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.

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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.

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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

June 2, 2023 at 10:14am

#1050418

Powerless

Throwing shade on solar power...

An activist group is spreading misinformation to stop solar projects in rural America

Hm, I wonder who would be motivated enough to fund such a group... let me think... nope, coming up empty.

Citizens for Responsible Solar is part of a growing backlash against renewable energy in rural communities across the United States.

It's important, when fighting against things that are net social goods, to use certain keywords to make people believe that you're not evil. It worked for various "democratic people's republic" dictatorships. "Family" is a perennial favorite. Or "children." But "responsible" and "citizens" are up there.

Citizens for Responsible Solar was founded in an exurb of Washington, D.C., by a longtime political operative named Susan Ralston who worked in the White House under President George W. Bush and still has deep ties to power players in conservative politics.

See? Evil.

And when Ralston was launching the group, a consulting firm she owns got hundreds of thousands of dollars from the foundation of a leading GOP donor who is also a major investor in fossil fuel companies.

Oh, wow. Wow. I never would have guessed, not in a million years.

Analysts who follow the industry say Citizens for Responsible Solar stokes opposition to solar projects by spreading misinformation online about health and environmental risks. The group's website says solar requires too much land for "unreliable energy," ignoring data showing power grids can run dependably on lots of renewables. And it claims large solar projects in rural areas wreck the land and contribute to climate change, despite evidence to the contrary.

Evidence? Who needs evidence when you have money?

And you know what form of energy has well-documented, proven health and environmental risks? I'll give you a hint: it rhymes with ossil uels.

People often have valid concerns about solar development. Like any infrastructure project, solar plants that are poorly planned and constructed can potentially harm communities. But misinformation spread by groups like Citizens for Responsible Solar is turning rural landowners unfairly against renewables, says Skyler Zunk, an Interior Department official under President Donald Trump and chief executive of Energy Right, a conservative-leaning nonprofit that supports solar projects that preserve ecosystems.

Any change in land use comes with potential hazards to the environment. Change the way runoff works, and you change erosion patterns and a watershed's hydrological characteristics. But there are ways to mitigate that.

Solar restrictions are gaining traction as the stakes for addressing climate change keep rising. Construction of more renewable energy is key to the country's plans to cut greenhouse gas pollution and avoid the worst damage from extreme weather in the years ahead.

Such is the nature of humanity that avoiding "the worst damage from extreme weather in the years ahead" will mean absolutely nothing, because the damage has been avoided. Kind of like people scoff at the giant nothing that was the Y2K problem, because they didn't notice efforts to keep it from happening.

Ralston reportedly said to E&E News in 2019 that no money from fossil fuel interests went to Citizens for Responsible Solar. Since then, she has declined NPR and Floodlight's requests to identify the organization's sources of funding.

I thought "why not tell the whole truth if you have nothing to hide" was a core conservative "value."

Many grassroots activists today credit their success in stopping solar projects to Ralston. One of them is Kathy Webb. Webb says she learned that a company wanted to clear a forest for a solar plant near her home in Rowan County, N.C., days before local officials planned to vote.

I gotta admit, I'm not sure that clear-cutting forests for solar energy is a net positive. Trees are kinda important for carbon capture. But I don't have any real data one way or the other.

Misinformation spreads easily online. It bounces around in echo chambers where dubious articles, videos and memes are posted and shared repeatedly. Longtime critics of the wind industry like John Droz cultivated opposition strategies that are now being used in the fight against solar, says Anderson, the renewable energy lawyer in Kansas.

And really, my point here isn't to rag on one group or another. I want to see decisions based on facts, not misinformation (or deliberate disinformation). Lots of people on both sides of the US political spectrum bang on about questionable or outright disproven studies or statements.

Every form of energy creation has its downsides. The question is whether the benefits outweigh them or not.

But, you know. Keep promoting coal and oil if you need to. By the time it all runs out and we can no longer sustain an industrial, technological society, I, too, will be long gone.

Sucks for your descendants, though.

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