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 8 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 10 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.

March 14, 2024 at 9:54am

#1066257

Current Events

Remember Rebecca Black's squirmy earworm song, "Friday?" No, I will not link it. Well, this morning, I was tempted to do a parody in honor of today called "Pi Day," but my cats talked me out of it, so we all get to keep our sanity.

Unlike this guy, apparently. From Big Think:

Tesla’s pigeon: How the great inventor fell for a bird
"She understood me and I understood her. I loved that pigeon.”

The main trouble with Nikola Tesla's legacy is that his name was appropriated by a mediocre band, and then run into the ground by one of the most successful con artists in history. The only way to reclaim what shouldn't be a laughingstock, I think, is to remember the inventor's life, accomplishments, and, yes, even his apparent madness.

On a February morning in 1935, a disoriented homing pigeon flew into the open window of an unoccupied room at the Hotel New Yorker.

That hotel, though it's been through a few changes over the decades, still exists. I've stayed there. I don't think the windows open anymore; with air conditioning, it's not necessary, and maybe it prevents some jumpers.

While management debated what to do, a maid rushed to the 33rd floor and knocked at the door of the hotel’s most infamous denizen: Nikola Tesla.

When I was there, I even made a pilgrimage to the 33rd floor.

“Dr. Tesla … dropped work on a new electrical project, lest his charge require some little attention,” reported The New York Times.

"Charge?" They just couldn't resist, could they? Revolting how some people just plug in the most obvious puns.

Nikola Tesla—the Serbian-American scientist famous for designing the alternating current motor and the Tesla coil—had, for years, regularly been spotted skulking through the nighttime streets of midtown Manhattan, feeding the birds at all hours.

He invented way more than that. Some say, rather poetically, that he invented the 20th century. While a bit hyperbolic, it's not that far off the mark.

In the dark, he’d sound a low whistle, and from the gloom, hordes of pigeons would flock to the old man, perching on his outstretched arms.

Look, all I'm saying is, that's a remarkable image and if someone hasn't painted that, someone definitely should.

Tesla said that he and his bird could speak to one another mind to mind, and that sometimes, as they silently conversed, beams of light would shoot from her eyes.

This is the sort of thing I meant by "madness" above. But is it really? Or was he operating on a different level of reality? With genius like Tesla's, there's always that seed of doubt: maybe he was right, and it's the rest of us who are blind.

Tesla’s love of pigeons was an obsession with a capital O. Likely followed by a capital C and a capital D. He seems to have suffered from obsessive compulsive disorder, and his case was severe.

Anyone with only a passing knowledge of Tesla's life, and of pop psychology, would immediately jump to the same conclusion. While it can be dangerous and rude to diagnose someone from some distance in space and time, all the signs were absolutely there. I'd even throw in the possibility of autism.

I won't quote more from the article, which is fairly long, but, and I can't emphasize this enough, beautifully written. It weaves quite the tapestry of history and science, and, of course, there's a bit of Mark Twain in there.

This is where I'd usually relate the subject of the article to Pi Day, but all I could think of was the relationship between pi and the sine waves of electrical current, but maybe that's a bit too obvious while at the same time too esoteric. So I'll just leave it at that and go find me some pie.

6 Like • Comment

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