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').

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.

September 15, 2019 at 3:07am

#966211

Lunacy

PROMPT September 15th

Today’s prompt is from reigning 30DBC Champion, Eric Wharton

!

In 1969, Neil Armstrong first set foot on the moon. Afterward, people commonly complained, “If they can put a man on the moon, why can’t they ______?” How would you finish that statement today?

"...shut up about moon landing hoax 'theories.'"

The ultimate expression of false credulity to finish that sentence was, "...put them all there?" usually uttered by neurotic women at wine-tasting events.

You wanna know why moon landing hoaxers piss me off so much? I'm just drunk enough to tell you, even though it'll give people ammunition to troll me with. So sit back and prepare for typo-laden rambling.

As I've noted before, landing humans on the moon was, by my definition, not only the most significant achievement made by humankind, but the most significant achievement that could ever be made by humankind. While there will, hopefully, be other firsts - first human on Mars, first human to leave the solar system, first human to set foot on a habitable planet other than Earth, first human to freeze to death on Pluto, etc. - the moon thing is significant because it's another world, and you only get to claim that first once.

Damn, I'm still drunk. What I mean is, for as long as we've been conscious, we've wondered about those other lights in the sky, made up stories about them, tried to figure out what they actually were, but were inevitably tied to this hunk of rock with its tissue-thin biosphere. At some point, our curiosity drove us to develop tools, languages, math, science, technology, and, inevitably, being what we are, we had to go and see these things in the sky for ourselves. And our first faltering step in that direction was to put someone under the gravitational influence of a world other than Earth. Considering that we evolved specifically for and alongside this world that we're on now, making the leap to a different world is kind of a big deal. The biggest deal, I think.

So when idiot moron retards try to minimize the accomplishment by claiming it never happened, that it was all a hoax done on a sound stage, that it was a vast conspiracy to play out international mind games, they are denying the greatest achievement of humanity. They are, in effect, denying that humanity can achieve anything great. It's the same kind of anti-hominid mentality that leads people to believe that the pyramids could only have been built with the assistance of space aliens, or that Stonehenge was a flying saucer landing pad or whatever fucking fringe theory that minimizes humanity's accomplishments in favor of believing in some "higher" power.

Other people have thoroughly debunked the moon-landing "hoax," so I'm not going to go into all that (besides... still drunk here), but I will point out that if we'd faked it, a) the Russians would have had to be complicit, meaning they would have had to agree to back us up in a big lie during the height of the Cold War; b) the thousands and thousands of people involved would have had to all manage to shut up about it; and c) it would have taken more advanced technology to make a realistic moon "set" than it did to actually go to the moon.

I kind of get the "denying humanity's achievement" thing. I mean, look at us - we're a bunch of brutal apes more generally concerned with our relative position in whatever social hierarchy than in achieving anything for its own sake. Hell, the Space Race was one massive dick-measuring contest; I can freely admit that. The motivations behind it, though, are irrelevant considering the results. Hell, you might say we achieved this momentous feat because of our primate nature, rather than rising above it. (Incidentally, I'd still consider it humanity's greatest achievement if the Russians had managed to be first in that as they were first in many other space milestones. It's not a nationalist thing; it's a humanist thing.)

But here's the thing about humans: we may be, collectively, a bunch of damn dirty apes, but occasionally we can do impressive things. To deny the achievement of July 1969 is to deny the promise of humanity itself, and I won't stand for it.

Damn, I'm starting to sober up, so I guess I'll quit rambling now.

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