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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10 Most Recent Blog Comments:

Write_Mikey_Write!

about 10 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 12 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.

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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

Turkey DrumStik

Nov 6, 2024 at 10:22am
In response to "Lo!"

Not gonna lie I need the holiday season chatter.

February 20, 2024 at 10:48am

#1064499

Dahlia Dreaming

And now here's another one for "Journalistic Intentions"

[18+]...

Dahlia Dreaming

A good number of my "why the hell is it called that?" questions popped up before there was an internet. As a kid on a farm, I got to know lots of different cultivated plants—whether fruit, vegetable, or flower—whilst perusing a seed catalog. A paper one. That came through the postal mail.

This led to a lot of "why the hell is it called that?" moments that my parents, who were what passed for Wikipedia for me in those days (they'd eventually buy me an actual encyclopedia volume set, which I actually read and then promptly forgot most of), had no answers for. "Go look it up." Where? We live on a farm.

But I do remember that one of these moments was for the flower known as a dahlia. It seemed an even odder moniker than most plants' names, but the seed catalog didn't have much to say about it. It probably listed the botanical binomial, but I don't remember that. It's dahlia pinnata, according to Wikipedia, and while it's native to Mexico and Central America, it's not to be confused with a piñata. But as with many other cultivated plants, there are several subspecies. It gets confusing and beyond the point of this entry.

Nor is Wikipedia

much help with the etymology of the name. It seems it might have been named after a botanist named Anders Dahl. Which seems bogus to me, a typical European appropriation of an American species. The least they could have done is mangle one of the native names for the thing, like they did with, say, the raccoon. Except for the French, who call them washing rats, which is unfair to rats, who are often quite fastidious.

Also, I can't be arsed to find out if Anders Dahl was ancestral to the far more famous Roald Dahl.

None of which was what I set out to write about; I just did my usual assuming that if I don't know something, then no one else does, either. I have my parents to blame for that, too.

No, what I wanted to note, apart from the excellent use of depth-of-field in the photograph the title links to, is the petal pattern.

Those aren't true petals, incidentally. Each one of those petal-like pieces is a flower unto itself. But that's a bit like arguing whether a tomato is a fruit or a vegetable. No, what struck me is that the arrangement of the florets is similar to other petal and/or leaf arrangements found in nature, such as in sunflowers or artichokes: an instantiation of the Fibonacci sequence and the Golden Ratio.

Once you notice that particular spiral arrangement, you can't ever miss it. You can even find it all over the Mandelbrot set (which involves complex numbers), if you know what you're looking for. There are solid reasons for plants taking that general form (some animals, such as mollusks, do it, too), and none of them is that plants can do math. No, it's a bit complicated, but, basically, it's because it's easy and efficient.

I can appreciate that.

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