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🎶

a day 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

February 9, 2023 at 12:02am

#1044486

Origin Stories

Everything's gotta start somewhere.

Fascinating Primitive Versions of Modern Technology

As the title suggests, this article (from Cracked) goes into some of those beginnings. Although I suppose an argument could be made that nothing ever begins; it's all on a continuum.

Many years ago, someone made a hammer. Afterward, following centuries of technological advancement and increased knowledge of human physiology, someone else made an updated version: a hammer with a rubber grip.

As useful as that is, I think the addition of a claw on the back is an even more important innovation. It's like putting an eraser at the end of a pencil, undoing what the other end does.

5. An Early Remote Control Used Little Hammers

But, sadly, not clawhammers. These are more like the hammers inside a piano.

The Space Commander used audio tones — ultrasonic ones, so you didn’t hear them and get annoyed. It created these tones by hitting aluminum rods with clappers. Pressing this remote’s buttons was like playing a xylophone.

Or that.

When I was a kid, we didn't have a remote at home. We had to get up and manually change the channel. While I'm sure these exercise sessions contributed to health, remember that we lived on a farm, so it was superfluous. When we'd go visit my aunt in New York City, she had a remote with all of two buttons, and I thought it was sorcery.

4. Vacuum Cleaners Used to Be Drawn by Horses

I'd make a joke about how it probably sucked, but you already thought of that. So did the article's writer.

3. The First 3D Movie Was Also Choose-Your-Own-Adventure

Movies have used 3D for longer than they’ve used sound. The first 3D cinema release was called The Power of Love and came out way back in 1922.

For the math-challenged, that was 101 years ago.

Audiences didn’t view the film using today’s polarized glasses. Instead, the glasses isolated different types of light by tinting one lens red and the other green. Some of you reading this may have grown up with similar glasses even long after the 1920s.

I think they used red/blue. 3D movies are kinda gimmicky, but not nearly as gimmicky as smell-o-vision.

In any case, this was kinda ahead of its time, even if it was a silent film.

2. Computer Games Used to Print Their Output on Paper

I have vague recollections of this. Mostly, lots of wasted paper.

But this is the cool one, because it verges on steampunk:

1. They Had Clockwork Coffeemakers in Victorian Days

I mean, literal steam.

They had clockwork teamakers in Victorian days, but we chose the word “coffeemaker” in the above title in case you’re unfamiliar with dedicated teamaking devices. Such devices, called teasmades, used to be popular in the U.K. relatively recently,,,

The article contains a helpful picture of a teasmade.

If this sounds somehow both primitive and overengineered, realize that these products were marketed toward British people who had been used to having servants but who no longer could afford them thanks to changing times.

Never underestimate the power of laziness. As I've said before, necessity may be the mother of invention, but laziness is the milkman.

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