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.

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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.

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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.

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April 25, 2023 at 9:56am

#1048752

That Newfangled Junk

By coincidence, today's article also talks about a late nineteenth century innovation, one which was apparently met with some pushback.

When the Push Button Was New, People Were Freaked
The mundane interface between human and machine caused social anxiety in the late nineteenth century.

Pun, of course, intended.

The electric push button, the now mundane-seeming interface between human and machine, was originally a spark for wonder, anxiety, and social transformation.

Because it's obviously sorcery.

As media studies scholar Rachel Plotnick details, people worried that the electric push button would make human skills atrophy. They wondered if such devices would seal off the wonders of technology into a black box: “effortless, opaque, and therefore unquestioned by consumers.”

Does this sound to you like almost every other complaint about new technology and/or ways of doing things? It does to me.

Right after we invented fire, a bunch of skin-wearing humans probably freaked out, too. "What is this 'starting a fire' bullshit? In MY day, we'd steal it from a burning tree after the gods smote it with lightning, as they intended. This is going to make people lazy and entitled, you mark my words."

“Some believed that users should creatively interrogate these objects and learn how they worked as part of a broader electrical education,” Plotnick explains. “Others…suggested that pushing buttons could help users to avoid complicated and laborious technological experiences. These approaches reflected different groups’ attempts at managing fears of electricity.”

One thing I remember about the show The Jetsons from when I was a kid (haven't seen it since) was that George Jetson's job in our utopian Eloi-like future involved pushing seemingly random buttons all day, to the point that his finger would cartoonishly swell and throb in pain.

Well. At least they got that part right (he typed on his computer keyboard). Still no flying cars, though. By the time we get those, it's all going to be touchscreens and voice interfaces, both of which are going to make people lazy and entitled, you mark my words.

Electric push buttons, essentially on/off switches for circuits, came on the market in the 1880s.

For those of you who didn't bother with yesterday's entry, that's more than a decade after the invention of blue jeans.

The word “button” itself comes from the French bouton, meaning pimple or projection, and to push or thrust forward.

I don't claim to be an expert in French, but I learned "bouton" to mean like the button on a shirt (or Levis), and that word doesn't look like a verb form to me. Perhaps archaic French; je ne sais pas.

Those who promoted electricity and sold electrical devices, however, wanted push-button interfaces to be “simplistic and worry-free.” They thought the world needed less thinking though and tinkering, and more automatic action. “You press the button, we do the rest”—the Eastman Company’s famous slogan for Kodak cameras—could be taken as the slogan for an entire way of life.

Like I said, lazy.

What I didn't say (at least not yet in this entry) is that I consider that a good thing. Necessity may be the mother of invention, but laziness is the milkman. Bill Gates supposedly once said that he hires lazy people because they'll figure out the most efficient way to do things. I have no idea if that quote is correctly attributed, but someone said (or wrote) it, and it rings true.

Ultimately, the idea that electricity was a kind of magic would triumph over a more hands-on, demystifying approach.

Guilty. For all I understand it on anything but the most theoretical level, it might as well be sorcery. Yes, I'm an engineer by training. No, that training didn't include electrical.

Plotnick quotes an educator and activist from 1916 lamenting that pushing a button “seems to relieve one of any necessity for responsibility about what goes on behind the button.”

Yep. History echoes once again.

150 years later, we equate button-pushing with "easy." That's the whole point of Staples' ad campaign (from a while back; I have no idea if it's still going on.) But I don't think they invented the concept; they just took an existing concept ("at the push of a button") and made it into a marketing gimmick.

No... technology is supposed to make our lives easier, make things more convenient. That's what it's for.

Kids these days with their simple-to-use devices, I'm telling you, mark my words...

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