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!

a day 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

a day 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').

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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.

3 Like

Turkey DrumStik

Nov 6, 2024 at 10:22am
In response to "Lo!"

Not gonna lie I need the holiday season chatter.

July 2, 2019 at 2:57am

#961908

Matter of Speculation

Write about something mysterious. Is the mystery ongoing, or has it been resolved?

There is always mystery, and there will always be mystery.

This is a good thing. I'd hate to think that, at some point, we humans will have figured everything out.

You know, at one point, just around 120 years ago, we thought we had. Physics, so the story went, was this close to being all sorted, with everything knowable about to be known.

Yeah... that worked out.

Thing is, the more you know, the more you realize what you don't know. Around the time I'm talking about - the turn of the last century, give or take a few years; I don't remember the exact timeline and can't be arsed to look it up - the prevailing thought about light was that a) it's a wave; b) waves have to propagate through something; c) let's call that something the "luminiferous ether." A lot of smart people studied this awesomely-named substance, which was presumed to infuse the universe.

The reasoning went something like this: They knew (or, well, thought they knew) that Newton's Laws applied to orbital dynamics, which predicted the motions of the planets to within the precision possible at the time. But if the planets had to move through some material, like a gas for instance, the predicted orbits wouldn't match up with observations. And yet, like I said, they reasoned that waves had to travel through some medium: like water, for ocean waves, or air for sound waves, or a Slinky for Slinky waves. Okay, Slinkies hadn't been invented yet, but you get the idea. So this... luminiferous ether (still an awesome name)... had to present no resistance to matter passing through it and yet allow for the propagation of light waves. Basically, they didn't know what it was but they had a good idea what some of its properties must be.

I might be getting some of the details wrong. I just got back from the midnight showing of Spider-Man and want to get to bed. But bear with me here; I'm pretty sure I've got the basic idea right.

Anyway, so a bunch of really smart people, including Einstein, did some experiments, made some calculations, and observed that behold! Photons were both particles AND waves; and, moreover, they didn't need to propagate "through" anything. Problem solved, and goodbye, awesomely-named substance that never really existed in the first place.

But.

(There's always a "but" in science.)

This raised more questions than it answered, and led to fun things that no one understands like general relativity, the photoelectric effect, and quantum mechanics. Okay, well, some people understand the first two, but they're still going back and forth on that last one - and probably will be until the heat death of the universe makes it all moot, or we find a way to blow each other up using, presumably, another of Einstein's discoveries.

What I'm really getting at here, though, is that we have our own "luminiferous ether," though it's not quite as awesomely named: scientists call it "dark matter" and "dark energy," because they don't fucking understand it. All they know is what properties these things would need to have in order to reconcile certain observations. Things like, in the case of DM, being massive and thus having gravity, but not interacting very much with everyday matter in other ways (such as electrical charge, the Pauli exclusion principle, and whatnot). As with the Luminous Ether Hypothesis, they know what some of its properties must be but have no real idea what it is.

With LE, they figured out what it was by changing the entire notion of what space and time are and, in the process, made it go away entirely as a concept. And I'm no expert - I'm just an observer, same as I can watch the Spider-Man movie but not make one - but it seems to me to be very possible that the answer will be something completely unexpected and definitely mind-blowing.

It's a mystery. It hasn't been resolved. When it is, even more questions will crop up.

I just hope I'm still around when it happens.

(roll credits)

Oh, and go see Spider-Man.

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