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

2 Like

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/

3 Like

WakeUpAndLive️~Happiness

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

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

2 Like

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 28, 2023 at 11:47am

#1054756

Law of Rules

Sorry, starry-eyed anarchists, there are always rules, and they're made to be followed.

Some of them, admittedly, don't make a lot of sense, like those old laws about having to get a license to wear a beard or whatever.

According to Cracked, however, some rules are just unjust. (Paradox!)

5 Logical, But Wildly Unjust Rules

Laws exist to maintain order in a chaotic society.

Ideally, yes. In practice, laws exist to protect the interests of the rich and/or powerful.

Sometimes, laws are more in place to keep everything moving smoothly and prevent headaches more than to punish Great Evil.

Because Great Evil makes the laws.

Here are five laws that make logical sense, but are deeply unjust…

As usual, it's a countdown.

5. No Dogs in Antarctica

Up until fairly recently, they were a central part of any activity on the continent, but in 1993 they were banned because of worries about them transmitting diseases like the thoroughly old-timey sounding “canine distemper” to native wildlife, like seals.

Sounds like Paradise to me. Well, except for the whole "desert" and "bitterly cold" and "dark six months out of the year" parts.

4. No Bread in Space

If you’re off to space as an astronaut, though, despite heading into a high-stress environment and situation, you won’t be able to rely on the airy, stretchy internal hug of a tasty sandwich or pastry to calm you.

Look, I like bread. Bread is probably my favorite food, when you consider all of its myriad forms, including pizza crust; as well as its cousin, beer (which I suspect is also a no-no in space). But it's hardly the worst thing you have to give up in space. Gravity comes to mind. So does the ability to use a toilet.

3. Getting Your Taxes Wrong

The American process of paying taxes is basically a massive system of mathematical gaslighting that seems designed to make April a forever horrible month.

Which might explain T.S. Eliot's famous opening line of The Waste Land, and why he abandoned the US in favor of England at the ripe old age of 25. Surely their bureaucracy is less impersonal and stress-inducing than our own.

If you get those calculations wrong, because percentages were probably taught to you by a depressed and underpaid English teacher during a staffing shortage, congratulations on committing tax fraud.

As the article notes, they do have to prove that it was intentional. But it still sucks to make a mistake.

2. Jaywalking

I do this on purpose in NYC just so I can be labeled a scofflaw. I also once spit on the sidewalk there. I'm such a rebel.

This all seems a little ridiculous, and, to be honest, unnecessary. Jaywalking already has its own, independent legal system of punishment, known as “physics.”

Except that the vast majority of jaywalkers never receive that punishment, and when they do, it can be traumatic for the vehicle driver as well.

1. No Putting Your Head Under the Soda Machine and Guzzling

Sadly, this one also applies to beer taps.

Maybe it's better in Antarctica, after all.

3 Like • Comment

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