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

September 29, 2023 at 11:19am

#1056392

Ring World

It's been a few weeks, I think, since the last time one of the solar system articles came up from my queue. This one's about Saturn.

Saturn Could Lose Its Rings in Less Than 100 Million Years
Recent discoveries suggest that the planet’s distinctive feature may be gone in the cosmic blink of an eye

I guess this is considered noteworthy because, with a few notable exceptions such as comets or sunspots, we tend to think of astronomical objects as relatively unchanging. And, from a human perspective, 100 million years is a very long time indeed. 100 million years ago, our ancestors were mouse-sized quadrupeds, hiding from dinosaurs. I did an entry a while back on Jupiter, that other gas giant in the solar system, and we've witnessed the shrinkage of its iconic Great Red Spot over the past couple hundred years—in comparison, lightning-fast. Now it might be more properly called the Okay Red Spot.

Anyway, back to Saturn...

If someone asked you to draw a planet other than ours, you would likely draw Saturn, and that is because of its rings.

Rings are very common in SF art, to indicate "hey, a different world." And yet, of all the planets we're most familiar with, only the one has easily identifiable rings.

It was Galileo Galilei who first spotted something there. His primitive telescope gave him only a slightly better view of the heavens than did the naked eye, and in 1610 he thought he saw two undiscovered bodies flanking Saturn, one on each side.

I think he called them "ears." Some of his, and others', early drawings are reproduced in the article. As the article notes, when they're edge-on, they basically disappear from Earth-based views.

The particles in the inner rings move faster than those in the outer rings, because they are fighting against a stronger gravitational pull.

That's... well, that's mostly correct. Except they're not "fighting against" anything.

The average thickness of the main rings is believed to be no more than 30 feet. A recent study showed that parts of the B-ring—the brightest ring of all—are only three to ten feet thick.

I've known this for a while, but it still amazes me. Considering the overall breadth of the rings, this makes them far thinner, proportionately, than a sheet of paper.

Most of Saturn’s rings lie within what’s known as the Roche limit—the distance a satellite can orbit a large object without the planet’s tidal force overpowering the object’s own gravity and tearing it apart.

Which might well be how they formed in the first place.

The article goes on to describe how one researcher figured out (probably) how the rings were disappearing. In 2012. Just goes to show that there's no apparent end to discovery.

There's also lots of cool pictures of the rings, of course. From close-up. Because we've sent robots there. We're also treated to what I consider an excessively long biography of the researcher in question, which, admittedly, I skimmed.

“Take a Good Look at Saturn Before It’s Too Late,” Time magazine cheekily warned, “Because It’s Losing Its Rings.”

And this is why I have issues with most science reporting. There's another article in my queue, which I'll get to eventually, about the recent discovery of potential life-indicating chemicals on an exoplanet. Popular media immediately jumped to techno-aliens in flying saucers.

But. This article shies away from such sensationalism.

Because the reality of it is sensational enough. I mean... just look at those fucking rings!

4 Like • Comment

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