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.

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').

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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

February 19, 2021 at 12:07am

#1004786

And the Spiders from Mars

Well, it looks like I didn't get as astronomical as I expected, so here's my usual just-after-midnight post. Last night, after completing my blog entry, I ended up getting completely danchu, sampling from many different bottles. Predictably, I felt like shit all day (worth it). Hence, tonight, I ended up just taking my traditional couple of shots of birthday tequila, and I also had some port with dessert. Consequently, I'm functional right now.

I'm sure you've all heard about the successful Perseverance rover landing on Mars yesterday. Technology works, and we can do some pretty awesome shit when we put our minds to it. By the time we finally land human boots on the fourth planet, there will be a thriving robot society there. They'll probably pass resolutions to send us back home, or maybe put us in concentration camps to discourage further immigration.

So with the success of the landing, which is designed to search for signs of past life on the Red Planet, it's appropriate that this subject came up at random from my stash.

Where is the informed discussion of alien life on the internet?

I'm somewhat sure that there is no alien life on the internet, politicians' tweets notwithstanding. Oh, wait, that prepositional phrase is meant to modify "discussion," not "alien life." English is fun.

Anyway, this blogger, PZ Myers, is one of the few that I follow regularly, because he usually has something interesting to say, even if he does have an obsession with arachnids. Fortunately, spiders are (we think) from Earth so there aren't any in this particular post.

I like speculations about alien life, just as I appreciate the diversity of life on Earth, the different forms of life in the past, and the prospect of evolution in the future, but every time I read about this stuff in astronomy-related journals, I feel like they’re making an effort to reduce my intelligence.

I know the feeling.

The problem is that they have no imagination and no biology, but they’re trying to imagine the nature of alien biology, and all they end up doing is running around in circles trying to figure out why little grey humanoids aren’t landing their flying saucers en masse to march out and shake hands with the president.

I know I've ranted about this sort of thing in here before, so here's another guy's take on the subject.

So, in this post, Myers does what I usually do, which is comment on someone else's article. So here I am commenting on an article that comments on an article. If someone quotes this entry and comments on it, that would be really meta.

This is my problem with the general tenor of these speculations. They all assume that we, that is human-like intelligences, are desirable, inevitable, and the only proper kind of life; they’ve read far too many science-fiction novels prophesying a colonialist destiny led by strong-jawed Anglo-Saxons with glinting eyes and a finger on the trigger of their blaster.

I mean, hey, I love those stories as much as anyone, though I prefer phasers.

They’re always going on and on about the likelihood of finding an intelligent civilization like ours. Why not speculate about finding a planet that has produced kangaroos? Or stomatopods? Or baobab trees? These are all unlikely outcomes of a contingent, complex process that produces immense diversity, and they’re all wondering what the “barrier” is that prevents our kind from winning the cosmic lottery every time. Get over it, we’re not a favored outcome, there’s no direction to evolution, and that’s why there aren’t smarty-pants bipeds tootling about the galaxy stopping by for tea.

I kinda hate that Myers puts this more eloquently than I have in the past. But I agree.

Anyway, I'm not quoting the quotes here, and I'm leaving out a lot of the commentary, so of course I recommend reading the source. Whether you read the article he's ragging on is up to you; I certainly didn't.

Point is, though, that this is a biologist's perspective on the question, but it's similar to one I came up with previously: that if we find life "out there" -- be it living life, or, as is hoped with the Mars robot, evidence of life that existed in the past -- it will, in all likelihood, be something other than a technological civilization. And yet, we've been so conditioned to equate "life" with "intelligent life" (as usual, please refrain from making silly dismissive comments about "intelligence," as the fact that you can make such a comment immediately contradicts it) that it's almost impossible to find someone who doesn't immediately think of bulbous-headed big-eyed gray guys, or Klingons, or other generally humanoid, tool-using, spaceship-flying aliens.

That said, I really hope this Mars rover finds unequivocal evidence of some sort of living, or once-living, organism on Mars. Because that by itself would be an incredibly important discovery, never mind what sorts of things we might learn from it.

Mini-Contest Results!

Thanks for all the mixed drink recipes! I definitely saw a few I'd like to try. Rolling the Virtual Dice for the nine commenters (sorry, one comment came in just after midnight), the result is 3, so the Merit Badge will go to... Cubby

for the mojito recipe (and I do love mojitos).

Still, I appreciated all the comments and we'll do this again soon!

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