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.

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

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

July 31, 2023 at 9:34am

#1053369

Underworld Underdog

Today, another bit about our solar system, this one certain to be controversial.

Schrödinger’s Planet
We’ll never have a final answer on whether Pluto is a planet.

The article, from The Atlantic, is two years old, but Pluto's label hasn't changed since then.

In 2006, astronomers gathered in Prague to consider a very basic question: How many planets are in our solar system?

They needed to Czech in with each other.

For many Americans, the names of planets were some of the earliest scientific facts we learned, and that there were nine of them seemed like a basic truth of existence.

This is a failure of science education in America.

There was no consensus among [astronomers] then, and there is none now. Ask one astronomer and they’ll sigh, before saying that it’s time to stop dredging up the past and move on. Ask another and they’ll say the matter desperately needs a do-over. They agree on so much about the cosmos, but on a matter that seems as though it should be straightforward, some of them might as well exist in two different solar systems.

You know, I bet you can get them to agree, within a significant figure or so, about the mass of Pluto, the length of its year, the period of its mutual orbit with Charon, and so on. This is, as I've noted before, a classification problem, akin to deciding whether to call a stone a "rock" or a "pebble."

In 1801, astronomers deemed Ceres, a rocky object they had spotted between Mars and Jupiter, a planet, but 50 years later, after further observations, they designated it as an asteroid.

This is the bit no one remembers. Unsurprising, as I don't think anyone now was around then.

Then, in the 1990s, astronomers started finding icy objects beyond Neptune. “People who were paying attention immediately said, Oh, we get it. This is what Pluto is,” Mike Brown, a Caltech astronomer who discovered one of these celestial bodies, told me. “Pluto is not this oddball at the edge of the solar system; Pluto is part of this larger population.” In 2000, when the American Museum of Natural History, in New York, opened a new space wing, curators tagged Pluto as part of the Kuiper belt, the disc of objects floating past Neptune, a decision that prompted a flurry of hate mail from Pluto fans.

Make Pluto Planet Again!

To Brown, the International Astronomical Union’s judgment was brave, considering how the public would perceive the loss of a beloved world.

Pluto isn't lost. We've even sent a space robot flying by it. It hasn't gone anywhere. Well, I mean, sure, it's progressed a bit along its orbit, but it's still in the orbit.

Some planetary scientists, whose work doesn’t center on such orbital details, say that there’s much more to a planet than that. They argue that Pluto has a host of characteristics that qualify it for planethood: an atmosphere, geological activity, even five of its own moons.

"A hot dog is meat between hunks of bread. Therefore, it is a sandwich, has always been a sandwich, and will always be a sandwich!" "No, it's a taco." "Heretic!"

"You call these things 'mountains'? Come out west, I'll show you mountains!"

"That's a river." "No, it's a stream." "No, it's a creek."

New Horizons flyby in 2015 make discounting Pluto difficult. The mission gave humankind its first close-up of Pluto, uncovering stunning topography, such as a massive heart-shaped glacier made of nitrogen. Fran Bagenal, a planetary scientist at the University of Colorado at Boulder, emailed me a picture from the flyby and wrote, “Doesn’t this look like a planet to you?”

This is Ceres.

It orbits the sun. It looks like a planet to me, too, being round and having surface features and shit, although admittedly not as visually striking as those of Pluto. As noted above, Ceres was a planet for half a century. As with Pluto, changing what it was called didn't change Ceres.

For all their disagreements, everyone I spoke with was on the same page about one thing: that the question of planethood has no bearing on whether Pluto is a fascinating place to study.

It could be argued that this "controversy" worked very well to get more people interested in space, which I absolutely approve of.

For anyone still wondering why it matters what we call a planet, dwarf planet, asteroid, etc., well, it really doesn't. But it's nice to have a working definition as we discover more and more bodies orbiting other stars. While, to the best of my knowledge, we don't have the technology to detect extrasolar dwarf planets, these questions of classification will go with us to the stars... assuming we make it that far.

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