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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Show Recent Comments

10 Most Recent Blog Comments:

tj wanderlust-words-in-motion

about 12 hours ago
In response to "Thwip"

Put it on ATVs so when they get stuck they can shoot out a web and winch themselves out.

2 Like

tj wanderlust-words-in-motion

a day ago
In response to "One Fine Thing"

Well, it was a fine idea.

Adhere - Definitely Writing

a day ago
In response to "One Fine Thing"

I immediately thought of "Imfamy, imfamy, they've all got it imfamy." But then my mind does that.

2 Like

tj wanderlust-words-in-motion

Feb 19, 2025 at 3:59pm
In response to "Fail to Succeed"

In one way, it's good to have people with boneheaded ideas, it provides something to blog about.

1 Like

Adhere - Definitely Writing

Feb 19, 2025 at 1:36pm
In response to "Fail to Succeed"

Having drunk soda on both sides of the Atlantic, I have to say that Mountain Dew with HFCS in America tastes a lot better than the 'same' drink made with sugar here in the UK. The experience is smoother and so is the taste.

2 Like

tj wanderlust-words-in-motion

Feb 18, 2025 at 6:10pm
In response to "Tater Tales"

Instant potatoes? Yes, I enjoy them, too. We also use them for thickening up soups and stews.

1 Like

Robert Waltz

Feb 17, 2025 at 11:23pm
In response to "Linguistic"

Annette

- I appreciate the recommendations, but I canceled Netflix when the choice was to put up with ads for a lower price or pay a shitload of money. If it were free with ads, I'd manage like I did in the days of broadcast TV, but I flat-out refuse to both pay money and watch commercials, which is why I never had cable TV.

The price for the ad-free experience also went up to the point where it wasn't worth it with the amount I watched.

Annette

Feb 17, 2025 at 3:40pm
In response to "Linguistic"

I refer to my three languages as three different brains. In France, I activate my French brain. Which made for some funny turns of phrase when I was speaking to my son in English with English words, but in French syntax.

Netflix has some really good multilingual TV series. I recommend

Monsieur Spade

Black Spot

Capitani

1 Like

Robert Waltz

Feb 17, 2025 at 12:00pm
In response to "Linguistic"

Kåre เลียม Enga

- yeah, I edited out the duplication, because it could be seen as a refutation of itself.

1 Like

Kåre เลียม Enga

Feb 17, 2025 at 11:32am
In response to "Linguistic"

I use translation tools all the time... and subtitles on youtube.

My French pronunciation is okay because I learned it as a child; my Spanish is okay because I practiced it. I slip into both with one key exception. French is easier when I'm depressed and Spanish can draw me out of depression.

Speaking multiple languages has also been linked to delayed onset of dementia symptoms.

Yes, I saw that you repeated that. It bears repeating.

Speaking multiple languages has also been linked to delayed onset of dementia symptoms. Or so I hope.

1 Like

February 21, 2025 at 8:25am

#1084172

Thwip

Here's a great example of science imitating art. From Wired:

Science Has Spun Spider-Man’s Web-Slinging Into Reality
When a US research lab accidentally created a sticky, web-like substance, it turned to Peter Parker and comic-book lore for inspiration on what to do next.

The original Spider-Man had Parker sciencing up his own web-shooters and web fluid, which was less a metaphor for puberty and more a comic book shortcut to giving him web powers that weren't inherently gross. Unrealistic? Sure, but so is everything else in comic books, including getting superpowers from being bitten by a radioactive spider, and that's okay.

Slowly but surely, we are making good on the gadgets we imagined, as kids, that the future would hold.

And yet, no phasers.

The Starfleet tricorder from Star Trek? Almost there. But web-shooting? Web-slinging? That wasn't one we really thought would make the crossover.

Yeah, I gotta agree on that one.

And it wasn't exactly in the plans for the scientist who has made the strong, sticky, air-spun web a reality either, Marco Lo Presti, from Tufts University’s Silklab.

Okay, but Silklab definitely sounds like a superhero hangout. Or maybe a supervillain lair. A very smooth one.

Fio is Fiorenzo Omenetto, professor of engineering at Tufts and “puppeteer” of the Silklab.

Oh, definitely supervillain lair.

“You explore and you play and you sort of connect the dots. Part of the play that is very underestimated is where you say ‘hey, wait a second, is this like a Spider-Man thing?’ And you brush it off at first, but a material that mimics superpowers is always a very, very good thing.”

What? No! It's a very, very bad thing in the hands of supervillains.

A lot of the Silklab’s work is “bio-inspired” by spiders and silkworms, mussels and barnacles, velvet worm slime, even tropical orchids—so working out whether this sticky web could become something useful might seem like an easy side-step for the team.

Velvet Worm Slime definitely needs to be the name of a band. Probably an emo/punk/EDM one.

In Stan Lee and Steve Ditko’s original 1960s comic books, starting with Amazing Fantasy #15, Peter Parker builds a “little device,” one fastened to each wrist and triggered by finger pressure, to produce strands of ejectable ‘spider webs.’ By the time of the mid-2000s Sam Raimi Spider-Man films, the web-shooting switched from a wrist-worn spinneret gadget to an organic part of his superhero transformation.

And we've never let Raimi hear the end of puberty jokes since.

The article describes the web-fluid development in more detail; of course, despite the hype, we're not going to get friendly neighborhood web-slingers anytime soon, if at all.

So, Spider-Man capabilities when? “Everybody wants to know if we're going to be able to swing from buildings,” says Omenetto with a wry smile. But we're not there yet—so far the Silklab team itself has speculated about some potential uses for the material: the retrieval of an object that’s lost underwater, perhaps, or a drone that captures something in a remote environment.

When I was a kid, there was this sticky goo that would stretch but hold together, and you could use it to pick up pennies off the sidewalk, at least until it got too dirty to be useful. This doesn't seem much more useful than that goo, at least not yet.

Lo Presti is interested to hear from anyone who has read his paper and thinks they might be in need of a web-shooting silk fiber.

"Hello, I am an aspiring supervillain and I am in need of a web-shooting silk fiber to achieve my plans of world domination."

Some humans are pretty clever, though, and I'll bet they'll find uses for this that don't involve costumed vigilantes swinging from skyscrapers. Still, I do appreciate the comic-book theme here.

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