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.

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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.

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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.

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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.

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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.

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February 1, 2021 at 12:00am

#1003364

Time and Time Again

Not long ago, in "Time After Time"

, I speculated about time travel.

And tomorrow is Groundhog Day, which is now more famous for the eponymous movie than for the rodent's weather forecasting abilities, which in turn displaced a much older purpose for observing the beginning of February, but that's not important right now - what's relevant today is that the movie Groundhog Day featured a time loop.

I should note once again that an episode of ST:TNG did the time loop thing before GHD did, and that GHD itself was inspired by a novel whose title I've forgotten. But Bill Murray is awesome enough that GHD is the only thing people can compare any time loop movies to.

This is fine. I love the movie, myself. I just have to be pedantic about it.

Anyway. So today's link, which as usual was chosen at random, relates to both of these things. Sort of.

A Physicist Has Come Up With Math That Makes 'Paradox-Free' Time Travel Plausible

I find that paradoxes tend to resolve themselves once semantic issues are resolved -- if, that is, they ever are.

No one has yet managed to travel through time – at least to our knowledge – but the question of whether or not such a feat would be theoretically possible continues to fascinate scientists.

"WHAT DO WE WANT?!"

"TIME TRAVEL!!"

"WHEN DO WE WANT IT?!"

"IT'S IRRELEVANT!!"

"...if you go back in time and stop your parents from meeting, for instance, how can you possibly exist in order to go back in time in the first place?"

People keep going on and on about the multiverse interpretation of quantum mechanics, and I'm still not sure how that works (neither is anyone else), but that would easily resolve such an apparent paradox.

"Classical dynamics says if you know the state of a system at a particular time, this can tell us the entire history of the system," says Tobar.

Okay, this guy's a physicist and I'm not, and I'm not going to contradict his statement, but the way this quote is presented is, in my opinion, misleading. First of all, classical dynamics doesn't rule here; quantum mechanics does, with the Uncertainty Principle and all that. Second, one would have to know the state of a system to an arbitrarily large number of decimal places, which is practically, if not theoretically, impossible.

What the calculations show is that space-time can potentially adapt itself to avoid paradoxes.

This is misleading, too, implying an intelligence for which there is as yet no evidence.

To use a topical example, imagine a time traveller journeying into the past to stop a disease from spreading – if the mission was successful, the time traveller would have no disease to go back in time to defeat.

Tobar's work suggests that the disease would still escape some other way, through a different route or by a different method, removing the paradox. Whatever the time traveller did, the disease wouldn't be stopped.

Hey, that sounds like a wonderful idea for a movie- oh, wait.

Incidentally, I once saw the movie that inspired 12 Monkeys, a short film called La Jetée. Actually, I saw the short film first, in a cinema class in college in the 80s. If you can find it, it's worth watching. Hell. I might try to find it in the original French now that I have some understanding of that language.

Tobar's work isn't easy for non-mathematicians to dig into, but it looks at the influence of deterministic processes (without any randomness) on an arbitrary number of regions in the space-time continuum, and demonstrates how both closed timelike curves (as predicted by Einstein) can fit in with the rules of free will and classical physics.

No, no, it's not just you; they lost me here as well. I mean, yeah, I know what a "closed timelike curve" is, sort of, but that doesn't say much. Anyway, the thing I take issue with here is "the rules of free will." There are no such rules. Free will is taken as a given, as a basic assumption, but I'm not convinced free will is anything other than an illusion. That is, it's not a matter of predestination, but that our consciousness is the result of physical activities in our nervous systems, which in turn are subject to the rules of determinism; our decisions aren't predictable, but they are deterministic -- probably with some quantum randomness thrown in; the details are above my pay grade, but that doesn't mean we can just accept "free will" is a real thing.

"The maths checks out – and the results are the stuff of science fiction," says physicist Fabio Costa from the University of Queensland, who supervised the research.

Technically, anything can be the stuff of science fiction. Just saying.

While the numbers might work out, actually bending space and time to get into the past remains elusive – the time machines that scientists have devised so far are so high-concept that for they currently only exist as calculations on a page.

Like I said: speculative. I still don't accept that time travel is likely on the kind of large scale that captures the popular imagination (that is, anything larger than subatomic particles).

But it sure is fun to read and think about. Especially if you're a writer.

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