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:

Writer_Mike

about 11 hours ago
In response to "Paris, I Guess"

Elisa: Middle Aged Stik

- I thought it very interesting, to say nothing of extraordinarily coincidental, that your video mentions a lock-adorned bridge in Paris. I just finished reading Spymaster by Brad Thor, in which the Russian mastermind takes strolls from Kaliningrad's mainland to Kant Island. The bridge he uses is also festooned with locks placed by couples declaring their undying love for one another.

1 Like

QPdoll

a day ago
In response to "Landmarks"

I hope you're enjoying every minute in France! What an opportunity.

2 Like

Wordsmitty ✍️

a day ago
In response to "Landmarks"

Hey! Why are the signs at that Mac's in English if you're in France?

With all that walking, I hope you stayed hydrated.

2 Like

Annette in France

a day ago
In response to "Paris, I Guess"

Bienvenue en France.

1 Like

WakeUpAndLive️~Happiness

a day ago
In response to "Paris, I Guess"

last one today:https://parisjetaime.com/eng/event/olivier-giraud-how-to-become-a-parisian-in-one-ho ur-e047

1 Like

WakeUpAndLive️~Happiness

a day ago
In response to "Paris, I Guess"

Could not resist sending the link: https://www.atlasobscura.com/things-to-do/paris-france

1 Like

Elisa: Middle Aged Stik

Sep 21, 2024 at 7:34am
In response to "Paris, I Guess"

Your remarks on Paris looking like many cities reminds me of this video (which I promise is short). It's a sassy look at Paris, and I'm here for it.

https://youtu.be/ByUPFYxy5vw?si=lT57RpTDntAhFdBs

2 Like

WakeUpAndLive️~Happiness

Sep 21, 2024 at 6:46am
In response to "Paris, I Guess"

if you have two hours to spare tomorrow? A beer bike tour in Paris. Check it out, still available. https://parisbeerbike.com/

3 Like

WakeUpAndLive️~Happiness

Sep 21, 2024 at 6:16am
In response to "Paris, I Guess"

Je suis sure, vous avez aimez la ville!!!!

2 Like

Aiva Raine

Sep 20, 2024 at 10:13pm
In response to "Get Baked"

FYI- basting them with olive oil before placing them directly on the oven rack- NO tinfoil- will allow the skin to crisp up a little as well and add a nice flavor to the skin. Recommend Idaho baking potatoes as well for best results.

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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