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.

2 Like

August 3, 2023 at 10:45am

#1053583

Warped Minds

Y'all should know by now that I'm a Star Trek fan. Not enough to dress funny and go to conventions, but I appreciate the various shows and movies (mostly).

But despite comments like the one I made the other day about Zefram Cochrane inventing the warp drive, I know it's fiction, mythology, and metaphor, not future reality. For one thing, I doubt very much that there are other tech-capable species in our vicinity (and there are almost certainly none that look like us but with forehead bumps), and for another...

Is Star Trek’s Warp Drive Possible?
The concept of the warp drive is currently at odds with everything we know to be true about physics.

And yes, I know that our understanding of physics is evolving and that lots of things that used to be considered impossible, aren't. So while some things are definitely impossible (counting to infinity, e.g.), I hesitate to give technological advancements that label.

Central to science fiction, and Star Trek in particular, is the ability to travel the galaxy at speeds far faster than light via a fictional technology called a “warp drive.” What is that, and will we ever have one?

Trek didn't come up with the concept; it's been around in various forms in SF for probably half as long as there's been SF (which roughly corresponds to the time frame in which we've known that matter cannot be accelerated to or past the speed of light).

A provisional answer is “no.” According to the accepted laws of science, nothing can travel faster than light.

That is, based on my admittedly limited knowledge, an oversimplification at best. But close enough for our purposes.

The distances are so vast that interstellar travel would take lifetimes.

Not if we extend our lifetimes by uploading our consciousnesses into machines... another thing that may not turn out to be possible.

That makes for a boring sci-fi plot, so the creators of Star Trek invented a convenient, but imaginary, technology.

Again, they didn't invent it, even conceptually. Early shows weren't even consistent in how it was supposed to work. They did, arguably, popularize it, though.

According to Star Trek canon, the warp drive works by creating a “warp bubble” around the spaceship, inside of which space is literally warped. In front of the spaceship, space is compressed, while behind the vessel, it is expanded. In this way, a spaceship never travels faster than light; it merely passes through a shorter distance.

The difference between "traveling faster than light" and "passing through a shorter distance" really needs more explanation here, but the article couldn't be arsed, and neither can I. Plenty of information out there, though, if you look for it.

Is this realistic? Maybe.

"Maybe" does a lot of heavy lifting here.

In 1994, theoretical physicist Miguel Alcubierre found a solution that distorted space in a way very similar to that originally envisioned by the creators of Star Trek. Under exactly the right conditions, it is possible to expand space behind an object and compress it in front.

The Alcubierre Drive is something I've seen stuff about before. There's still a difference between something being theoretically possible, and actually attaining it. For example, from what we know about quantum mechanics, it's theoretically possible for an object of arbitrary size to suddenly appear seemingly out of nowhere. The chance of this happening, though, would be akin to that of an individual winning a fair lottery... every day for a billion years.

If the universe really is infinite, though, this has happened. Hell, it might be the actual origin of the universe. Doesn't mean we can make it happen.

There is a problem, though. To accomplish this distortion, researchers would have to use negative energy—that is, reduce the energy of empty space to below zero. According to Einstein, a warp drive requires an impossible premise.

That, the negative energy bit, is definitely a serious problem. That doesn't excuse the article's appeal to authority. Einstein was right about a great many things, but we don't just take his word for it; we do experiments to back up the theory. He was human; he was wrong about a great many things. Just ask his wife.

Therefore, while scientists try to find loopholes in the conditions required for Alcubierre’s solution, most think that a warp drive will not be created this way; negative energy is a mathematical artifact and not a physical phenomenon.

On the flip side of that, for a long time, people considered negative numbers to be mathematical artifacts and not representative of physical phenomena. It took a paradigm shift to start thinking of them as "real" numbers. And don't even get me started on that extension of negative numbers, complex numbers.

Altogether, our current understanding of the laws of nature neither allows for faster-than-light travel nor an Alcubierre-like solution to warping space. However, those who dream of traveling the stars should not give up. Scientists do not have a full understanding of the laws of nature.

So yeah, I'd hesitate to call it impossible. Within the timeframe of the Star Trek series? Unlikely. Though probably not as unlikely as that other Trek mainstay, the transporter.

But that's a story for another time. I have a new episode of Strange New Worlds to watch.

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