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.

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').

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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 17, 2019 at 12:26am

#967982

Of Time and Black Holes

http://nautil.us/issue/68/context/how-black-holes-nearly-ruined-time

Time for some physics!

How Black Holes Nearly Ruined Time
Quantum mechanics rescued our understanding of past and future from the black hole.

Is there anything a black hole cannot ruin?

As exotic structures of spacetime, black holes continue to fascinate astronomers, physicists, mathematicians, philosophers, and the general public, following on a century of research into their mysterious nature.

And don't forget nearly a century of science fiction authors getting the science wrong and thus contributing to the general public's misunderstanding of black holes.

The most striking feature of a black hole is its event horizon—a boundary from within which nothing can escape.

No, the most striking feature of a black hole is the singularity at its heart, a place where relativity breaks down and strange physics takes over. Without the singularity, there's no event horizon - and the event horizon isn't anything physical; it's a mathematical boundary, albeit an important one.

But, okay, I'll grant that the opinion of an actual physicist is probably more relevant than my own.

Entropy and the Arrow of Time

The article goes into some depth about how the second law of thermodynamics defines the arrow of time. I've been reading up on this sort of thing recently, and I have to say, the explanation leaves me unsatisfied. This is not to say that I think it's wrong - again, this was proposed by people way more knowledgeable and intelligent than I am - just that it's either a) a failure on my part to understand or b) a failure on their part to explain it well.

Basically, to me it sounds like a circular argument: Entropy can never decrease over time; therefore, time is the direction of non-negative entropy.

But, to be fair, I've never heard a better definition of time. After all, it may be something we can measure, but it's not exactly something we can touch, like a rock or a tree. So maybe there are just things we still don't understand. That's cool.

So, after that (necessary) detour into entropy, the article comes back around to black holes.

The resolution to this problem is to add quantum physics into the mix.

You know, back when the British were doing their colonial empire building thing in India, they had a problem with malaria. It turned out that an effective treatment, if not cure, for malaria was quinine, which is found in tonic water. The difficulty is that quinine tastes a lot like ass, so they added gin to it to make it more palatable - and threw in a lime for scurvy while they were at it. Thus was born one of the greatest inventions of history, the gin and tonic. The point here is that it's rare that you find phrases like "the resolution is to add gin" or "the resolution is to add quantum physics."

Anyway, I thought the essay explained things well and is largely accessible, so if you ever wanted to know shit about black holes, time, and entropy, well, it's a good introduction.

Might help to have a gin and tonic first.

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