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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Turkey DrumStik

about an hour ago
In response to "Barley There"

And approximately 15 seconds later, someone's going to screech about genetically engineered crops.

That would be me...screeching for more genetically engineered crops.

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

February 27, 2019 at 1:10am

#953251

Life

To follow up my last "there are probably no spacefaring aliens" post, here's some speculation about what extraterrestrial life might look like.

http://www.cracked.com/article_26213_4-awesome-theories-about-what-real-aliens-w...

It's Cracked, so this is more about amusement - and if you're cynical, clickbait - than science. But there's still some interesting stuff.

And if we ever do run into something along the lines of what's been dubbed the "Octomite," we'll have to make sure we don't show them all of our television shows, in which our noble heroes immediately destroy anything that looks remotely like that.

Too late. One line of reasoning speculates that we haven't met aliens because they know damn well we're xenophobic. Hell, they probably are, too. It's kind of a byproduct of natural selection.

Humanity, along with all other life on Earth, is carbon-based. Carbon is great at bonding and forming long molecular chains, and it's often been assumed that carbon is a requirement for metabolism and other important functions that make life work. But "carbon chauvinism" has been disputed more and more over the years, which we're sure has led to some sarcastic comments about Silicon Justice Warriors.

Yeah, look, what's methane? CH₄. The C stands for carbon. Methane might - might - stand in for water (though it lacks some interesting properties that make water unique), but it's still fucking carbon.

The next alkane after methane is ethane (hey, I learned this stuff before I knew how to read). Ethane is two carbons and six hydrogens. Replace one of those H atoms with an OH radical and what do you get? Ethanol. You know what would be cool? If we found ethanol-based life forms. Hey, it's not as crazy as it sounds. Astronomers have legitimately found entire clouds of ethanol floating out there in the galaxy. I bet it's even aged nicely.

One theory is that life was brought to Earth by a meteorite that slammed into the planet, and so the only genetic material we could work with was what was available on that rock.

Again with the life-came-from-elsewhere hypothesis. Still doesn't answer the question of how life got going in the first place. There are some thoughts on that subject (and they don't involve the supernatural, the presence of which would ask more questions than it answers), but there's no consensus I'm aware of.

In any case, the speculation is fun.

5 Like • Comment • View 2 comments

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