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Sunday, Nov. 17, 2024

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Complex Numbers

by Robert Waltz

Rated: 18+ · Book · Personal · #1196512

Not for the faint of art.

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/

Signature for Honorable Mentions in 2018 Quill Awards

A signature for exclusive use of winners at the 2019 Quill Awards

Signature for those who have won a Quill Award at the 2020 Quill Awards

Show Recent Comments

10 Most Recent Blog Comments:

Turkey DrumStik

a day 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').

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

June 14, 2024 at 9:44am

#1072638

And Time Again

There are lots of subjects we don't fully understand. This is a good thing and it makes life interesting.

We Still Don’t Fully Understand Time

Interesting, but also amusing, because here's an article about time from Time. Or is that from Time to time?

In our everyday lives, time is a precious commodity. We can gain or lose it. We can save, spend or waste it. If our crimes are revealed, we risk having to do time.

Notice the phrasing in that last sentence: "If our crimes are revealed..." Not "If you do crime and get caught," no, straight to "You know what you did even if the authorities don't, yet."

To scientists, time is something we can measure. Clocks have, over the centuries, been the high tech artifacts of their era—the water clock, the pendulum clock, Harrison’s chronometer, and so forth up to the incredible precision of atomic clocks—marvels of modern technology, albeit without the evident aesthetic quality of more traditional timepieces.

You know, it occurred to me the other day that the first computers were clocks. When you think about it, they have the ability to compute the current time based on an input value (the time you set it at). I got this insight when I saw an astrolabe—a device that computes the movement of sun, moon, and planets—described as an early computer. But what's a clock besides a simpler astrolabe, one that only focuses on the sun with respect to the earth?

I guess it's a categorization thing, like with hot dogs or Pluto.

The article spends a few paragraphs going into the history of our conception of time, delving into philosophical, religious, and scientific matters, but not telling me anything new.

Then:

The traditional view, even among those who accept Darwinian evolution, is that we humans are necessarily the culmination of the evolutionary tree.

Yeah, that may be the "traditional" view, but there's no evidence for this egocentric idea. Not that there's any compelling evidence against it, besides pedantically wondering what "culmination" means, when we could be wiped out while other species (cockroaches being the most popular example) continue to exist.

But even in the immensely concertinered timescape that modern cosmology reveals, extending billions of years into the future as well as the past, this century is special. It’s the first in the 45 million centuries of Earth’s history when one species, ours, can determine the entire planet’s fate. We’ve entered what’s sometimes termed the ‘anthropocene.’

Actual scientists have denied the idea of the anthropocene,

but the name's stuck in public consciousness now, and good luck digging that tick out.

Perhaps our remote descendents will have a much-enhanced lifespan; they might even become near-immortal.

Or perhaps they'll lose capacity for cognitive function. We're already seeing that happen. Assuming, of course, that the collective "we" will have descendants. That "traditional view" up there assumes that evolution is about gaining functions such as what we call intelligence, but really, it's about genetic survival (that's simplistic, of course, but close enough for my argument). When "intelligence" stops being a survival trait, evolution starts to select against it.

So this is more of a philosophical essay on the state of humanity than anything meaningful about time. That's okay, though. I'd comment on more of it, but it seems I'm out of time.

3 Like • Comment • View 2 comments

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