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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October 30, 2022 at 12:01am

#1039942

Uncertainty Principle

With no comments or feedback of any kind on yesterday's entry, I can only assume that no one actually wants me to be positive. Which is fine, because I don't want to be positive anyway.

Heisenberg's Bathtub
Copenhagen University
Copenhagen, Denmark
Did he come up with a theory of cosmic-ray showers while sitting in this bath?

There's an old tourism cliché: "George Washington Slept Here." The idea was that historic places would put the sign up as a way to draw in tourists, whether or not the father of our country (or the worst traitor ever if you're British) actually lodged there or not. It's the American secular equivalent of a church having a "piece of the True Cross."

This especially amused me as a kid because we lived in the area where Washington grew up, and visiting a place where we know (to a high degree of certainty) that he slept was pretty commonplace for me.

Werner Heisenberg may not be as famous as Washington; after all, he doesn't have a city, a state, several schools, and a giant phallic symbol named after him. But his position in quantum theory is similar to that of old George in the US: a prominent founder. Heisenberg didn't even own slaves, so he has that going for him. Perhaps his most famous contribution, from a layman's perspective, is the Uncertainty Principle: the mathematical formulation of the idea that you can know a particle's velocity, or its location, but never both (it's somewhat more complicated than that, but whatever).

So there's a joke floating around in nerd circles: theoretical signs in historical places that read "Heisenberg may have slept here."

It's a good joke, a fine joke, albeit one that requires a certain amount of knowledge of the history of science. But sometimes in-jokes are extra amusing because you know that out-groups won't get it.

What's truly maddening to me, though, is that apparently there was a character on a popular TV show who was also named Heisenberg, and, just like one cannot mention the Big Bang Theory without someone making a Sheldon comment, the Heisenberg joke is now ruined for all time.

It's a real shame.

In any case, here we have a place where Heisenberg probably didn't sleep, but close enough.

On the top floor of the Niels Bohr Institute at Copenhagen University, tucked away in a room that now serves as a kitchenette, is a famous bathtub. The simple porcelain tub has become a well-known object in certain circles because it was used by at least one famous physicist: Werner Heisenberg.

Bohr was, of course, another founder of quantum physics. He apparently got stuff named after him, though.

Though it has likely been years since anyone bathed in this tub, it has been kept around for about a century. Most physicists that work there will not really have an answer for it. Most will say that the institute did not get around to removing it yet, or that it’s kept for potential future use.

Or maybe it hasn't been hit with a high-enough energy photon.

Amongst students the story is slightly different, most of them know of its existence and connection to Heisenberg and other famous physicists who may have bathed in it at some point.

See? "may have bathed in it." Uncertainty Principle. Look, I can't overstate just how damn funny that is to science nerds.

Much like Albert Einstein’s Sink at Leiden University, this item can be called an “academic relic,” or a mundane item that gained value from its connection to a prominent scientist.

Einstein's Sink sounds like it should be something related to black holes, but alas, this is apparently an ordinary plumbing fixture.

Similar relics connected to saints or Christ himself are very common in religious circles, but many scientists would not be too happy with the comparison.

Like I said way up there.

Despite that, they might be reluctant to throw the item away due to sentimentality and perhaps because it offers a connection to great minds of the past. Everyone’s reason may be different but still, these items somehow persevere.

Nothing wrong with preserving a connection to the past. It gets people thinking and talking about it.

Who knows what is being kept in your local university?

Considering that my local university is the one Jefferson founded, there's a lot. And probably a lot that's been kept secret, too. But we'll never know the full story. You know. Uncertainty Principle.

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