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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Turkey DrumStik

about 5 hours ago
In response to "Barley There"

And approximately 15 seconds later, someone's going to screech about genetically engineered crops.

That would be me...screeching for more genetically engineered crops.

Write_Mikey_Write!

Nov 14, 2024 at 2:04pm
In response to "Global Domination"

A little girl wearing a Hello Kitty shirt in Middle-of-Nowhere, Siberia will certainly make you wonder how it got there. My grandson has a souvenir t-shirt with a Terex 33-19 "Titan" on it. When it was built in 1973, the Titan was the largest, highest payload capacity truck ever built, a title it held for 25 years. The thing is, the truck on that t-shirt is on display in Sparwood, BC, Canada - over 1000 miles from here, and his mom got the shirt at a local thrift store. Makes me wonder.

Oh, yeah - I thought 'Schrödinger's Hello Kitty' was funny, too.

1 Like

Adherennium Plotting something

Nov 14, 2024 at 12:10pm
In response to "Global Domination"

For whatever warped reason, 'Schrödinger's Hello Kitty' has tickled my funny bone.

3 Like

Brandiwyn🎶

Nov 12, 2024 at 10:50am
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.

3 Like

December 29, 2018 at 12:46am

#948396

Brains

Today's science article is, I'm afraid, fairly long - but it's written to be easily understood:

https://www.nytimes.com/interactive/2018/11/15/magazine/tech-design-ai-predictio...

Randy Buckner was a graduate student at Washington University in st. Louis in 1991 when he stumbled across one of the most important discoveries of modern brain science. For Buckner — as for many of his peers during the early ’90s — the discovery was so counterintuitive that it took years to recognize its significance.

Now, there's a lot to unpack at that link. The title, I think, is a bit sensationalist: "The human brain is a time traveler."

Well, as it turns out, kind of. What he means is only that we have the ability to contemplate both past and future, sometimes skipping between the two in less than a heartbeat. Okay, fine. Then he proposes that, perhaps, this ability is what makes humans human and not, say, sponges or dogs.

Jury's out on that. Every time someone comes up with a single thing that claims "This... this is what gave us the ability to go to the moon / create reality TV / contemplate quantum mechanics / build skyscrapers / whatever," someone else comes up with an example of someone in the animal world having the same ability. Language. Tool use. Opposable thumbs. Etc. Apparently, it's not any one attribute, but the whole complex of them.

But, anyway, let's take this at face value for now. Like I said, there's a lot to read in that article, but I'd like to pay particular attention to something that's more between the lines.

I've been hearing a lot more lately about "mindfulness," usually in the context of "living in the moment" or in the present. This idea has never sat very well with me. Based on the information in this article, it turns out I may be onto something.

First of all, I don't think there is a "present." Or, if there is, it's something akin to the infinitesimals they use in calculus - something arbitrarily small that helps us to calculate a broader function. And I've just lost at least half my audience, so I'll move on to my next point.

Seems to me that, given this information about how our brain works in its resting state, any attempt to live in the present makes us less than human. And while there are a lot of things that humans do that we ought to be ashamed of, is the answer really to attempt to revert to some sort of pre-cognitive state?

I don't know. I did say I was biased. But I'll continue to contemplate the past and the future, because, as the author points out - that's the essence of storytelling.

Sure, the future is uncertain, and the past can be murky because our memories aren't exactly precise. But they're all we've got.

1 Like • Comment

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