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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QPdoll

about 15 hours ago
In response to "Landmarks"

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

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Wordsmitty ✍️

about 18 hours 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.

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Annette in France

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

Bienvenue en France.

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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

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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

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Elisa: Middle Aged Stik

a day ago
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

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WakeUpAndLive️~Happiness

a day ago
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/

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WakeUpAndLive️~Happiness

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

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

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Aiva Raine

a day ago
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.

Sum1's In Kalispell Montana

Sep 20, 2024 at 9:08am
In response to "Get Baked"

I have long advocated using only the oven for baked potatoes, and am guilty of punching holes in the tops of one. I don't use Aluminum Foil either. But my other half likes to nuke them first, or only nuke them. It's the speed thing for her. I prefer to take my time when I cook, and do things the way I do them, not her way. But, sometimes, ,ya gotta keep the peace in the house. She won't buy Russet potatoes, but those are the only ones to use for this. I may have to exert a little effort an buty Russets just for Baked Potatoes. She loves the English Chips (really enjoyed those in England when I was there, so maybe she'll like the link I sent her (from you!))

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August 1, 2023 at 9:35am

#1053424

Pretty Petty

A couple of weeks ago, I posted a Cracked piece about ideas that started as jokes: "The Step-Parent of Invention"

Today's is similar, only instead of jokes, they started as a giant middle finger.

Four Important Things That Were Invented Out of Sheer Pettiness

Unless you’re an actual deity (in which case, greetings, Ms. Knowles), you’ve almost certainly pulled some petty shit in your day.

Actual deities invented petty. Or, well, as they don't exist, we made up stories about them that projected our own penchant for pettiness upon them.

Slowed to a crawl in front of a tailgater, helpfully transferred your neighbor’s dog’s poop from your yard to theirs, squeezed all your roommate’s fruit after the third day of seeing their dishes in the sink — nothing life-ruining, just a risk-free outlet for your anger.

I'd like to say I've never done anything like that. But I can't.

But have you ever been so annoyed, you changed the world?

One person cannot change the world. Not even Beyoncé.

4. The Myers-Briggs Personality Type Indicator

I've ragged on this before: astrology for psychology nerds.

It was also a fairly useless exercise, as the MBTI has been largely dismissed as pseudoscience, probably because it was invented by a stay-at-home mom with zero psychological qualifications to throw shade on her son-in-law.

Hence the "petty" part.

She took over entire sections of the library as she embarked on a lifelong study of exactly what factors made her son-in-law suck so much, which her daughter finished after her death, which might have stung for poor Mr. Myers if he had any capacity for emotion.

Both of my former mothers-in-law had their issues with me, but not enough to ruin the lives of everyone who ever applied for a job.

3. The Automated Telephone Exchange

Let's be real: this was going to get invented eventually anyway.

Back in the early days of the telephone, you had to call the operator, who would then put your call through to your mom or your side piece or whoever.

And then, if they were bored, they'd listen in. You think internet surveillance is invasive? Ha.

They knew who was calling who at all times, and they might even decide for you who you were calling, which must have been frustrating for people who asked for their side piece and got their mom. That was a problem Almon Brown Strowger kept running into in 1878. He was an undertaker living in Kansas City, and he just couldn’t figure out why his business was, uh, not as lively as it could be.

You want to piss off the undertaker, it's your funeral.

It turned out one of the local telephone operators was the wife of another undertaker, and every time someone called for Strowger, she patched them through to her husband instead.

As evil plans go, that one's rather amusing. Of course, when the operators were replaced by machines, few of them likely found that amusing.

There's also a nice plot twist, but I gotta save something for the article.

2. The Lamborghini

This one, I knew about. What can I say? I don't talk about cars much, but I retain some of Teen Me's fascination with them.

Not going to quote this one; the article is funnier than I can be.

1. The Dishwasher

I've said before (including in that entry I linked above) that while necessity may be the mother of invention, laziness is the milkman. A lot of labor-saving devices came into being not out of pure necessity, but a burning desire to do less work, which of course is a worthy goal that unfortunately only means you end up filling all the time you saved with other busy work, instead of writing poetry or doing philosophy or whatever the labor-saving advocates imagined our current utopia to consist of.

Today, it’s a mark of being Kardashian-level spoiled to not even know how to use a dishwasher, but it was invented by a socialite essentially as an insult to the help.

But this one, if a joke site can be believed, wasn't even to save the inventor time, but to diss the servants.

The socialite in question was Josephine Cochrane, the wife of a wealthy businessman living in Illinois in the mid-1800s.

Must run in the family; their distant descendant, Zephram Cochrane, will invent the warp drive, the ultimate time-saving device.

In summary, the dishwasher was invented because servants tended to chip the china. Which is ironic, because rare is the dishwasher machine today that doesn't chip whatever is banging around in there.

I don't know about you, but if I were a servant, I'd chip the china sometimes just to be petty.

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