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 19 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 21 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.

February 2, 2020 at 12:30am

#974712

Paradoxxed

It's Groundhog Day, and later I'm going to a screening of the eponymous movie. In it, Bill Murray relives the same day over and over. It's not a trope that originated with that movie - I eventually read the book that inspired it, though it has an entirely different premise and cast of characters. Besides, ST:TNG did a time-loop episode first. Not by much; "Cause and Effect" aired in 1992, while GHD The Movie came out in 1993. Why, yes, I am a nerd; why do you ask?

Since then, the time-loop trope has become a staple of science fiction and superhero shows. There seems to be an unwritten rule (which is now written) that every show gets to do that exactly once, unless the show is about time travel.

In my life, though, there are times I seem to be looped, myself, like when I have to keep explaining the true meaning of the "blue moon," or, as illustrated by today's linked article, the question of alien life.

http://nautil.us/issue/63/horizons/our-attitude-toward-aliens-proves-we-still-th...

Our Attitude Toward Aliens Proves We Still Think We’re Special
Why we downplay Fermi’s paradox.

Um, the fact that we can conceptualize extraterrestrial life and write about it using picture-sound-symbols demonstrates that we are special.

One summer’s day in 1950, the great Italian-American physicist Enrico Fermi was having lunch with the physicists Edward Teller, Emil Konopinski, and Herbert York at Los Alamos... Then, after some delay—and, one might imagine, in the midst of some tasty dish—Fermi allegedly asked his famous question. Where, indeed, is everybody? Where are the extraterrestrials?

I'm not imagining "some tasty dish;" I'm imagining copious quantities of booze.

The Milky Way galaxy is about 100,000 light-years from edge to edge, Fermi reasoned, which means that a star-faring species would need about 10 million years to traverse it, even if moving at a very modest velocity of 1 percent of the speed of light. Since the galaxy is more than a thousand times older than this, any technological civilization will have had a lot of time in which to expand and colonize the whole galaxy. If one species were to fail in this endeavour, another wouldn’t. Consequently, if intelligent species were out there in any appreciable numbers, they would have been here already. And yet, we do not see them on Earth or in the solar system. For Fermi and many thinkers since, this constituted a paradox.

Most philosophical paradoxes are failures of using picture-sound-symbols, and not reflective of reality. Consider the ancient Greek Zeno's "paradox," which, to grossly simplify, asks the question: To get from Point A to Point B, you have to traverse half the distance. And then half again. And then half again, and so on ad infinitum. One can, consequently, never get there. If I recall the story, Aristotle at that point got up and walked out the door, disproving the "paradox." There are sound mathematical reasons why it's not a paradox, anyway, but they're irrelevant to my discussion. What is relevant is that just because we can formulate a brain-teaser doesn't mean it's a philosophical brick wall, any more than an optical illusion shows that all of reality is an illusion.

When you consider fiction and movies, it’s clear that Fermi’s paradox has become an important part of contemporary culture, challenging us to think more deeply about our place in the cosmos.

I referenced Star Trek above; that franchise can be viewed as an answer to Fermi's Paradox: its basis is that aliens quarantined us until we developed warp drive on our own, a kind of coming-of-age ritual necessary to become a galactic adult. I love Star Trek, but that doesn't mean I don't think that premise is utter crap, or that we can expect to find Vulcans and Klingons and whatnot when we do go out there.

If we believe that humans and the planet we live on are not particularly special compared to other civilizations on other planets, we would conclude that the stage of the biosphere and technology on other occupied planets must be, on average, older than the corresponding stages we see on Earth. If we humans are now on the cusp of colonizing our solar system, and we are not much faster than other civilizations, those civilizations should have completed this colonization long ago and spread to other parts of the galaxy.

And here we get to both an explicit assumption and an implicit assumption. The explicit assumption is the "If..." clause in the above quote. What hubris can make us think we're not special?

The implicit assumption is that life, necessarily, produces technological civilizations.

I'll go into a bit more detail on this. First of all, my standard definition every time I get Groundhog Dayed into a discussion like this; this definition is explicitly to forestall smartasses like me from saying "hurr durr but there's no intelligent life on Earth either." So for our purposes, I'll simplify "intelligence" to mean something like "the ability of some members of the species in question to build spacecraft." By that definition, we're intelligent. Any discussion about detecting traces of alien life implies that said life is intelligent by this definition; an otherwise sentient species who might be swimming in an ocean somewhere won't leave telltale traces for us to find with SETI or whatever.

I bring this up to segue from cosmology to biology, to wit: there is nothing inherent about the process of evolution that requires the development of intelligence; intelligence is not the end product of evolution; and there are plenty of other modes of being that facilitate species survival without the need to develop technology, as one can see by noting that, given the vast number of species on this planet, the number of intelligent species is approximately 0.

Yes, I've just said there's no intelligent life on Earth, contradicting my above definition. No, not really; I'm just saying that if there are 8.7 million species on the planet, and we're one of them, the difference between 1 and 0 is a rounding error. Point is, we haven't been around that long compared to the 4+ billion years of life on Earth, and there are other ways to survive and propagate a species than through developing bigger brains, opposable thumbs, and Netflix.

In other words, from an evolutionary standpoint, pigeons are just as successful as we are. Civilization, technology, rocket science, etc. may well be a fluke. And I'm not even going to go into the argument about how eukaryotic life was almost certainly a fluke, and how unlikely it would be for prokaryotic life to build spaceships; that would take me all night.

Basically, the idea that if life exists, then it must necessarily produce a spacefaring civilization, is spurious and about as egocentric as it gets. It's entirely possible that we won a lottery. What were the odds of the lottery? No one knows, but it's irrelevant; even if the odds are one in 10¹⁰⁰, once you win the lottery, your chance of having won it becomes unity. We exist, obviously, and even the most navel-gazing of philosophers would be hard-pressed to argue otherwise.

The consequences are rather dramatic: If life is quick to form after its host planet has formed, we get good probabilistic support for the existence of simple life on many planets in the Milky Way, and potentially complex life on some of them.

And here we come to another implicit assumption, in addition to the "complex life must produce intelligence" assumption: the conflation, ubiquitous in popular culture, of "life" with "intelligent life." I don't doubt that other planets harbor life, though we have no evidence for any besides on our own planet. But, again, life could very well stay simple.

I'm not going to rehash the author's other points here, but if you're at all interested in this sort of thing, it's worth it to read the article. I mean, yeah, I don't believe its premise or its arguments, but I've never let a little thing like vehement disagreement stop me from hearing others' opinions.

Just one more quote, from near the end:

We should not miss that opportunity by fighting for an outdated vision of ourselves as pinnacles of complexity in the universe.

As Douglas Adams noted, "Space is big. You just won't believe how vastly, hugely, mind-bogglingly big it is." I'd be surprised if there wasn't intelligent life somewhere else in the universe, if not in our own galaxy. I'd be really, vastly, hugely, mind-bogglingly surprised if there wasn't non-intelligent life, however simple or complex, in our own galaxy. But the further out you go, in intergalactic terms, the farther back in time you see - so the "argument from time duration" our author puts forth starts to fall away there. Besides, we have absolutely no evidence that superluminal travel is achievable, and overwhelming evidence that it is not, so any meaningful discussion about extraterrestrial intelligences must be limited to our local neighborhood. And let me reiterate for emphasis: "life" should not be confused or conflated with "intelligent life" (as I defined it above).

Fermi's paradox isn't a paradox at all. It's an argument against the ubiquity of alien rocket scientists.

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