laws changed. Lots of peoples' minds did not.

I was too young to watch it when it came out, but of course when I got old enough to catch reruns, I saw that and all the other episodes of Trek. I don't think it ever occurred to me to note the "interracial" aspect, and I was certainly ignorant of the history behind it. Kid Me was too busy being bothered by other aspects of that particular episode, as we'll see in a bit.

The episode’s plot is bizarre: Aliens who worship the Greek philosopher Plato use telekinetic powers to force the Enterprise crew to sing, dance and kiss.

You know, as far as Trek episode summaries go, that's pretty tame. I'm trying to imagine being Roddenberry or Coon or whichever producer was responsible for hearing all the elevator pitches (yes, there were elevators back then), and having to go, "No... weirder."

Anyway, that was the part that I recall being scarred by as a child: the idea of aliens using mind powers to force others to do their bidding. Dance, my puppets! DANCE! (The episode, by the way, is worth watching just to see Nimoy doing his "I'm being forced to do this" dance.)

It was only later that the cultural significance of the Shatner/Nichols scene was pointed out to me. I mean, I lived in the South, not far from where the "Loving" in "Loving v. Virginia" came from, so I wasn't exactly sheltered from racism; I just thought of it as something only ignorant people believed in.

I guess I still do.

The smooch is not a romantic one. But in 1968 to show a black woman kissing a white man was a daring move.

As I see it, one of the main purposes of science fiction is not to show how things will be, but how they could be. This includes social change.

But just as significant is Nichols’s off-screen activism. She leveraged her role on “Star Trek” to become a recruiter for NASA, where she pushed for change in the space program. Her career arc shows how diverse casting on the screen can have a profound impact in the real world, too.

The rest of the article pretty much focuses on Nichelle Nichols, which as far as I'm concerned is a good thing. But it's impossible to talk about Nichols without mentioning Uhura, and especially that episode of Star Trek. So it's good to get that bit out of the way.

Nichols’ controversial kiss took place at the end of the third season.

Flag on the play: it was the middle of the third (and final) season.

It's worth taking a look at the bits I'm not quoting here; the article is pretty short and details Nichols' efforts to advocate for inclusion.

Star Trek is, of course, fiction, but the stories we're told influence the way we see the world. The show was far from perfect, but it tried to demonstrate the benefits of diversity at a time of great social change. And who knows; maybe someday life will imitate art.

4 Like • Comment • View 6 comments

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