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 an hour 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 3 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.

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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.

May 26, 2023 at 11:34am

#1050116

Thirty Years After

Feeling much better today, thanks.

30 years ago, one decision altered the course of our connected world

This article, from NPR, is dated April 30 of this year, the actual anniversary in question.

"Imagine being able to communicate at-will with 10 million people all over the world," NPR's Neal Conan said...

9,999,990 of whom are idiots, trolls, bots, or some combination thereof.

..."Imagine having direct access to catalogs of hundreds of libraries as well as the most up-to-date news, business and weather reports...

Mostly fake, and/or stuck behind paywalls.

...Imagine being able to get medical advice or gardening advice immediately from any number of experts...

And having it turn out to be (sometimes dangerously) wrong.

"This is not a dream," he continued. "It's internet."

And it was fun for a couple of years.

On April 30, 1993, something called the World Wide Web launched into the public domain.

The web made it simple for anyone to navigate the internet. All users had to do was launch a new program called a "browser," type in a URL and hit return.

And then troubleshoot the url typed in, because most people can't spell, and were hopelessly confused by the // thing.

This began the internet's transformation into the vibrant online canvas we use today.

Jackson Pollock's paintings can also be described as vibrant canvases. And any attempts to make sense of those are doomed to failure, or at least false success.

Anyone could build their own "web site" with pictures, video and sound.

And malware.

They could even send visitors to other sites using hyperlinked words or phrases underlined in blue.

Such as this one.

Okay. I'm being deliberately cynical. There's a lot to like about the internet, not the least of which is this website. But the web certainly has its downsides... just like everything else. Later, the article goes into some of these downsides.

CERN owned Berners-Lee's invention, and the lab had the option to license out the World Wide Web for profit. But Berners-Lee believed that keeping the web as open as possible would help it grow.

Well... he wasn't wrong. Fortunately for all of us, CERN was—and is—an international government science research organization, not a private company like ExxonMobil, AT&T, or Apple.

Today, nearly two-thirds of the world's population uses the web to visit hundreds of millions of active websites. Some of those pages belong to companies that are among the most valuable in history like Facebook, Amazon and Google.

None of which would exist without the internet as we know it.

With all of its problems, it's still one of the most influential inventions in history, if not the most. It's difficult to put an exact date on "this is when the internet was born," because there were all sorts of remote networks available before 1993, with many different individual inventors (none of whom were Al Gore), and of course the internet continues to evolve.

But the beginning of the World Wide Web is as good a milestone as any to commemorate.

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