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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Write_Mikey_Write!

a day 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

a day 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.

October 11, 2019 at 12:01am

#967619

I'll Drink To That

So, speaking of drinking...

https://www.atlasobscura.com/articles/things-invented-in-bars

How Prohibition Tossed a Wet Blanket on America’s Inventors
New research reveals the link between bars and new inventions.

There have been dark times indeed in the US. But the darkest, most uncivilized period of the 20th century was certainly Prohibition - with McCarthyism running a close second.

I've known this on an instinctual level all my life; after all, my country pays lip service to "freedom" without actually allowing freedom, but Prohibition was the most egregious example of this.

We also pride ourselves, sometimes to an unreasonable extreme, at innovation. So it shouldn't be surprising that booze - specifically, the public consumption thereof - is linked to innovation.

“Researchers tend to think that informal social interactions—people bumping into one another and swapping ideas—is vital for innovation,” Andrews says... “If you press economists on this when they’re giving talks, and ask why it matters that everyone’s in the same city or within a few blocks, they’ll say something like, ‘People get together and talk at the bar,’” says Andrews. “I’ve actually heard this multiple times. [But] I don’t think direct evidence of that has ever existed before.”

Hypothesis: Bars are essential for innovation. Here comes the science.

He downloaded patent data, compared the number granted to inventors in the wet and dry counties before and after statewide prohibition began, and came up with a measurement of the importance of slightly drunken discussion to invention.

The result? A 15 percent decrease in the number of patents. The areas whose saloons shuttered had become less inventive.

Dark. Times.

Prohibition disrupted much more than nightlife. In a tradition dating back to the country’s origins, Americans had long favored beer and bars over coffee and cafés. “The American Revolution was basically plotted in local taverns in Boston and Philadelphia,” says Andrews. Repudiating the tea-loving Brits they’d fought against, Americans viewed bars as venues of culture.

And rightly so.

For his part, Andrews came up with the idea for his research while chatting with other graduate students at a bar, Joe’s Place, in Iowa City. “I find the bar to be a great place to air out lots of ‘bad ideas’ in a setting that is safer than the seminar room,” he says

So, again, it's not drinking but social drinking that's important. Being around people tends to curb the worst effects of overindulgence, provided you're among people who are looking out for you. And presumably, it doesn't matter if an individual drinks or not; what matters is the free exchange of ideas - yet another miracle brought to you by the wonder compound, ethanol.

1 Like • Comment

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