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

October 26, 2020 at 12:02am

#996764

Such Stuff as Dreams Are Made On

I sometimes find SyFy articles to be interesting. I started following them so I could keep up with Phil Plait, whose Bad Astronomy feature is always educational. But they have other science articles, and this is one of them.

Yes, it turns out if you die in a dream, you could die in real life. No one tell Freddy.

If you die in a dream, you'll die in real life.

Me: Oh, I guess I'm actually dead and this is the afterlife.

*looks around at all this*

Me: Wow, I must have been truly evil.

It’s one of those urban legends most of us have heard, the sort of knowledge that gets passed around the playground without being questioned. It was a meme before memes, like the knowledge that Marilyn Manson scooped his eye out with a spoon. Or that Marilyn Manson played Paul in The Wonder Years. Or that Marilyn Manson removed one of his ribs for... reasons. Holy hell, we liked to tell rumors about Marilyn Manson.

Does this guy also write for Cracked?

The "dying in your dreams" rumor persists for similar reasons, not because there’s no internet, but because it’s nearly impossible to fact check. Dreams are nebulous and fleeting and, after all, if someone did die as a result of dying in their dream, how could we know?

Fact: since we became recognizably "human," something on the order of 100 billion humans

have died.

Fact: Some percentage of people who die do so in their sleep. I'm unable to find a good figure for this. I suspect it's changed over time, anyway. Even if it's only 1%, though, that's 1 billion deaths during sleep over the last 50,000 years or so.

Fact: We don't dream during the entire sleep cycle.

Still, out of the assumed 1 billion, I wouldn't be a bit surprised if at least some people just happened to die during an REM phase that featured a dream about death. But that says nothing about causation.

The legend, as I originally heard it, was definitive. If ever you die in a dream, you will absolutely die for real. It wasn’t a suggestion or a could-be, it was presented as irrefutable fact.

Like I said, if it's fact, then I'm actually dead right now. So let's dismiss that legend like all the others.

However, if we reframe the question to whether or not it’s possible for you to die in real life if you die in a dream or, even more loosely, whether it’s possible for a dream or nightmare to kill you, the answer seems to be a qualified... yes?

Finally, an answer! Oh, wait.

First, it is possible (though unlikely) for a person to be scared to death. When we’re frightened, the body flings itself into fight or flight mode, which is triggered by a flood of adrenaline. The heart beats faster and blood flow is rerouted to major muscle groups. Particularly in those who are already predisposed, the influx of adrenaline can cause a cardiac event, which could lead to death: An ironic result from a process that is meant to keep us alive when sensing danger.

From what little I understand, rabbits apparently react badly to being cornered by a predator. If they can't fight or flee, sometimes their heart just gives out, presumably so they won't have to actually feel the teeth sinking into their little fluffy bellies. As a friend of mine once pointed out, "Worst superpower ever."

It’s unclear, and in fact unknowable, if reported SUNDS cases were the result of dreams in which an individual died, but there is some correlation between parasomnias (sleep disorders) like night terrors, and the sudden onset of death during sleep.

One of these days, it's possible that we might have the technology to record one's dreams. I wouldn't want to be the one writing the grant proposal for a study on dream-death, though. "Yes, we're going to hook sleepers up to these electrodes and study if they die when they dream about dying." Even absent such technology, though, I suspect we could have the technology to be able to tell if someone died in their sleep while experiencing the symptoms of fear:

We also know that the mechanisms exist for the heart to be catastrophically impacted by overwhelming emotions, like fear. All of which is to say, while dreaming of death is not in and of itself a death sentence, it probably doesn’t help.

It's been suggested, though I can't be arsed to look it up, that one of the purposes -- or at least one of the results -- of dreaming in humans is a kind of emotional rehearsal. Dreams are not real in the sense that the clown standing behind you right now is real, but they can certainly provoke emotions -- which are, at base, electrochemical cascades in the body. So I could certainly imagine it happening that a sufficiently scary dream about death could elicit a surge of adrenaline that might stop your heart.

I just wouldn't lie awake at night worrying about it. That clown, though... you might want to duck right about now.

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