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 10 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 12 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.

May 9, 2022 at 12:01am

#1032143

Shooting Tsar

"Invalid Item"

So on Wednesday, you told Andre about great places to visit where you live. Now he'd like to know about the people who live or have lived where you do.

Sundays's prompt; Tell us about a famous person or persons who lives or came from your Home Sweet Home

When I was at college, the dorms were arranged with several dorm rooms around a common area. Someone had done a country-looking cross stitch as a throw pillow: "Suite sweet suite."

That always pops back in my mind whenever I see "home sweet home."

Anyway. I've talked enough about Jefferson, and I mentioned Dave Matthews last time. I've also noted before that Poe spent some time here. Then there was the one lady who claimed to be the last Romanov...

You know what? I'm going to go with that one.

Hopefully you all know that in 1918, some revolting Russians rounded up some revolting Russians. That is to say, the revolutionary Bolsheviks executed the entire former royal family of Russia, who, let's face it, were pretty revolting, being autocrats and all. The youngest was Anastasia Romanova, also known as Grand Duchess Anastasia

despite being, like, totally a teen.

A couple of years later, a woman in a mental institution in Germany (should have been the first clue) claimed to be Anastasia. Apparently, some people believed her, but certainly not everyone. Personally, I'd figure that claiming to be the last royal Russian in early 1920s Europe couldn't possibly be good for one's health, but okay, whatever, there were enough believers that her claims were taken seriously by a bunch of people.

Apparently, she spent the next few decades in and out of asylums, which I suppose could be ordinary mental illness, or it could have been, I don't know, watching a bunch of revolting peasants destroy your entire family.

It wasn't until the late 60s that she became associated with Charlottesville, having married a history professor here (should have been the second clue).

I never met the lady; she died shortly after I came to town (not my fault). My ex-wife's father claimed to know her, but he claimed to know everyone (and, to be fair, he knew a whole lot of people). Apparently, Anna and her husband were described as "eccentric," which is code for "crazy, but rich enough to get away with it."

It was only after her death that the final nails were driven into the coffin of her claims to royalty. I mean that metaphorically; she was cremated and didn't have a coffin.

First, DNA testing became a thing, and apparently there was enough of her left (ew) to test, and behold, she was not Russian.

And second, after the end of the USSR, archeologists found the remains of the Tsar's entire family, including Anastasia.

A local brewery occasionally makes a beer called Anastasia's Chocolate Fantasy, which is a Russian Imperial Stout brewed with cocoa, because of this local connection. It is absolutely delicious. But Russian Imperial Stout is, as I've noted before, not actually Russian... much like Anna Anderson herself

So Anna Anderson was just a crazy lady, but apparently she made the best life out of it as she could, and, Duchess or not, she's absolutely a part of Charlottesville history. Though few people here actually believed she was Romanova, they humored her anyway, because that's what we do here. It really should have been blindingly obvious in retrospect, though.

I mean.

If you're the only heir to the Russian throne, and you saw a bunch of peasants rise up and grind your family into borscht, and you need to find a place to live...

...would you really choose the state whose motto is "sic semper tyrannis?"

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