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 13 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 15 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.

May 19, 2020 at 12:26am

#983901

The Bucket

The response to today's prompt is one of those that changes for me pretty much daily.

PROMPT May 19th

If you knew that in one year you would die suddenly, would you change anything about the way you are living now? If so, what would you change and why?

I'm going to give one of my non-COVID-19 answers, because the way I'm living and the things I can (or am willing to) do are, and for the foreseeable future will continue to be, restricted by the presence of the pandemic. So the following assumes that there's no pandemic. Because otherwise the simple answer is "No."

Long ago, I guess right about the time The Empire Strikes Back came out, someone told me that the plan for Star Wars was to finish Episodes 4-6, then show 1-3, then release 7-9. For a while there, it didn't look like it was going to happen. And then it did, and I was, for the most part, disappointed. But I remember thinking, back then, "Okay, after they release Episode 9, then I can kick it."

Well, here we are.

Since that time, I've come very close to death at least three times that I know of. Probably a lot more, given the uncertainties of life. And don't get me wrong -- I'm not suicidal or in any hurry to shuffle off, but I'm done. I've done the things I set out to do, experienced almost all of the things that I wanted to experience, and at this point for me it's all about being comfortable until the end.

It's that "almost" that bugs me, though.

As I noted in a recent entry, I became a professional engineer, and co-ran a business. I've flown an airplane. I went horseback riding in a Central American rainforest. I've stood at the top of the Eiffel Tower, and heard the deep tones of Big Ben from up close. I've written a novel -- three, actually, though they're not exactly in finished form. I experienced a week at a dude ranch in Colorado. I've played poker in Vegas (and lost profoundly) and also blackjack (and won). I've driven across the country and back multiple times. I went to Springsteen concerts, and made a holy pilgrimage to Asbury Park (a few times). I've been to both the easternmost and westernmost points of the continental US, and I've floated in the Dead Sea. I looked out over New York City from the top of the World Trade Center, back when it was two towers. I've sampled Pappy Van Winkle bourbon, and crossed the Golden Gate Bridge (not at the same time). I spent a month on Maui. I watched C-beams glitter in the dark near the Tannhäuser Gate. Oh, wait, that last one was Roy Batty from Blade Runner, not me.

But there are still things I want to do: visit Belgium for the beer, Scotland for the scotch, and Japan for the whiskey (there's a particular brand of whiskey called Yamazaki; the 12 year and 18 year styles are available in the US, but for the 25 you pretty much have to go to Japan). I'd really like to take the Trans-Siberian Railroad from Vladivostok to Moscow. A river cruise on the Danube. Iceland. Hell, there are still three US states I haven't been to: Michigan, Nebraska, and Alaska. And I still haven't actually seen the Grand Canyon. If I knew without doubt that my demise would occur in exactly one year, I'd make those things happen (again, barring international pandemic restrictions).

Probably you've noted, as I have, that most of these things, and all of the fuck-it list items, involve travel.

And not a single one of these things will matter after I'm gone, so sometimes I ask myself: why bother? Well, you can say that about anything, and that leads one down a rabbit hole of nihilistic philosophy that I'd rather steer clear of. Everest isn't on my list -- too bloody cold -- but the words of George Mallory (I looked it up and it wasn't Sir Edmund Hillary) resonate with me: "Because it's there."

Hell, if I could, I'd take one of those promised trips to space, just to say I did it. And actually, I want my lifeless corpse to be launched into space when the time comes. It's not going to happen; the best I could realistically hope for would be lofting my ashes into orbit, and even that might be beyond my means. But the idea of being out there, even after death, has appealed to me for a long time, and maybe someday whatever civilization replaces ours would find my remains and go, "...huh, look at that."

No, it's not going to happen.

But I can dream.

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