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 9 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 11 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.

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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