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:

Write_Mikey_Write!

about 8 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 10 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.

March 20, 2024 at 10:30am

#1066614

Not My Baili-Wiki

Well, "Spring is Sprung"

, as they say, so for no reason other than random chance, here's a Wired article in praise of Wikipedia.

Wikipedia Is the Last Best Place on the Internet
People used to think the crowdsourced encyclopedia represented all that was wrong with the web. Now it's a beacon of so much that's right.

Of course, Writing.com is really the Last Best Place on the Internet, but I'll accept Wikipedia as a close second.

Remember when Wikipedia was a joke?

In its first decade of life, the website appeared in as many punch lines as headlines.

I've noted before that WDC came into being prior to Wikipedia. Not by much, mind you. September 2000 and January 2001.

I remember when those years seemed so futuristic. We expected interplanetary travel, jetpacks, flying cars, suborbital transport, and asteroid mining. What we got was the internet.

The article is long, but here's a few select comments.

To confess that you've just repeated a fact you learned on Wikipedia is still to admit something mildly shameful. It's as though all those questions that used to pepper think pieces in the mid-2000s—Will it work? Can it be trusted? Is it better than Encyclopedia Britannica?—are still rhetorical, when they have already been answered, time and again, in the affirmative.

I remember back then, someone took a random sampling of articles from both Wiki and EB and fact-checked the hell out of them. They came in about equal. Of course, that was 20 years ago, and a few things have changed.

It does not plaster itself with advertising, intrude on privacy, or provide a breeding ground for neo-Nazi trolling. Like Instagram, Twitter, and Facebook, it broadcasts user-generated content. Unlike them, it makes its product de-personified, collaborative, and for the general good.

It does do a fundraising event every year, kind of like public radio. I'm not the least bit embarrassed to say that I contribute. Money, that is. I don't have the will to contribute content there.

More than an encyclopedia, Wikipedia has become a community, a library, a constitution, an experiment, a political manifesto—the closest thing there is to an online public square. It is one of the few remaining places that retains the faintly utopian glow of the early World Wide Web. A free encyclopedia encompassing the whole of human knowledge, written almost entirely by unpaid volunteers: Can you believe that was the one that worked?

I have, at times, noted that it's the modern equivalent of the Library of Alexandria. Like that ancient institution, I fully expect it to be overrun and destroyed by the know-nothings, at some point. Until that happens, I continue to use it.

Wikipedia and Britannica do, at least, share a certain lineage. The idea of building a complete compendium of human knowledge has existed for centuries, and there was always talk of finding some better substrate than paper: H. G. Wells thought microfilm might be the key to building what he called the “World Brain”; Thomas Edison bet on wafer-thin slices of nickel.

Wells was a science fiction writer, and Edison was a hack. They were both products of their time and prognosticated the best they could with the technology available to them. Wells couldn't foresee some storage medium denser than microfilm, and there was no way Edison or anyone else (not even Tesla) could have predicted semiconductor technology (or, if they did, it was by mere chance).

There is, as I said, lots more interesting information in the article. One reason I'm sharing this is that I have a tendency to hyperlink Wikipedia entries here, and I wanted to justify that.

Two things can happen with crowdsourcing. Well, more than two, as usual, but at the extremes, you can get the worst of humanity, or the best. As the article notes, Wikipedia isn't perfect. Nothing is. But it represents the best of humanity, while social media usually represents the worst.

6 Like • Comment • View 1 comment

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