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:

Turkey DrumStik

about an hour ago
In response to "Barley There"

And approximately 15 seconds later, someone's going to screech about genetically engineered crops.

That would be me...screeching for more genetically engineered crops.

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

February 2, 2019 at 12:31am

#950956

Hav-a-Tampa

https://www.tampabay.com/visual-arts/top-things-to-do-in-tampa-bay-for-saturday-...

Now this... this is more my speed.

Full disclosure: I had ties to Tampa. My uncle lived there (died in the early noughties). He was kind of a big deal there, has buildings named after him and shit. Nice place. Never snows. Oh, sure, you get the occasional hurricane, but what place is completely safe?

Sadly, I haven't had a chance to go back since his death. I need to - there are a lot more breweries there, now, and plus there's this:

https://yborcityonline.com/

And this, across the bay:

https://thedali.org/

And of course, these:

https://foursquare.com/top-places/tampa/best-places-cigars

People are always asking me about Cuban cigars, since I make no secret of my hedonism and decadence. We still can't legally get them in the US - stupid embargo that long ago outlived its usefulness. But of course, Florida has a thriving Cuban population, so you get things like Ybor City, above, and one of Tampa's major beer factories is called Cigar City Brewing.

Don't tell my government, but I've had Cuban cigars anyway. I've visited Canada and the UK and purchased (and smoked - couldn't risk schlepping them back to the US) them there. And I'm here to tell you: Cuban cigars aren't all that.

Supposedly, when Kennedy initiated the embargo, he filled a giant humidor with Cuban cigars before anyone else knew the embargo was to take effect. The perks of leadership, I suppose. You know where that got him. I have no idea what happened to the cigars. Possibly, Johnson smoked them all. I like to think it was Nixon.

But since then, the major manufacturers and growers have gone to places like the Dominican Repulic, Guatemala, Nicaragua... and Florida. What we get from them is every bit as good, possibly even better, than what comes out of Cuba. Hell, there are excellent sticks coming out of India. The only things left of the Cuban cigar legacy are those smokes, and for people in the US, a cachet of lawlessness that surrounds the very idea.

I'm not a fan of Florida in general. I mean, look at the place on the map - what is it, but the Wang of America? (Yes, I got Turkey DrumStik

, who wisely moved out of the state, to start calling it that.) And the Keys are... well, let's just say the place could use a few antibiotics. Disney World is like a giant chancre sore in this metaphor, as are The Villages, a sprawling retirement community about an hour north of Orlando where golf carts have the right-of-way and the houses all look like they only need one good Cat I hurricane to end up in the Gulf. So, there's no danger of me joining the herds of other retirees living there.

Doesn't mean I don't want to visit again.

I'll end this with a joke. It's an old one, and I'll try to render it faithfully. I promise it's relevant on a couple of levels:

Guy dies and goes to Hell. One of the demons notices that, despite the high temperatures, he's smiling all the time, so he reports to his Dark Lord. Satan goes up to the guy and goes, "You know you're in Hell, right? It's as hot as... well, it's hot here. Why are you smiling?"

"Well," says the guy, "This is just like back on the farm in August. I'm used to it."

So Satan puts him, alone, in an even hotter place. But a week later, he's still smiling and laughing.

"This is Hell," Satan reminds the guy. "You're supposed to be tortured here. What's with the smiling?"

"This is just like being in the silos back home," says the guy.

So Satan leaves and turns to the demon in charge of solitary confinement: "Okay, turn the heat all the way down."

So a week later, it's sleeting and icing up in there, and Satan looks in to see the guy laughing and dancing a jig, occasionally throwing his arms up in victory. "Dammit," he says. "What does it take? And don't tell me this reminds you of your farm."

"Look around you," the guy explains, smiling and gesturing expansively at all the ice and snow. "The Bucs have won the Super Bowl!"

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