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 9 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 11 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.

June 15, 2021 at 12:02am

#1011896

I Expected This

One of the cool things about science fiction is that we SF readers are rarely surprised by new tech.

This tech was science fiction 20 years ago. Now it’s reality.

This is not always a good thing.

This article describes some things that some of us use every day, or at least is just over the horizon, but almost all technology was first imagined in science fiction.

Not only is today’s tech far more powerful than it was 20 years ago, but a lot of the gadgets we thought of as science fiction have become part of our lives. Heck, in some cases, this technology has become so ubiquitous that we don’t even think about it as being cutting-edge tech.

This "we" you're talking about, I do not believe it means what you think it means.

Self-driving cars

As recently as 2004, well-respected economists Frank Levy and Richard Murnane of the Massachusetts Institute of Technology and Harvard, respectively, argued in their book The New Division of Labor that driving a vehicle was a task only a human could do.

Just as you can almost always find a science fiction prediction for any given invention, you can often find examples of people who claim it's unlikely or even impossible. Breaking the sound barrier comes to mind. Hell, even cars -- I remember seeing someone's prognostication that cars will never go above some arbitrarily low speed because the human body just can't deal with higher velocities. Well, it's true there's a limit, mostly having to do with our perceptions and reflexes, but it's much higher than the naysayers predicted.

A caution though: the reverse isn't true. Sometimes when someone says something can't be done, it can't be done.

Now, I could write an entire blog entry - hell, a series of blog entries - on the subject of self-driving cars, but this isn't the place for it. Suffice it to say, for now, that I, for one, welcome our new autonomous vehicle overlords.

Universal translators

The thinking person’s ray gun, the universal translator is the pocket-sized gadget seen in many a science fiction movie. Today, devices like smart earbuds can do a pretty great job of real-time, speech-based translation. A person can also point their camera phone at text written in another language and get it translated back to them right away.

Admittedly, this has been getting better. I've noticed that, for instance, the old standby Google Translate has improved over the years, and there's no logical reason why improvements couldn't continue.

However, there are real advantages to learning other languages. Even if a UT could be as perfect as the ones on Star Trek (except, of course, when the plot calls for it to be less than perfect), it's still a boon for personal growth to immerse oneself in another language.

Personalized advertising

This one’s not quite as sexy as the others on this list. But that also just goes to show how ubiquitous it’s become. In the 2002 movie Minority Report, there’s a scene in which Tom Cruise’s protagonist John Anderton walks past a billboard that screams out to him, “John Anderton! You could use a Guinness right about now.” A couple of decades later, the idea of an ad that addresses us personally, using our name and targeted based on our expressed likes and dislikes, is an everyday part of our lives.

The other purpose of science fiction is to warn us away from undesirable outcomes. Stories about cloning, or flubbing First Contact situations, or developing fully autonomous AI androids (you know, like in that other PKD-inspired movie), are meant as warnings.

However, I'm not as opposed to targeted advertising as some people are. While there are definitely privacy concerns, I'd rather see ads for men's clothing, beer (I wonder how much Guinness paid for that particular product placement), video games, travel services and the like than for things like laxatives, tampons, or monster truck rallies. Not that there's anything inherently wrong with those things; I just have no use for them myself, so don't waste your money and my time trying to sell me them.

That said, yesterday I was having lunch with a friend. At one point, we discussed maybe going to Vegas (if, you know, there's still water there when we're able to go). Our phones dinged a couple of minutes later, and we both got promotional emails from casinos.

Probably a coincidence.

Probably.

Touchscreens everywhere

A touchscreen used to be as much shorthand for “you’re watching a sci-fi show or movie” as form-fitting jumpsuits and aliens with weird foreheads. While touchscreens did exist in the real world (the first resistive touchscreen technology was shown off in 1982), it was nowhere near as buttery smooth or intuitive as filmmakers liked to suggest.

Touchscreens annoy me. Give me a laptop with a keyboard and mouse; I can never get my finger to put a cursor in the right place on a touchscreen. The screen keyboards are clunky as fuck, and autocorrect was clearly programmed by Loki. Still, there's no other way to do smartphones, really, and the rest of the interface is intuitive enough.

This one might be a cheat, though. I don't remember seeing much in the way of touchscreens in SF before ST:TNG, which came out after 1982. Also, I distinctly remember prototype touchscreens in meatspace before that year, but they may have used different tech.

Voice assistants

Rarely do as much as I wish they would, and half the time they don't understand what I'm saying. That said, when I'm sitting at a bar and hear a song I like that I don't recognize, being able to go "Hey Google, what song is playing?" makes me feel like I'm living in the future.

Video calling

Meanwhile, most people have become more accustomed to video chat platforms like Zoom, Hangouts, and Skype over the past year than they ever thought they would. Video calling is so ubiquitous that, in the age of working from home, it’s quite possible to unironically say, “Not another video call? This is my fifth this week,” prior to picking up a tablet computer thinner than a deck of cards, and having a conversation with people sometimes located on the other side of the planet.

I might have said this before, but they tried rolling out video calling back in the 70s, if I recall correctly. Even had prototypes. People freaked the hell out. I mean, you do them in SF movies so you can see the actors' facial expressions without having to do that annoying split-screen shot that was ubiquitous in movies and TV of the time, or just to make something look like The Future. But I remember when they tried to promote it, there was this general consensus of "No way! What if the phone rings while I'm in the shower?" Because at the time we were conditioned to a) always answer the phone (no voice mail, no answering machines, could be something important) and apparently the idea that you could flip the camera to "off" (or even cover it like we all do with our webcams now, just in case) never occurred to anyone.

Robot exosuits

No, thanks.

I'm kind of surprised GPS didn't make this list. It might be the most technically advanced, in some ways, of any of our everyday gadgets, and it was a niche thing at best 20 years ago. Of course, some people still freak out about it ("I heard about someone who followed their GPS and drove into a lake!" as if no one ever got lost or stuck trying to follow a paper map), but it's essential for a lot of things right now.

Unless something even worse than a pandemic occurs, though, things will keep being invented and perfected. Want to know what things? Read more science fiction.

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