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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Annette in France

about 3 hours ago
In response to "It's Hot in The South, Once Again!"

The quote is too small and low contrast to be read.

Elycia Lee ☮

about 13 hours ago
In response to "There Is A Such Thing As Coincidence"

I loveeee the photo! You guys look good!

Coincidences are great! So happy you met!

2 Like

Wordsmitty ✍️

a day ago
In response to "Two Castles and a Cave"

That's definitely a wonderful picture that shows the whole story. When enlarged, it's a little fuzzy as if after quite a few bottles of wine.

1 Like

Annette in France

a day ago
In response to "Two Castles and a Cave"

After being in Tours for an afternoon/evening and only, my favorite part of my time there was to see you.
Old Tours with its medieval houses is charming to stroll through, but France has many charming old towns.

The most impressive cave system that I went to this vacation is under the Chateau de Brézé. It is where the Troglodytes lived. Their caves didn't only house people, they even had underground horse stables.

Interestingly, Chateau de Brézé is also a wine castle complete with an attached tasting room. Even the gift shop sells mostly wine along some postcards.

1 Like

Elisa: Middle Aged Stik

a day ago
In response to "Two Castles and a Cave"

Oddly enough, Tours is a sister city to Minneapolis. That still isn't enough of a reason for me to visit France, but it's still a nice bit of trivia.

2 Like

Kåre Enga in Montana

a day ago
In response to "There Is A Such Thing As Coincidence"

That's awesome. Enjoy your visit of the Loire.

TGV = tren à grande vitesse = HST. I only took it once, Paris > Luxembourg. Everything was a blur.

2 Like

Annette in France

Sep 25, 2024 at 3:20pm
In response to "There Is A Such Thing As Coincidence"

That was a total coincidence. I didn't even plan to go to Tours at all. I was there because of the way I planned out a travel leg. To run into you on the other side of the Atlantic in a random town was pure luck.
It was fun to catch up with you.

2 Like

Brandiwyn🎶

Sep 25, 2024 at 2:49pm
In response to "There Is A Such Thing As Coincidence"

Love it when stars align!

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Wordsmitty ✍️

Sep 24, 2024 at 5:15pm
In response to "All Out of Louvre"

I agree. You should always check the "Five Must Dos in ..." before you go, so you know what to avoid from crowds.

Smurfs are universal.

1 Like

Elisa: Middle Aged Stik

Sep 24, 2024 at 11:30am
In response to "All Out of Louvre"

That street art is definitely a step up.

1 Like

November 16, 2019 at 12:05am

#969794

If I Had A Million Dollars

PROMPT November 16th

If you had to spend one million dollars in one day, what would you spend it on?

Shares of publicly-traded companies.

Usually I lead with the joke answer, but no, that's my actual answer.

So, what if stocks aren't a legitimate answer? Well, I'd also answer "bonds" or "real estate." Maybe even "fine art" or "coin collections."

But okay. I think the spirit of the prompt is about blowing through the money, not investing it in assets that have some chance of appreciating in value.

Can't resist embedding this video.

It's harder than you might think to spend a million dollars, especially in a limited period of time. The median annual income of US residents is somewhere in the neighborhood of $50,000 a year (it's actually less than that, but I can't be arsed to look it up, and $50K is a nice round number.) Assuming you live hand-to-mouth on a $50K annual income, it will take you 20 years to spend a million dollars.

On the other hand, a million ain't what it used to be. Retirement counselors often quote a "4% rule," which can get a little more complicated, but basically once you're retired you should live on 4% of your net worth per year. 4% of a million is $40K, also close to the median annual income for the US, but hardly "fuck off" money.

I like to work with the concept of "effective net worth," which is something I came up with, but someone else probably already has because I always seem late to these things. Basically, using the 4% rule, what is your effective net worth considering all your other income (salary, bonuses, royalty checks from those sweet publishing gigs, etc.)? It's not hard, just work backwards: if your income is $40K then your effective net worth is $1M; if it's $80K, then your effective net worth is $2M, etc. This gives you an idea of how much you'd need to have saved for retirement if you want to continue your current lifestyle. Of course, there are other considerations.

Shit, I could write about this sort of thing all day. Back to the actual topic.

If we leave out assets like houses and such, I'd be hard-pressed to come up with a way to blow through a million dollars in one day. Give me a couple of hours on Amazon and I could probably do it, though. Or, more likely, I'd find some experience to spend it on. Rich people go on these round-the-world cruises, I've found, and they get expensive. The cruises last for months, but you have to give them a deposit; I'd imagine they wouldn't say no to getting the whole price up front.

Or, it'd be tempting to sign up for one of those space tourism services. For something on the order of a million dollars, you can reserve a spot on a suborbital flight that technically takes you past the imaginary line separating atmosphere from space. The flight lasts maybe an hour, though, and all you really get to show for it is bragging rights. "I have slipped the surly bonds of Earth!" "Yeah? I've been scuba diving at the Great Barrier Reef."

I should note here that I'm also deliberately leaving out any variation on "giving it to charity." That's not spending it, by my definition, and usually all you're doing is enriching the charity itself, and its organizers. I'm not suggesting we shouldn't give to charities, but I am explicitly leaving them out of this particular discussion.

And then, of course, there's the old standbys. I wrote the following around this time last year, so it's appropriate I trot it back out now. If I had to spend a million dollars in one day, I'd spend most of it on...

Alcohol, Hookers, and Blow

(18+)
A Thanksgiving thanks giving
#2175291 by Waltz en France

The rest, I'd probably waste.

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