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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Show Recent Comments

10 Most Recent Blog Comments:

Write_Mikey_Write!

about 15 hours 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

about 17 hours 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.

April 22, 2020 at 12:01am

#981725

Sit Back

No.

https://www.outsideonline.com/2391763/stop-reclining-your-seat-airplanes

Stop Reclining Your Seat on Airplanes
You heard me. On cramped flights, the person who reclines their seat in front of me really chafes.

You know what really chafes? Complainers.

I love airline travel.

You're a monster.

The predictable flow of airports soothes me, and I get giddy during takeoff because I still don’t quite understand the science (read: magic) that keeps planes aloft.

And you're willfully ignorant.

Let me be clear: reclining is perfectly acceptable on flights longer than, say, four or five hours, especially if it’s an overnighter.

And who appointed you Arbiter?

So when we all pile into our clown car in the sky, we do so knowing that for whatever time we’re at cruising altitude, we’re going to sacrifice some personal space and our legs are going to fall asleep. Which is why my fellow economy-class members who recline their seats are the worst.

So don't fly.

Let’s get real. Those extra three degrees of slope on your reclined seat back aren’t going to make you more comfortable.

Yes. Yes, they are. And you're engaging in hyperbole.

Did they have a drink on their tray table? You didn’t think to check before you reclined onto their warm Diet Coke, did you? Were they working on their laptop? Now their screen is hunched forward at an angle that makes productivity impossible.

How about: Be aware that the person in front of you will recline their seat, and act accordingly. And for fuck's sake, if you need to spend every waking minute being "productive," you're the one with the problem.

The thing is, you know you suck when you hit that recline button, but just in case you truly are ignorant, hear this: your actions have direct consequences for the people around you—any space you take is taken from someone else.

Damn right. I have one magic button on my shitty airplane seat that makes me marginally less uncomfortable, and I am going to use it.

So if you want to lounge back and make life a little bit worse in an already untenable situation, I suggest you get rich, leave us common folk behind, and live it up in business class, which United is expanding while, you guessed it, shrinking economy.

Oh, I've done that. Not just business class, but first class. And believe me, nothing -- nothing -- gives me more pleasure than sitting in a wide, comfortable first-class seat, sipping my complimentary Bloody Mary, and watching people like you struggle past with disease-vector offspring and more stuff than will ever fit into an overhead compartment, let alone under the seat in front of you.

But when I do fly cattle-class, I always make sure to take off my shoes, fart silently, eat fragrant food, and, yes, recline my goddamned seat.

Want to make air travel more comfortable? Take it up with the airlines, and be willing to pay more for the privilege. You know, if there are still airlines after all this.

2 Like • Comment • View 2 comments

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