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').

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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.

April 1, 2022 at 12:02am

#1029828

Steely Gazelle

The theme for this month's "Journalistic Intentions"

[18+] is Earth. It's a well-rounded topic, and I hope it doesn't leave you flat. Feel free to click that link and enter. I know it can be intimidating to have such transcendent competition as Me, but I didn't win any prizes in the last round there, so you do have a chance.

Gazelle Conservation Achievements

Gazelle Conservation Achievements is the name of my grunge Toto cover band.

I picked up the habit, elsewhere on the internet, of turning unlikely phrases into band names. After all, a lot of band names are unlikely phrases. Pearl Jam (yes, I know what it means). Steely Dan (that too). Goo Goo Dolls. Green Day. It's like they sit around after practice, get stoned, and brainstorm the weirdest, most surreal names that they can.

The next day, when they're sober, they pick one out of a hat and run with it. Then, when they become all famous and shit, someone will inevitably ask, "So where did the band name come from?" In order to avoid seeming random and/or stoned, they'll always say something like, "Oh, our drummer, Pete, woke up with the name in his head from a dream." Or, "It was the name of Chester's first dog." And then Chester has to run around the internet changing all his challenge questions for password recovery.

I heard somewhere that the band Jethro Tull went through a bunch of different names before settling on that tribute to a person most often described as an eighteenth century agriculturist

who probably no one who's not a farmer would have ever heard of were it not for Ian Anderson's band.

An agriculturist (not to be confused with an aggro cultist) is apparently not the same thing as a farmer, though I suppose it's possible to be both. I guess it's kind of like in physics: you have theoretical physicists to think of weird experiments and do math, and then you have experimental physicists whose life's work is to make the theoretical physicists look like idiots by proving them wrong.

That's because theoretical physicists, and presumably agriculturists, work in the realm of (wait for it...) theory. And more often than not, theory goes against the concept known as "common sense." So there's always pushback from common-sensers whenever something theoretical pops up that doesn't, in their limited worldview, make any damn sense. Like the idea that spacetime is curved, or that our bodies are basically collections of quarks and electrons, neither of which can be actually seen.

But then someone tries the new theory, and behold, it works, and a couple of generations later, that becomes common sense.

If there's one thing I've learned in my time on the planet (which is round), it's that I've learned more than one thing. If there's another thing I've learned, it's that common sense is neither. Never trust a politician that runs on a "common sense" platform. Inevitably, they're so stuck in their ways that they'll ignore all evidence that is contrary to their interpretation of common sense, much to the detriment of their constituents.

Maybe they should take a page out of the band playbook: Get stoned. Brainstorm some new ideas. When sober, pull one out at random and try it.

Sure, much of the time it won't work. But sometimes it will, and that's how you get new stuff.

Because, remember, the idea that the Earth is flat is just common sense. I mean, look at it. It looks flat, doesn't it? Unless, I guess, you're in the mountains, and then it looks all craggy and shit. But then you do some simple measurements and think about it for about 20 seconds with an open mind, and the truth reveals itself.

None of which explains why we need to conserve gazelles, I'm afraid. Or why. I guess it's so that lions will have something to eat tomorrow. I mean, that's just common sense.

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