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

about 8 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 11, 2023 at 11:14am

#1046235

Cache Is King

Today's financial advice is brought to you by Lifehacker. Lifehacker: You're Doing It Wrong.

Here’s How Much Emergency Cash You Should Keep at Home
Having some cash salted away in your home is a really useful thing.

Wait, wait... we're supposed to have cash that we haven't spent yet?

In many places, cash is no longer king. While you still can manage to live a cash-only lifestyle if you want to, few of us do.

Yeah... not really. Depending on what you mean by "lifestyle."

Some of us don’t even carry cash at all, relying on our phones and credit cards, tapping and swiping at stores and using apps like Venmo or Cash App for personal transactions that would once have involved passing a wad of crumpled bills to someone.

Beggar in the median the other day: "Can you help out?" Me: "No cash." Beggar: "I have Venmo."

Look, I'm not trying to rag on the legitimately homeless here, but these median beggars generally aren't.

Even when we talk about the necessity of an emergency fund, we’re talking about money in a bank account, not cash stuffed under a mattress.

If I were robbing a house, under the mattress is the first place I'd look. Followed by sock drawer. You should express some creativity in cash caching, like, say, the toilet tank. (Note: I do not keep cash in my toilet tank; this was just an example.)

Money in a bank will earn interest, be insured against robbery, and will be accessible in some way no matter where you are.

This comes up in my queue the day after a bank fails. In fairness, the depositors' money was insured and the FDIC did its job. Shareholders are boned, but that's the risk they take.

But cash can still be useful. And in an emergency, cash can even be necessary. That’s why you should definitely keep a modest amount of cash at home. But how much?

I've said this before and I'll say it again: Come the collapse of civilization, bitcoin will be worthless (well, more worthless). Gold will be worthless, because other than being shiny, it's useless in an apocalypse. Cash, though, will still have value, because they won't be making any more of it. But your best bet is to stash away coffee, chocolate, and cigarettes.

As for how much cash, well, to some, a modest amount is a million dollars. If you have that stashed away, please let me know. I'll check your toilet tank.

It’s true that in most day-to-day operations no longer require cash, so keeping a stack of bills in a home safe or hidden away somewhere might seem like a waste.

Just be aware that there's an opportunity cost involved.

When it comes to cash, it’s best to have enough to get you through a week or so of paying for fundamentals like gas for your car and generators, groceries, necessary prescriptions, and maybe a night or two of emergency housing like a hotel or motel room.

I wasn't aware that you could even use cash at a hotel.

Anyway, the article goes into specifics. I'm not actually snarking on the idea; it's a good idea. It's just that, these days, for many people, the concept of "extra cash" might as well be "rainbow unicorn" or "pixie dust."

3 Like • Comment • View 5 comments

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