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').

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.

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

1 Like

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

June 1, 2024 at 9:13am

#1071970

Border Dispute

Wars have been fought over less than this.

The Man Who Went to War With Canada
For centuries, the United States and Canada’s only remaining land border dispute has been kept alive by a single family.

Article is from Atlas Obscura and 2019, but the Wikipedia article

suggests that the issue hasn't been resolved over the last five years.

After probably a few hours at sea, they reached the island’s rocky shore, and managed to land among the island’s main residents: puffins, razorbills, murres, and Arctic terns.

If I were that guy, I'd have pointed at one of the latter birds and proclaimed: "It's my tern now!"

Depending on whom you ask, Machias Seal Island is either off the coast of Maine or of Grand Manan. It’s also either American or Canadian. It is the only place with this particular unsettled identity that you can actually stand on top of. Although the ownership of some stretches of water is still contested, this island—and neighboring North Rock, which is even smaller and barer—are the last crumbs of their land the two countries don’t agree on.

Just wait a few years, and climate change will settle the debate for us.

I feel obliged to point out that while I've never been to this island, it sits within easy distance of the easternmost point of the continental US: Quoddy Head Lighthouse. I visited that spit of land when I decided to travel from it to the westernmost point, in the state of Washington. Quoddy Head was the easier of the two to reach, requiring significantly less hiking and way fewer bears.

The article delves into the source of the land dispute, then:

If the world were more just, this would all be moot: The people of the Passamaquoddy Nation likely used the island long before anyone else even knew it existed. (“Machias,” also the name of precipitous local river, is a Passamaquoddy word that means “bad little falls.”) Instead, even as their identifications and affiliations have shifted, the neighbors have kept squabbling over it, like a pair of growing siblings in a shared bedroom.

Look, let's hope it stays on the "siblings" level of quarrel. We don't want to go to war with Canada. We have a history of losing said wars.

The rest of the (rather long) article delves into the American side of things, focusing, as the headline implies (look at that, a headline matching up with the text), on one person's, or more properly one family's, quest to keep the US claim to the island.

I won't quote further, but it's a way more interesting read than the Wiki page.

2 Like • Comment

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