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 6 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 8 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.

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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').

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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.

July 30, 2021 at 12:02am

#1014738

Tower

Unlike those about the band or the car company, articles about Tesla the man never get old.

The Rise and Fall of Nikola Tesla and his Tower
The inventor’s vision of a global wireless-transmission tower proved to be his undoing

I know I've talked about him before, but I thought this article shed some additional light on what I knew of his history.

By the end of his brilliant and tortured life, the Serbian physicist, engineer and inventor Nikola Tesla was penniless and living in a small New York City hotel room. He spent days in a park surrounded by the creatures that mattered most to him—pigeons—and his sleepless nights working over mathematical equations and scientific problems in his head.

Too bad the pigeons didn't have answers.

Tesla believed his mind to be without equal, and he wasn’t above chiding his contemporaries, such as Thomas Edison, who once hired him. “If Edison had a needle to find in a haystack,” Tesla once wrote, “he would proceed at once with the diligence of the bee to examine straw after straw until he found the object of his search. I was a sorry witness of such doing that a little theory and calculation would have saved him ninety percent of his labor.”

Edison was probably like, "Theory? Calculation? That's for pussies!"

He managed to convince J.P. Morgan that he was on the verge of a breakthrough, and the financier gave Tesla more than $150,000 to fund what would become a gigantic, futuristic and startling tower in the middle of Long Island, New York. In 1898, as Tesla’s plans to create a worldwide wireless transmission system became known, Wardenclyffe Tower would be Tesla’s last chance to claim the recognition and wealth that had always escaped him.

I've never been clear on whether wireless electricity transmission was a fever dream or an actual possibility. It's not my field (pun intended). I am sure that if it were, we'd never know because there's no way to monetize it.

He would spend the next six years of his life “thinking” about electromagnetic fields and a hypothetical motor powered by alternate-current that would and should work.

And of course, AC electricity is how we get most of what powers us now. This is mostly thanks to Tesla. If Edison had had his way, well, we'd be in an even worse dystopia than we're in now.

According to Tesla, Edison offered him $50,000 if he could improve upon the DC generation plants Edison favored. Within a few months, Tesla informed the American inventor that he had indeed improved upon Edison’s motors. Edison, Tesla noted, refused to pay up. “When you become a full-fledged American, you will appreciate an American joke,” Edison told him.

I'm also unclear on whether this story is apocryphal or not. But it does capture Edison's essential dickishness.

His work with electricity reflected just one facet of his fertile mind. Before the turn of the 20th century, Tesla had invented a powerful coil that was capable of generating high voltages and frequencies, leading to new forms of light, such as neon and fluorescent, as well as X-rays. Tesla also discovered that these coils, soon to be called “Tesla Coils,” made it possible to send and receive radio signals. He quickly filed for American patents in 1897, beating the Italian inventor Guglielmo Marconi to the punch.

Other fans of Tesla have made a great deal out of it being him, and not Marconi, who invented radio (suck it, Starship). While this certainly has a kernel of truth, the invention of radio as we know it today was somewhat convoluted and there wasn't just one inventor, the way we like to think of things.

On the tower which is ostensibly the focus of this article (keeping in mind that this was more than 50 years before satellites, and even before airplanes):

“As soon as completed, it will be possible for a business man in New York to dictate instructions, and have them instantly appear in type at his office in London or elsewhere,” Tesla said at the time. “He will be able to call up, from his desk, and talk to any telephone subscriber on the globe, without any change whatever in the existing equipment. An inexpensive instrument, not bigger than a watch, will enable its bearer to hear anywhere, on sea or land, music or song, the speech of a political leader, the address of an eminent man of science, or the sermon of an eloquent clergyman, delivered in some other place, however distant. In the same manner any picture, character, drawing or print can be transferred from one to another place. Millions of such instruments can be operated from but one plant of this kind.”

Well, that sounds familiar. Sure, it was implemented differently (and incompletely), but the vision was pretty clear.

Unfortunately, it also led to social media.

Wardenclyffe Tower became a 186-foot-tall relic (it would be razed in 1917), and the defeat—Tesla’s worst—led to another of his breakdowns. ”It is not a dream,” Tesla said, “it is a simple feat of scientific electrical engineering, only expensive—blind, faint-hearted, doubting world!”

For a while there, Musk had the idea to rebuild Wardenclyffe. Like a lot of what Musk does, that seems to have gone by the wayside. It wouldn't have done anything much, but stand as a monument to one of the greatest geniuses of all time.

“The present is theirs; the future, for which I really worked, is mine.”
― Nikola Tesla

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