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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Writer_Mike

32 minutes 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.

QPdoll

about 17 hours ago
In response to "Landmarks"

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

2 Like

Wordsmitty ✍️

about 20 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.

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

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

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.

July 7, 2023 at 10:24am

#1052279

Thought for Food

Here's a liberating take on food.

There Is No Such Thing as “Junk” Food
Food hierarchies are, in truth, sorting mechanisms. It does not make you a worse person to eat “junk” food, and it certainly doesn’t make you a better person to eat whole grains.

Except that food has been linked to morality and virtue-signaling for a long time.

When I was in elementary school, I filled out a well-intentioned worksheet that asked me to sort foods into boxes.

What you were really learning was that humans like to stick everything into neat little boxes.

The worksheet didn’t consider how to feed your child when you’re a single parent working swing shift and a Happy Meal or a frozen pizza is the cheapest and most reliable way to feed your kids, or if the nearest fresh broccoli was an hour’s drive away.

And this is the often-overlooked dirty little secret of food snobbery: eating foods considered healthy is expensive and time-consuming. Being able to do so is a product of privilege.

My understanding that certain foods were so “bad” they were junk was complicated by the peculiar diet culture of the ’90s, which convinced me that cheese and avocados were high-fat and to be avoided, bananas had too much sugar and should also be avoided, but Starburst Jellybeans (lot of sugar, sure, but fat-free!) or Snackwell Devil’s Food Cookie Cakes (cake...but diet?) were somehow okay.

I've ragged on nutrition science in here before; no need to repeat it.

I entered my 20s with deeply illogical ideas about food and nutrition, which became even more contradictory when injected with the sustainability gospel of Michael Pollan, anti-GMO politics, the locavore movement, and the rise of “fast casual” cuisine.

But the problem isn't just nutrition science; it's that, as evidenced by the anti-GMO bit, people will believe whatever suits them.

Some people, regardless of age, need more salt in their diet, not less; some people need more fat, or caffeine, or dairy, or none at all. And others just need more things in their lives that are delicious—that remind them of the true bounty and delights of being human.

And really, as a hedonist, that last part resonates strongly with me. Food isn't penance. Food isn't medicine. Food isn't virtue. Food is about not being hungry, and, ideally, enjoying life.

People weren't nearly as neurotic about food before we got all these choices. But now that we have them, sure, it's good to be healthy, but not at the expense of eating cardboard.

The article goes on to link food snobbery with racism and classism, and it makes some good points, but it's tangential to my purpose in linking this.

This is, to me, the important bit, right at the end:

Contrary to what those worksheets might tell us, food does not have moral character, and consuming it does not influence or infect our own character. Food is delightful, and food is fuel, and food is culture. It becomes shadowed with shame—often, the sort that can distort our eating habits for years to come—not when we eat it, but when we restrict it, and attempt to spread that shame to others who do not.

I noticed this in particular with the low-carb or gluten-free fads. There has not been a food that I'm aware of in the entire history of agriculture that has been more culturally significant than bread. It is symbolic to many religions, but even just culturally, it's something that brings people together. I can't help but feel that the attempts to demonize it are just another way to separate us from each other, and give us yet another thing to feel shame about—shame that can only be ameliorated if you Buy My Product.

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