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 14 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 16 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.

April 26, 2022 at 12:01am

#1031393

Engage Tractor Beam

Every so often, I'll see an article relating scientific advancement to ideas from science fiction in general and Star Trek in particular. This is one of them.

Star Trek has tractor beams. So do we.
But so far, they can't grab anything bigger than a dime.

And, like all the rest of them, it's disappointing and ends up increasing, rather than decreasing, misconceptions about physics.

Like, a while back, there was a slew of articles about teleportation. Of a single particle, but with all the hype attached to it, I'm sure people thought transporters were just a few years away.

Spoiler: they're not.

This is similar.

The “tractor beam” has been a reliable narrative device in science fiction for nearly 100 years, deployed whenever the plot requires seizing a runaway spaceship or manipulating objects at a distance.

That's right; Trek didn't invent the concept. As an aside, as a kid, the phrase "tractor beam" in Trek confused me. Being on a farm, I knew damn well what a tractor was. It was only later, after four years of Latin in high school, that I finally figured out that it comes from the same root as "attractor."

Okay, maybe it didn't take me that long, but remember, I started watching Trek at a very young age.

Anyway, the root, trahere, means "to pull," and we can already do that mechanically, or with magnets.

But most sci-fi fans probably know it from Star Wars and Star Trek — the sinister Death Star and the mighty USS Enterprise each boasted a frequently convenient tractor beam system.

Ugh.

Star Wars is great, too. I don't take sides. What I do insist upon, though, is that Star Wars isn't science fiction; it's fantasy with SF props.

The first thing to know about real-life tractor beams is that they work more like another sci-fi concept: force fields.

No. No, they really do not. As the article explains, it's sorcery with light and sound waves. The completely fictional idea of force fields is about energy without mass. One could argue, of course, that creating a barrier of focused sound waves would do a lot of what a force field is supposed to do; that is, keep people and things from crossing it.

Anyway, look, I'm not knocking the science, and I'm only knocking the article for being a little misleading up front; the author does go on to explain the technology in some detail (without getting too technical). As the author points out,

We won’t be using tractor beams to yank around Corellian-class smuggling ships, but we could certainly use them to accomplish touchless manipulation of tiny objects in places where precision is at a premium.

The abdominal cavity, for instance. Subramanian says that in the future, acoustic tractor beams could be used to move tiny nanobots inside the body for targeted drug delivery, when a particular medicine needs to be delivered to a particular area of tissue.

That seems to be a very niche application, but I think it's an important one, and pretty cool.

Drinkwater concedes that it will be “a fair few years” before any of this happens — and many more before we can start scaling up to spaceship-sized tractor beams.

Yeah, more like centuries for the latter, absent some sort of massive breakthrough.

Just don't expect to see these at your neighborhood Tractor Supply Store.

3 Like • Comment • View 3 comments

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