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 6 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 8 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.

August 7, 2020 at 12:02am

#990157

Lox Pie

The way language evolves and changes over time is pretty cool.

http://nautil.us/blog/the-english-word-that-hasnt-changed-in-sound-or-meaning-in...

The English Word That Hasn’t Changed in Sound or Meaning in 8,000 Years

So linguistics is something I'm interested in and I wish I knew more about it, which is why I read articles like this one -- which is more about what doesn't change, which is equally fascinating.

One of my favorite words is lox,” says Gregory Guy, a professor of linguistics at New York University. There is hardly a more quintessential New York food than a lox bagel—a century-old popular appetizing store, Russ & Daughters, calls it “The Classic.” But Guy, who has lived in the city for the past 17 years, is passionate about lox for a different reason. “The pronunciation in the Proto-Indo-European was probably ‘lox,’ and that’s exactly how it is pronounced in modern English,” he says. “Then, it meant salmon, and now it specifically means ‘smoked salmon.’ It’s really cool that that word hasn’t changed its pronunciation at all in 8,000 years and still refers to a particular fish.”

Well, now I'm all hungry. I haven't had a lox bagel in months. There's a place near me that makes a good one but... well, you know.

So, I've known about Proto-Indo-European, or PIE, for some time. The idea is that a large number of languages originated from one source, a language that is so lost in the mists of time that we don't even know what it was called, or if it had a name at all. But I like pie almost as much as I like lox, so that will do.

In modern English, well over half of all words are borrowed from other languages.

I'd argue that all of our words are borrowed from other languages, but, well, I'm not a linguist.

Analyzing the patterns of change that words undergo, moving from one language to another, showed how to unwind these changes and identify the possible originals.

I can't help but notice the similarities to genetics, where biologists can trace common ancestry through genetic drift. The really obvious difference there is that horizontal gene transfer is rare in organisms with mitochondria in their cells, whereas horizontal meme transfer happens in language all the time. (I'm using "meme" in the original sense proposed by Dawkins.)

The word lox was one of the clues that eventually led linguists to discover who the Proto-Indo-Europeans were, and where they lived.

This is where it starts to get really cool, because it's never been clear, at least to me, where the people who spoke PIE lived, or how their language, in the form of its descendants, came to dominate half the world.

In reconstructed Indo-European, there were words for bear, honey, oak tree, and snow, and, which is also important, no words for palm tree, elephant, lion, or zebra. Based on evidence like that, linguists reconstructed what their homeland was. The only possible geographic location turned out to be in a narrow band between Eastern Europe and the Black Sea where animals, trees, and insects matched the ancient Indo-European words.

No reason to name something that you don't have in your area. But that alone is a hypothesis that requires further investigation. So they did further investigation.

In the 1950s, archaeological discoveries backed up this theory with remnants of an ancient culture that existed in that region from 6,000 to 8,000 years ago. Those people used to build kurgans, burial mountains, that archaeologists excavated to study cultural remains. In that process, scholars not only learned more about the Proto-Indo-Europeans but also why they were able to migrate across Europe and Asia.

And to summarize the rest of it: horses and the wheel made this culture extremely mobile, so they were able to spread themselves, or at least their language, far and wide.

Which still doesn't explain why you have some very different words for things in languages, like "cheval" in French and "horse" in English. But I suppose they have theories for that, too.

And yet the word for "cat" is so similar across so many different languages that I have to wonder if they brought their feline companions with them as they traveled and/or conquered.

Which would also be pretty cool. After all, someone has to eat all the leftover salmon.

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