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 9 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

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

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.

March 23, 2023 at 10:35am

#1046875

Unnatural

I've said before that "natural" as a food or other product label is bullshit.

5 ‘All-Natural’ Products That Are Complete and Utter Garbage

Good to know Cracked has my back.

You don’t have to be an Amish man, churning butter and spitting distastefully at the idea of an electric can opener, to admit that the grip tech has on our lives is a pretty tight one.

I like it that way. Could do with less surveillance, but other than that, bring on the convenience.

So what is there to do? What humans do best: violently overcompensate for something that upsets them by swinging back in the other direction.

See also: tiny houses as a reaction to McMansions; "cluttercore" as a perfectly reasonable response to Marie Kondo.

This sort of panic pops up in all sorts of ways, from going back to flip phones, to deciding to eat like cavemen (despite the fact they were never known for their health).

Health, no. Probably stronger than fuck, though. But that wasn't because of what they ate, but how they caught it.

As such, there’s been a massive movement of “natural” products that have come along to capitalize on people’s distrust of words that start with “di-” and end in “-ate”, and, looking at their price tags, I do mean capitalize.

Everything you eat or drink is made of chemicals.

So, in usual fashion, five items on a countdown list.

5. Natural Peanut Butter

A jar of classic Jif or Skippy is an absolute delight in every possible way. Cutting into that perfect plateau atop a new jar’s contents, watching it curl and collect along the blade of a knife and then spreading it like sweet mortar onto your delivery method of choice.

Dammit, now I'm hungry. Goddamn porn writers...

I guess this one's a matter of opinion. I kind of like the "natural" labeled kind. I prefer almond butter, though.

4. Natural Deodorant

Natural deodorant. That old borderline oxymoron. Here’s a tip: If anyone ever asks if you’re wearing natural deodorant, it means you smell like an upside-down porta potty.

This. Once we were thoroughly sold on the idea that we should smell like soap and not like human, there was really no going back.

For most of us though, the reason given for using natural deodorant is mostly avoiding aluminum, for antiquated reasons that plenty of scientists now say aren’t proven.

I thought aluminum was just used in anti-perspirants, which I know come bundled with deodorants, but you can get the kind of the latter that's not also the former.

3. Oil Pulling

I first heard this referenced maybe a year or so ago? I don't know; my time sense is warped. And my mind went, "bullshit."

It comes from ancient medicine dating back to hundreds of years B.C. You know, back when they had medicine super figured out, and everyone was the epitome of health. Somehow, it’s come roaring back, mostly thanks to the kind of people who spend $200 a month on crystals because dying is scary. Unsurprisingly, the general medical consensus on the practice is, “Look, if you want to dump a bunch of oil in your mouth, we’re not going to stop you, but also, this is probably bullshit.”

Yeah, maybe they gave up on it because it wasn't doing squat.

Doubt it's harmful, though, so whatever.

2. Hair-Growth Cures

There are only two things proven to slow or reverse hair loss, and neither of them grow on trees: They’re the medications finasteride and minoxidil, and even they aren’t guaranteed. If you’re wondering if some new natural, simple hair regrowth solution works, the answer’s right in front of you, in the form of a world full of balding men that sure wouldn’t like to be.

It's questionable that one can consider most hair loss to be something that must be "cured." Want to be "all-natural?" Accept male pattern baldness; it's called that because it's genetic in origin. Now, I understand wanting to not be bald. But I also understand leaning into it like Jean-Luc Picard.

1. Vitamin C

I know, I know. I’m sorry to come for your precious packet of no-sick powder, especially in a time of public health panic. But mega doses of vitamin C, in whatever form you ingest them, have been proven time and again to be psychosomatic more than scientifically helpful.

Funny thing about the placebo effect (which is psychosomatic): if it works, why not take advantage of it?

I'd always heard that you can't overdose on Vitamin C, that excess isn't bad for you. And that's probably true in doses that won't explode you first, though there is a such thing as too much of anything. But what is bad for you is spending a bunch of money on something that doesn't work as advertised, especially when you need to save that so you can pay at least part of your inevitable medical bills.

In the end, "all-natural" is a marketing strategy, nothing more. Plenty of natural things will make you sick or otherwise mess you up, like poison ivy or the mushrooms I've talked about in here before. While there are certainly human-developed chemicals that are dangerous, too, having a knee-jerk reaction to all of the stuff they put in food is probably taking things a bit too far.

6 Like • Comment • View 4 comments

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