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:

tj-Merry Mischief Maker

about 15 hours ago
In response to "Extro Special"

I'll also use the spelling extrovert because if I use extravert my spell checker flags it. In college psychology our professor termed his own words, innervert and outervert because he thought they fit better. I, however, thought they sounded like definitions of belly buttons.

1 Like

Kåre เลียม Enga

about 21 hours ago
In response to "Extro Special"

I probably fit omnivert best. I'm temperamental.

Words change... definitions change... we change.

1 Like

tj-Merry Mischief Maker

a day ago
In response to "Sensational"

They say carrots can improve your eye sight, but it's been proven that drinking alcoholic beverages can make you see double.

After stating this, I find myself wondering why hasn't anyone tried fermenting and distilling carrots and marketing it as a vision tonic?

2 Like

🌖 HuntersMoon

a day ago
In response to "Sensational"

OK, look around you and observe (where possible) the reality of the newly revealed "facts." It's pretty clear that "Smartphones make you a hell of a lot of folks stupid."

2 Like

Elisa: Snowman Stik

a day ago
In response to "Sensational"

Thank you for reminding me to check on Retraction Watch.

https://retractionwatch.com/

3 Like

tj-Merry Mischief Maker

Dec 26, 2024 at 4:20pm
In response to "Gray Matters"

I wonder how many nine year old people have gray hair? I also wonder how much stress most nine year olds have?

3 Like

Wordsmitty ✍️

Dec 26, 2024 at 1:07pm
In response to "Gray Matters"

"who were between age nine and 39," - so the old folks need the rejuvenation formula first to turn back the years.

I wonder if anyone has studied why hair goes gray in certain places and not everywhere. Maybe different kinds of stress?

3 Like

🌖 HuntersMoon

Dec 26, 2024 at 10:00am
In response to "Gray Matters"

I've always believed "better gray than gone away..."

2 Like

tj-Merry Mischief Maker

Dec 25, 2024 at 3:02pm
In response to "A Mammoth Discovery"

This makes think of a book, "Evolution and Genomic Diversity" by Gene Poole.

1 Like

tj-Merry Mischief Maker

Dec 24, 2024 at 5:16pm
In response to "Hot Shoulder"

Maybe if we say, "Betelgeuse" three times in a row we can witness it.

1 Like

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.

5 Like • Comment • View 2 comments

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