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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Brandiwyn🎶

a day ago
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').

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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.

Aiva Raine

Nov 5, 2024 at 3:03pm
In response to "It's Hard Not To Scare"

Shaking head. Aside from the inevitable Simpson's spider goat song playing in my head which I will now share with you below, whenever I come across a story that has all these scientists doing ridiculous experiments for no logical reason, I must go- ah- scientists- CANCER? You do remember that problem right? Can we maybe cure that before we make cockroaches magnetic and elephants see pink elephants?

spider pig clip- and if you all now are singing the spider goat song- you're welcome! grin
https://www.youtube.com/watch?v=BARjPuUN36Y

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Aiva Raine

Nov 5, 2024 at 2:55pm
In response to "Anomalies"

One of your more humorous entries- I especially enjoyed- "Lofoten Vortex can be the name of my Bjork cover band."

Thanks for the laughs. Much appreciated.

2 Like

July 26, 2023 at 9:33am

#1053066

It Ain't Wright, but It Ain't Wrong

I wanted to share this article, not just because the story isn't that widely known, but because of a tenuous personal connection to it.

This Odd Early Flying Machine Made History but Didn’t Have the Right Stuff
Aerodrome No. 5 had to be launched by catapult on the Potomac River on May 6, 1896, but it flew unpiloted 3,300 feet

The vessel floated in the shallows of the Potomac River on the leeward side of Chopawamsic Island, just off Quantico, Virginia.

It's fairly common knowledge that Maryland claims the Potomac. In contrast with standard surveying practice, which usually puts boundaries in the river's thread or main channel, Maryland gets the whole expanse, up to some defined line (I think it's the mean low tide line where it is tidal, which is all the way up past DC) on the opposite side.

This is in spite of the fact that the river is named for an Indian tribe that lived, to the best of my knowledge, exclusively in what is now Virginia. The reason for this isn't relevant to this story, though, except to note that because of this historical oddity, there just aren't that many islands in the Potomac that are part of Virginia. And this one is also claimed by the county where I (arguably) grew up, exactly (I checked) 10 miles from my childhood home.

They had to do some creative surveying to include it within Stafford County, but it's there, in a kind of pseudopod-like extrusion.

So, that's the personal connection. I told you it was tenuous.

History would be made that day, May 6, 1896, as this apparatus—a flying machine, known as Aerodrome No. 5—was started and then launched from a spring-loaded catapult.

An "aerodrome" is what we (well, mostly the British) call what's basically a small airport today. In 1896, no airports existed, so I guess the word meant something different.

The Aerodrome would take off and travel for 90 seconds some 3,300 feet in an effortless spiral trajectory and then gently land in the river.

Both time and distance exceed my best efforts at paper airplanes when I was a kid. To be fair, those experiments usually ended at a classroom wall.

The third Secretary of the Smithsonian Institution, Samuel Pierpont Langley, an astronomer who also enjoyed tinkering with his own creations, was aboard the boat. His winged invention had just made the world’s first successful flight of an unpiloted, engine-driven, heavier-than-air craft of substantial size.

Yeah... those are some specific definitions. Not quite the level of success the later Wright Brothers enjoyed, but why let North Carolina get all the credit?

As for Langley himself, yes, it was that Langley, the one the famous Air Force base is named for. He was also a Secretary at the Smithsonian (a position similar to Royal Astronomer of Britain), so of course this article is from Smithsonian.

With Langley that day, was his friend Alexander Graham Bell, the inventor of the telephone...

Another dubious claim, but that's what history insists upon, though that's really irrelevant right now.

It should be emphasized again that this flight wasn't quite as significant as the Wright one, and the article does so:

The world rightly remembers that in 1903 the Wright brothers achieved human flight at Kitty Hawk in North Carolina. “Langley’s Aerodrome No. 5 wasn’t practical and it wasn’t a working prototype for any real flying machine,” says Peter Jakab, senior curator at the museum. But the largely forgotten unpiloted flight that took place seven years before Kitty Hawk did move motorized flight from the drawing board into reality.

Langley was also apparently one of the last great polymaths:

“Langley’s real accomplishments in research were in astronomy,” says Jakab. “He had done a great deal of significant work in sun spots and solar research, some of that while at the Smithsonian.”

He had a bit of mad scientist about him, apparently:

“This was still a period when people didn’t think flight was possible,” Jakab says. “If you were a young person in the 1890s contemplating a career in engineering, flight was not exactly an area you would go into. It wasn’t taken seriously by a lot of people. The fact that someone like Langley was starting to study flight gave the field credibility.”

The article goes on to describe his fundamental errors in design, and how he lost out to those brothers from Ohio a few years later.

“Those two catastrophic failures in 1903 ended Langley’s aeronautical work,” Jakab says. “He was a broken man because he took a lot of ridicule. He spent a lot of money and did not achieve a great deal in this field.”

Perhaps not, but we also learn from failures. And by all accounts, he accomplished great things... just not so much in aviation. Still, he did what few others even dared to try, and isn't that worth remembering?

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