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:

Write_Mikey_Write!

a day ago
In response to "Global Domination"

A little girl wearing a Hello Kitty shirt in Middle-of-Nowhere, Siberia will certainly make you wonder how it got there. My grandson has a souvenir t-shirt with a Terex 33-19 "Titan" on it. When it was built in 1973, the Titan was the largest, highest payload capacity truck ever built, a title it held for 25 years. The thing is, the truck on that t-shirt is on display in Sparwood, BC, Canada - over 1000 miles from here, and his mom got the shirt at a local thrift store. Makes me wonder.

Oh, yeah - I thought 'Schrödinger's Hello Kitty' was funny, too.

1 Like

Adherennium Plotting something

a day ago
In response to "Global Domination"

For whatever warped reason, 'Schrödinger's Hello Kitty' has tickled my funny bone.

3 Like

Brandiwyn🎶

Nov 12, 2024 at 10:50am
In response to "Blond Man's Bluff"

That's interesting about the two spellings of blonde. I always thought blonde referred to the person and blond referred to the hair. But I never bothered to look it up, So you've catered to my lazy today.

3 Like

Kåre เลียม Enga

Nov 9, 2024 at 1:02pm
In response to "It Was a Dark and Starry Night"

Ah... but why did he see those swirls when many of us don't? A totally different discussion... out of thin air... which is a frozen fog today when I gaze out the window.

Humans have great skills of observation; some of us extraordinary skills. Is The Scream depictive or evocative? 😱

Turkey DrumStik

Nov 9, 2024 at 11:04am
In response to "It Was a Dark and Starry Night"

Frankly, I'm more surprised to see reasonably good reporting from CNN. I'd given up on their ability to rein in their tendency toward loaded speech.

2 Like

JayNaNoOhNo

Nov 8, 2024 at 12:00pm
In response to "Bungalow Bash"

For the love of God, what did I just read? That is by far the worst use case of a bungalow I've ever seen, and a fundamental misunderstanding of the value of front porches—and the size of bungalows.

Was this his point? Nope, I know what his point was. For a guy so worried about presentation, he certainly undercut his own arguement.

1 Like

Turkey DrumStik

Nov 8, 2024 at 9:47am
In response to "Bungalow Bash"

"People don't change their minds on the internet. They usually do that in books, battlefields, or not at all.

Yeah, that's a little bleak."

Bleak and also inaccurate.

1 Like

Kåre เลียม Enga

Nov 7, 2024 at 10:19pm
In response to "Past and Future Intents"

I must live in the present. In the past I daydreamed a lot... probably to escape the bruises of reality; however, I need to be more aware crossing streets or even how I get out of bed (no rapid movements without thinking of consequences).

I can envision a future more peaceful than the present because this time-and-place isn't the only "present" that's ever existed, and I do try to see the larger view (emphasis on 'try').

1 Like

Lilli 🧿 ☕

Nov 7, 2024 at 10:58am
In response to "Past and Future Intents"

If the aforementioned exercises don't help; there's still liquor.

3 Like

Turkey DrumStik

Nov 6, 2024 at 10:22am
In response to "Lo!"

Not gonna lie I need the holiday season chatter.

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 Robert Waltz

The rest, I'd probably waste.

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