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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Show Recent Comments

10 Most Recent Blog Comments:

Write_Mikey_Write!

about 14 hours 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

about 16 hours 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').

1 Like

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.

April 2, 2020 at 12:14am

#979969

All Wet

There was a Shyamalan movie a while back, Signs, that was all about (spoiler here) aliens whose only weakness was water. Everybody (except, apparently, Syamalan) wondered why aliens would even bother visiting a planet whose surface is mostly covered in the stuff, and where it regularly falls out of the sky. Decent movie, stupid premise. And don't get me started on what the Wicked Witch of the West must have smelled like if all it took to melt her was water.

Anyway, the point is, water's kind of important for us.

http://nautil.us/blog/-why-water-is-weird

Why Water Is Weird

Water can appear to be “fine-tuned” for life.

Well, duh. Anthropic principle: if it weren't, we wouldn't be here to gawk at it.

In their 2018 study, Hajime Tanaka, John Russo, and Kenji Akahane—all researchers in the Department of Fundamental Engineering at the University of Tokyo, in Japan—tried to tease apart what makes water unique among liquids.

Whatever the technical reason for it, it is unique as far as we know. One weirdness that they touch on in this article: ice is less dense than water, so it floats. What the author doesn't mention is how this protects life in the water; as the ice forms a barrier, the liquid water underneath becomes more temperature-stable, keeping aquatic organisms from freezing (usually). This probably helped evolution along.

The ancient Greeks thought water was one of the four “essential” elements, the others being earth, air, and fire. Homeopathy, which purports to cure illness using small doses of disease-causing substances dissolved in water, evolved out of this, Saykally said. But there are more modern magical claims about so-called “structured” or “hexagonal” water. Some “wellness” practitioners claim humans age in part because we don’t replenish our stock of structured water. Depending on water’s structure, they say, it can penetrate your cell walls more effectively and has all kinds of health benefits.

Well, water does have all kinds of health benefits. As the article points out, though, "structured water" is concentrated bullshit.

The ancient Greeks may have been wrong about water being an essential element, but Saykally says it’s no coincidence that water is essential for life on Earth.

I wouldn't say "wrong," exactly. Just that this was the state of knowledge at the time. Aristotle was wrong a lot, too, but we still study his stuff because it helped form the foundation for modern thought.

Anyway, some people speculate that other liquids might help form life on other worlds. Methane, for example. But anyone positing such a thing is going to have to explain how methane, which has rotational symmetry, can do all the fun things that water, which does not, does. I'm not saying it's impossible, of course; it's a big universe. But I don't think it's very likely.

Maybe there are sentient aliens that are based on ethanol instead of water. I wonder if they get pulled over if they have too much water in their bloodstream...

5 Like • Comment • View 3 comments

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