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!

about 13 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 15 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 15, 2022 at 12:02am

#1030742

Getting Cabbaged

One of the many euphemisms we used for "drunk" (as an adjective) in college was "cabbaged."

I haven't used it in a long time, but today's article reminded me. The word also has the distinction of being the longest one I can think of that can be spelled with musical notes.

Grilled Cabbage "Steaks"

Admittedly, I'd have to be pretty cabbaged to make these, which would pretty much contraindicate the use of a grill for cooking. Or anything else, for that matter.

Okay, we know these are not quite grilled steaks made out of meat.

Not "quite?" They're not even in the same universe. But considering the price of dead bovine (and everything else) lately, alternatives may need to be considered.

I'm just not convinced that this should be one of them.

But whether or not you're vegetarian, these cruciferous "steaks" are potentially even better.

"Potentially" being the key word here. I wonder if they might steer us wrong?

(Bad cow pun)

Cabbage is a chameleon ingredient, good and good for you and affordable.

Econ 101: the more people buy cabbage, the less affordable it's going to be.

To make sure your cabbage steaks stay in one piece on the grill, pick the most compact head of cabbage when you're out shopping.

Or, if you're me, try to get the Instacart shopper to pick a compact head of cabbage. Don't be surprised when they don't.

All you need to make grilled cabbage steaks is cabbage, oil, salt. Everything else is optional, but it's the optional things that make life what it is.

Another "optional" thing is to try something else.

We top this grilled cabbage à la wedge salad with bacon, blue cheese, AND ranch dressing for good measure.

Thus negating any health benefits one might derive from cabbage. But with all of that, it sounds... well, not terrible, anyway. Still, if you're going to fake French for the "à la wedge salad" part, the least you could do would be to spell it "bleu cheese."

Other fantastic options would be a buffalo-style cabbage, a Caprese-style cabbage, or a lemon-garlic dressed grilled cabbage with lemon and garlic. You really can't go wrong!

Except that you already have.

The article, of course, goes on to provide the actual recipe. Which, in their defense, looks dead simple to prepare -- even easier than actual steak, assuming of course that you take the time to prepare a marinade for said steak like a civilized person instead of just flopping it from the package onto the grill like a barbarian. Which I've done, because when it comes to cooking, for me the operative word is "lazy."

In fact, the hardest part would seem to be actually slicing the head of cabbage, which I definitely don't recommend doing while cabbaged.

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