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 11 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 13 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.

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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.

May 22, 2022 at 12:10am

#1032703

Speculations

Sigh. Here we go again.

There Used to Be Aliens in Our Galaxy, but They Killed Themselves Off
ETs have a thing for self-annihilation.

No. Just... no. That shit makes it sound like an established fact, which it very definitely is not. It does, however, appeal to the cognitive biases of many people, so of course people (other than me) nod sagely and go "Yep, exactly as I thought. And we will, too."

To be clear: There is not one shred of credible evidence that there is, or was, or will be, other technologically proficient civilizations in the galaxy. Speculating that there could have been is fine. Boldly declaring in a clickbait headline that there were is egregious pandering.

In a new study, researchers suggest the answer to the Fermi paradox could be pretty bleak: Maybe all the intelligent civilizations have annihilated themselves. Jeez, 2020, that’s a little on the nose.

Oh yeah, the article is from 2020, which I suppose partially explains the pessimism.

And sure, "maybe" all the intelligent civilizations (please do me the courtesy of sparing us any snark about how we're not intelligent either, hurr durr) have annihilated themselves. Maybe they never existed in the first place. Maybe they ascended to a higher plane of being. Maybe they visited Earth in the past and are now thought of as gods. Maybe they take joy rides in our atmosphere just to fuck with Navy pilots.

Maybe. Maybe. Maybe.

I'm pretty sure I've said this before, but the Fermi paradox isn't a paradox.

This is the Fermi paradox stated at its most succinct: The universe is unfathomably gigantic, but so far, we’ve never seen any sign that there’s intelligent life anywhere else.

Let's change "intelligent" to "technological," okay? It avoids a joke that was funny once, when Monty Python made it, and it better describes what we're actually looking for Out There. For instance, there are several other species on Earth -- macaques, corvids, cetaceans, felines -- who demonstrate what we call intelligence without displaying the slightest tendency to go and build spaceships or Dyson spheres. We'd never see signs of alien crows or dolphins on planets thousands of light-years away, because they'd be happily hanging out in their evolutionary niches, just like ours do, without sending radio signals or forcing their sun to become a message generator. Or whatever.

In short, there is not a single element in the process of evolution that requires a species to build rockets. For all we know, the chance of it could have been one in a trillion. One in a trillion trillion. One in a googolplex. We don't know because we have no baseline for it, only one data point -- us -- and the persistent gut feeling that we built rockets and we can't be all that special.

Once you've won the lottery, the chance of having won the lottery is unity. The chances of winning the lottery are irrelevant at that point, except as a way to contemplate just how lucky you got.

Why aren’t they sending telescope satellites through our part of space? And how can it be that out of all the planets and systems we’ve peeked into so far, we’ve seen nothing?

There are as many individual theories as there are theorists, and these run a huge gamut.

These aren't theories. These aren't even hypotheses. These are speculations. Which, again, is fine, but let's not pretend that we know anything about extraterrestrial life other than we don't know anything about it.

They could even be extinct. In fact, the research includes parameters for extinction and the idea of “self annihilation,” a probability that could be extraordinarily high.

Or it could be extraordinarily low. Come ON.

Here's another thing to consider: we only came about because of unique characteristics of Earth's history. No life-ending catastrophes. Some pretty big ones, sure, but, for instance, the dinosaur-killer paved the way for the rise of mammals and birds (which are descended from some dinosaurs), but, for example, we weren't fried by a nearby supernova, pushed into interstellar space, or swallowed up by a rogue Jupiter-class planet.

Any of these things could have happened, and certainly did happen to other planets in the galaxy. That, we know for as close to a fact as we'll ever get. And any of them could stop right in its tracks any nascent technology-using beings. No need to invoke self-annihilation when we live in a dangerous universe that's actively trying to kill us.

Invoking self-annihilation is just playing to the fears and fever dreams of the readers. Again, could it happen? Sure. I'd postulate -- speculate, that is -- that if there is another tech civilization in the galaxy (I'm leaving out the rest of the universe here because it's big, far away, and long ago), that only they would have the ability to self-annihilate. Dolphins don't look to be on the verge of causing the Aporpoiselypse (okay, that's a stretch, I admit it), and ravens seem to be content with messing with our heads rather than murdering us or themselves.

In other words, I suggest that technology is a prerequisite for deliberate self-annihilation, but we have no data on how likely it is.

“Since we cannot preclude the high possibility of annihilation, [this result] suggests that most of the potential complex life within the Galaxy may still be very young,” the scientists explain. That means there could be a proliferation, but it’s of other civilizations that can’t push out into the galaxy yet—just like us.

Translation: we don't know shit, but we're gonna speculate anyway.

And just to reiterate, nothing about evolution requires the emergence of a species with technological capabilities. I have no doubt we'll find "life" elsewhere. Just not, you know, Klingons or Vulcans.

So. Anyway. I know I've ranted about this sort of shit before, but it's one of my prime annoyances, so you get to hear about it again. And probably will also in the future.

3 Like • Comment

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