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:

Brandiwyn🎶

about 22 hours ago
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.

Aiva Raine

Nov 5, 2024 at 3:03pm
In response to "It's Hard Not To Scare"

Shaking head. Aside from the inevitable Simpson's spider goat song playing in my head which I will now share with you below, whenever I come across a story that has all these scientists doing ridiculous experiments for no logical reason, I must go- ah- scientists- CANCER? You do remember that problem right? Can we maybe cure that before we make cockroaches magnetic and elephants see pink elephants?

spider pig clip- and if you all now are singing the spider goat song- you're welcome! grin
https://www.youtube.com/watch?v=BARjPuUN36Y

1 Like

Aiva Raine

Nov 5, 2024 at 2:55pm
In response to "Anomalies"

One of your more humorous entries- I especially enjoyed- "Lofoten Vortex can be the name of my Bjork cover band."

Thanks for the laughs. Much appreciated.

2 Like

November 6, 2024 at 9:03am

#1079566

Lo!

Well, that's finally over. Now I just get to grump at holiday season chatter.

Today's article, from BBC, has nothing to do with politics or seasons, and everything to do with the invention you're using right now.

'We were just trying to get it to work': The failure that started the internet

There have been differences of opinion concerning the actual beginning of the internet. It's not like a human birth, or Armstrong's boot on the moon: a clear and obvious transition point. In my view, this article is more about a precursor technology, but a vital one for what the internet became.

On 29 October 1969, two scientists established a connection between computers some 350 miles away and started typing a message. Halfway through, it crashed.

1969 would have been long before "try rebooting and reinstalling all your drivers" would become tech support's second suggestion, after "make sure it's plugged in."

At the height of the Cold War, Charley Kline and Bill Duvall were two bright-eyed engineers on the front lines of one of technology's most ambitious experiments.

I should note, for context, that this was the same year as the aforementioned moon landing. Unlike NASA's stated mission, though, this early attempt at remote networking was in service of more military pursuits.

Funded by the US Department of Defense, the project aimed to create a network that could directly share data without relying on telephone lines. Instead, this system used a method of data delivery called "packet switching" that would later form the basis for the modern internet.

Like I said, not the actual invention of the internet, and military.

It was the first test of a technology that would change almost every facet of human life. But before it could work, you had to log in.

Some things don't change.

But Kline didn't even make it all the way through the word "L-O-G-I-N" before Duvall told him over the phone that his system crashed. Thanks to that error, the first "message" that Kline sent Duvall on that autumn day in 1969 was simply the letters "L-O".

And that's what I find amusing about the story: it's very Biblical. "Lo!" As in "Lo and behold."

On the other talon, I want to think that once they got it working (which, as the article notes, they did, after about an hour), the second message sent over this proto-internet was "Send nudes."

The BBC spoke to Kline and Duvall for the 55th anniversary of the occasion.

The rest is a transcript of that interview. It goes into more depth over what happened (or didn't happen) at the time, but there's no reason to repeat it here.

As compelling as this origin story is, I had this vague memory of a different origin story for the internet, one which took place some years later. In a rare case of me actually looking something up, I found this entry from 2019: "Birthed in Beer"

So if I had to choose which one was the actual invention of the internet, I'd pick the 1970s one, because it involved beer.

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