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.

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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 13, 2021 at 12:03am

#1004357

That's My Story and I'm Sticking To It

Sometimes, I just like to have fun with these things.

Our Brains Tell Stories So We Can Live
Without inner narratives we would be lost in a chaotic world.

And sometimes those inner narratives lead to nutjob whackaloon conspiracy "theories."

We are all storytellers; we make sense out of the world by telling stories.

Occasionally, that story is "Aliens did it."

And science is a great source of stories. Not so, you might argue. Science is an objective collection and interpretation of data. I completely agree. At the level of the study of purely physical phenomena, science is the only reliable method for establishing the facts of the world.

Facts which can then be twisted into science fiction. Hey, I'm not knocking science or science fiction, but it's important to keep the two separate in one's head.

There are no naked facts that completely explain why animals sacrifice themselves for the good of their kin, why we fall in love, the meaning and purpose of existence, or why we kill each other.

Not "completely," but we have pretty good ideas about some of these. That last one, for example. It's because you're playing what you think is "music" too loudly.

For all of the sophisticated methodologies in science, we have not moved beyond the story as the primary way that we make sense of our lives.

To get serious for a moment, I happen to agree with this. But then, I fancy myself a writer.

Let’s begin with an utterly simple example of a story, offered by E. M. Forster in his classic book on writing, Aspects of the Novel: “The king died and then the queen died.” It is nearly impossible to read this juxtaposition of events without wondering why the queen died.

It's because she said, "Let them eat cake."

Okay, look, Marie Antoinette probably never uttered those words (not even the French equivalent). Supporting this, there was a story written by Jean-Jacques Rousseau long before the French Revolution where he had a princess utter those words (except it was "brioche," not "cake," but whatever). Here.

(That wiki page also has a very helpful photograph of a brioche, and now I'm hungry.)

That's a story we tell ourselves. Some stories are based on fact. That one appears to be the other kind.

Once a relationship has been suggested, we feel obliged to come up with an explanation. This makes us turn to what we know, to our storehouse of facts. It is general knowledge that a spouse can die of grief. Did the queen then die of heartbreak? This possibility draws on the science of human behavior, which competes with other, more traditional narratives. A high school student who has been studying Hamlet, for instance, might read the story as a microsynopsis of the play.

Nah, I'm sticking with the French Revolution. Vive la guillotine!

The pleasurable feeling that our explanation is the right one—ranging from a modest sense of familiarity to the powerful and sublime “a-ha!”—is meted out by the same reward system in the brain integral to drug, alcohol, and gambling addictions.

Which goes a long way toward explaining why so many writers have drug, alcohol, and/or gambling addictions.

The article goes on to discuss the science behind this. I couldn't think of any jokes about it, but it's a good read.

An efficient pattern recognition of a lion makes perfect evolutionary sense. If you see a large feline shape moving in some nearby brush, it is unwise to wait until you see the yellows of the lion’s eyes before starting to run up the nearest tree.

I'm too old to run up trees, and besides, who runs up a tree? I'd want to pet the kitty. It may be the last thing I ever do, but it'd be worth it.

Science is in the business of making up stories called hypotheses and testing them, then trying its best to make up better ones.

That's... actually a really good synopsis of science.

But there is also a problem. We can get our dopamine reward, and walk away with a story in hand, before science has finished testing it. This problem is exacerbated by the fact that the brain, hungry for its pattern-matching dopamine reward, overlooks contradictory or conflicting information whenever possible.

Hence why I'm always rambling on about confirmation bias in here. There are other biases and fallacies, of course, but that's the one I know I'm inclined toward.

Because we are compelled to make stories, we are often compelled to take incomplete stories and run with them. With a half-story from science in our minds, we earn a dopamine “reward” every time it helps us understand something in our world—even if that explanation is incomplete or wrong.

And now they've just explained 90% of the internet.

Good science is a combination of meticulously obtained and analyzed data, a restriction of the conclusions to those interpretations that are explicitly reflected in the data, and an honest and humble recognition of the limits of what this data can say about the world.

Nowhere is this illustrated better than in the insistence of certain authors to ascribe mystical properties to quantum phenomena.

The article has a lot more than I'm quoting here, and I think it's important to read. It shows, at least in part, why I'm sometimes in here ragging on science reporting, while still praising (most of) the science on which it reports.

And it helps me to remember that, believe it or not, I, too, am not always right.

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