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 15, 2024 at 10:28am

#1064218

PIE in Your Face

Landed on another article about language. But this one's about the common root of many languages:

A new look at our linguistic roots
Linguists and archaeologists have argued for decades about where, and when, the first Indo-European languages were spoken, and what kind of lives those first speakers led. A controversial new analytic technique offers a fresh answer.

The source, Knowable, is not one I've linked before, and I don't know much about it because I'm way too lazy to find out. As for timing, it's a recent article.

Almost half of all people in the world today speak an Indo-European language, one whose origins go back thousands of years to a single mother tongue.

Another way to put that is that less than half of the world speaks an Indo-European language. Depends on one's perspective. As I speak one, yes, its shared origin with other languages is interesting to me.

Over the last couple of hundred years, linguists have figured out a lot about that first Indo-European language, including many of the words it used and some of the grammatical rules that governed it.

I've mentioned this language group in here before. Repeatedly.

This is the heart of the discussion:

Most linguists think that those speakers were nomadic herders who lived on the steppes of Ukraine and western Russia about 6,000 years ago. Yet a minority put the origin 2,000 to 3,000 years before that, with a community of farmers in Anatolia, in the area of modern-day Turkey. Now a new analysis, using techniques borrowed from evolutionary biology, has come down in favor of the latter, albeit with an important later role for the steppes.

Now, look, I don't claim to be an expert in either linguistics or evolutionary biology. But I've seen the parallels between language evolution and biological evolution. There's at least one big, huge, major difference, though: in the latter, at least for eukaryotes, you don't get a lot of significant horizontal gene transfer. That is to say, organisms' DNA depend primarily on their ancestors' DNA. I understand there are exceptions. But with language, there's little barrier to horizontal meme transfer; that is, languages can liberally borrow from other languages. English is perhaps the most obvious of these; it's stolen pretty heavily from non-Indo-European languages.

I'm not saying they're wrong. They know more about this shit than I do. (There is some discussion of that sort of thing near the end of the really quite long article.)

I'd also like to point out that it's not like IE sprang into existence from nothing. It developed from an earlier language. I understand linguists call the earlier language Proto-Indo-European, or PIE, which cracks me up.

Anyway, the article goes on to describe the process by which they figured out that IE existed, and that's pretty fascinating by itself, but too much to quote here.

But then they mention the thing I find most intriguing:

For example, the Proto-Indo-European language had a word for axle, two words for wheel, a word for harness-pole and a verb that meant “to transport by vehicle.” Archaeologists know that wheel and axle technology was invented about 6,000 years ago, which suggests that Proto-Indo-European can’t be any older than that.

I knew I'd mentioned "the wheel" before, so I went and looked. It was way back in 2020: "As the Turn Worlds (or whatever)"

The article speculates about the kind of people who would have used axles and wheels in prehistoric Eurasia: pastoralists or agriculturalists. Personally, I find it glaringly obvious that one speculation is missing. Once you have horses and carts, you can more effectively wage war, which is an even older human occupation than herding or farming. And tends to spread faster and further.

This, to me, is a more likely origin for the proliferation of Indo-European languages: a conquering people, not only taking over vast tracts of land, but imposing their language on the cultures they encounter. It happened with the Greeks and Romans, in historic times. Not so much with the Mongols, but not for lack of trying. And let's not forget how English got so widespread.

But, again, I'm far from an expert on these things. It just seems obvious that it's at least a hypothesis they can test. Maybe it's wrong. But given what I know of human nature, it could well be right.

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