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:

Write_Mikey_Write!

about 8 hours ago
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

about 10 hours ago
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

Turkey DrumStik

Nov 6, 2024 at 10:22am
In response to "Lo!"

Not gonna lie I need the holiday season chatter.

January 29, 2024 at 11:13am

#1063143

Ice Sea Moons

Mostly just because I find this stuff fascinating, an article from Quanta:

These Moons Are Dark and Frozen. So How Can They Have Oceans?
The moons orbiting Jupiter and Saturn appear to have subsurface oceans — tantalizing targets in the search for life beyond Earth. But it’s not clear why these seas exist at all.

Usual disclaimer here: by "life" they mean probably simple life; complex life is unlikely, and sentient aliens are pretty much off the table. So let's not go imagining little green women.

For most of humankind’s existence, Earth was the only known ocean-draped world, seemingly unlike any other cosmic isle.

Counterpoint: we barely knew other worlds existed, what the stars and planets really were, and certainly not that they had satellites.

Now Earth’s oceans are no longer unique. They’re just strange. They exist on our planet’s sunlit surface, while the seas of the outer solar system are tucked beneath ice and bathed in darkness. And these subterranean liquid oceans seem to be the rule for our solar system, not the exception. In addition to Europa and Enceladus, other moons with ice-covered oceans almost certainly exist as well.

Really, it shouldn't be so strange that there's a lot of water out there. Even before they made these discoveries about outer solar system worlds, we knew that comets were largely water (in the form of ice).

These moons have existed in the frosty reaches of our solar system for billions of years — long enough for residual heat from their creation to have escaped into space eons ago. Any subsurface seas should be solid ice by now.

This sort of thing is, of course, what science is for.

These oceanic moons also offer a grander possibility. Temperate, potentially livable oceans could be an inevitable consequence of planet formation. It may not matter how far a planet and its moons are from their star’s nuclear bonfire. And if that’s true, then the number of landscapes we might explore in our search for life beyond Earth is nearly limitless.

I should note that I saved this article to my queue before I started playing Starfield.

“Oceans under icy moons seem weird and improbable,” said Steven Vance, an astrobiologist and geophysicist at NASA’s Jet Propulsion Laboratory.

And yet, defiantly, these alien seas remain liquid.

Side note: Every time these days that I see "defiantly," my assumption is that someone doesn't know how to spell "definitely." This is not the case here.

Now, it might seem, on the surface (pun intended), not so odd that an ice-covered moon would have liquid water underneath. After all, Earth is rocky (though much of those rocks are underwater), and yet has a molten core. Water is molten ice. But it's not that simple; we're talking about globes much smaller than Earth, worlds whose inner heat has had more of a chance to dissipate.

The article goes on to explain some theories that fit the evidence, but I won't go into more detail here. In summary, other factors produce heat, such as tidal friction.

As one of the prime candidates for under-ice life in the solar system is Europa (moon of Jupiter), the only thing I have left to note is that it's a good thing Arthur C. Clarke was writing fiction when he penned: "All these worlds are yours - except Europa. Attempt no landing there." Because telling humans they can't do something is a sure way to get them to do it.

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