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 3, 2023 at 12:01am

#1044108

Rants At A Picnic

My second-biggest gripe here is: "Only four?"

4 Bad Foods We Can Move on From in the Modern World

Because I can think of a lot more than four.

But this is Cracked, so you gotta account for short atten- SQUIRREL!

The sheer variety and volume of food available in the world is enough to make both your eyes and mouth water simultaneously.

Especially if you're one of the millions who can't access it.

Dark humor is like food: not everyone gets it.

And yes, guy whose go-to-move at parties is to shut down casual chats by bringing up the saddest possible parts of the human condition, I do realize that starvation and food security is still a massive problem both in the U.S. and abroad. Can we move on now, and you can resume your job as a professional hitman for conversations?

Nah, I reserve that for the blog. But that's only because no one invites me to parties or casual chats.

Here, then, are four foods that, as a modern civilization, we can finally kick to the curb.

4. Necco Wafers

Congratulations, you've identified something more hated than candy corn.

The world has changed around you, while you’ve been chopping up sticks of sidewalk chalk and wrapping them in the sort of wax paper that feels like it was collected from an Egyptian tomb. We’re two months away from Gushers with LEDs in them, and you’re still trying to sell us a candy most monkeys would spit out.

Stop giving candymakers ideas.

3. Baby Corn

Not to be confused with candy corn, because as hated as candy corn is, at least it has a flavor.

I, like many American Jews, have a deep, insatiable love for Chinese food. Fried pork dumplings are very possibly my favorite food of all time, and if not, they would at the very least make it handily through the primaries.

I'm just leaving this in here to avoid comments/jokes about keeping kosher. It's not a thing for most American Jews.

However, there is one consistent, unwelcome invader in many entrees at Chinese restaurants: Those fucked-up little corns.

Truth.

2. Plain Cheerios

YOU SHUT YOUR WHORE MOUTH.

But the light is dimming for the default Cheerio in the modern world — the new cereals are faster, sweeter and more colorful.

That is exactly what makes Cheerios great.

Not that I eat cereal much anymore, but on those rare occasions that I do (usually as part of a free breakfast at a cheap motel), the default is Cheerios, not Chocolate Frosted Sugar Bombs.

1. Brazil Nuts

Unlike the other items on this list, I have absolutely no opinion about Brazil nuts. But I tend to avoid them because the idiots I knew as a kid had a rude name for them that I shan't perpetuate by noting it here.

I am sure there was a point in the history of the human race where the precious food inside a Brazil nut was needed for survival, but I highly question their continued relevance past the invention of fire. The value proposition of the Brazil nut is like breaking into a safe in order to retrieve a piece of dry toast.

I've never had to actually break into a Brazil nut. But you can't make that argument about Brazil nuts when there are so many other foods that are difficult to access, like coconuts or pomegranates (pretty sure that's a word meaning "apple made of granite").

Not to mention, there are so many other better nuts that are begging you to eat them! You’re telling me you’re putting on the blinders and digging past peanuts, pistachios and cashews just to draw blood trying to access the non-prize that is the meat of the Brazil nut?

Peanuts are barely food, pistachios are too much work (or too expensive if you get the pre-shelled kind), and the only purpose of cashews is to provide a unique shape in the trail mix.

Anyway, as I said, why stop at four? I can think of lots of foods whose time has passed, especially since Cheerios doesn't belong on that list at all.

For example: mass-produced milk chocolate. It's basically edible plastic with just enough cocoa so the food police can let them call it "chocolate." There are many tastier alternatives. Sure, they're probably more expensive, but we're not talking about price.

Your turn. What food would you want to see relegated to the circular file of history? (If you're vegetarian or vegan, saying "animal products" is cheating.)

3 Like • Comment • View 4 comments

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