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!

a day 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

a day 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.

June 24, 2019 at 2:29am

#961433

Very Superstitious

Looks like it’s my turn to host your final Summer Sprint of June This week, we’ll be exploring my home island: the Big Island of Hawaii!

I’ll be taking you to visit several special places over the next few days, but before we go, there’s one major rule you should know: “Take nothing but photos and leave nothing but footprints.” It is a well-known legend that anyone who removes lava rock from the island will be cursed by the guardian goddess, Pele. All manner of ailments and misfortune has befallen those who’ve taken sacred lava rocks off the island, often so disruptive to their lives that they’ve been led to mail their theft back to the island in an attempt to placate the goddess.

In your entry today, write about superstitions, legends, and curses. Are there any legends in your culture that you take to heart? Are you superstitious? Do you believe in Pele’s Curse?

New York City is home to well over 8 million people. This sprawling metropolis - the Platonic ideal of "metropolis" - concentrates that many people into just over 300 square miles, or, if you insist, a bit less than 800 km².

That's a lot of people. And the diversity is stunning - it's the world, concentrated. One particular neighborhood in Queens - called Flushing - is dominated by people of Chinese origin. I mean, everyone knows about Chinatown in Manhattan. Entire movies have been made about the place. It's world-famous. But no one seems to know or care about Flushing, the end of the 7 line, a stop on the LIRR, a few square blocks that might as well be Beijing.

I mention this because Flushing is one of the cheapest places to stay in NYC. While a hotel room in Manhattan could cost upwards of $200 in the off-season, rates in Flushing are pretty close to what you'd find in, say, Paducah, KY. I know this because I've stayed in Paducah, too. And with its proximity to the NYC subway system, one can be in Manhattan in less than an hour. It's also very close to La Guardia airport and the place where the Mets lose. Anyway, point is, all of the hotels in Flushing are owned by people of Chinese origin, and none of them have a fourth floor.

Anyone familiar with Chinese culture will find this unsurprising. As I understand it, the number "four" in Mandarin or Cantonese is very similar to the word "death." I'm no expert in Chinese languages, but this is what I've been told. Consequently, "four" is considered bad luck, so the floors are numbered 1.. 2... 3... 5... like some Monty Python skit.

It's tempting to scoff at this cultural oddity, but hold on a minute - New York is also home to world-famous skyscrapers, buildings containing dozens of floors. And most of them, built and owned and maintained by people of European ancestry, skip the 13th floor.

So, basically, shut up.

The Western superstition surrounding the number 13 is ancient and well-known, though stories of its origin are uncertain. Personally, I think it harks back to the Mesopotamian culture that pretty much kick-started Western civilization. Sumer basically invented civilization, which can be defined as "the art and science of living in cities," and they also had a great influence on how we count. But they positively sucked at fractions, so they used a numbering system that minimized the need for anything that isn't an integer. We still count time in the Sumerian way, with a base-60 division of time. Hours are base-12, while hours themselves are divided into 60 minutes of 60 seconds each. 13 is just out of bounds; it's a prime number with no integer divisors, and it freaked the Sumerians right out.

Hell, the fact that we have special names for "11" and "12" is evidence that we aren't as base-10 as we like to think we are. In a world that made sense, those numbers would be like "oneteen" and "twoteen" or some such. Thirteen is the first "teen" number in English, betraying its origin.

Not as interesting as the association of a certain number with death, perhaps, but the result is the same: an abiding superstition embedded in the entire cultural zeitgeist.

I've gone on about all this because 1) I attended a beer festival earlier, and am still a bit drunk; and 2) it speaks to the very human need for superstition. There are cultural superstitions, and individual ones. For instance, when I gamble, I always wear a Hawaiian shirt. I suppose these days the proper term is Aloha shirt, but mine are actually made in Hawai'i, not Taiwan or Hong Kong. Not that there's anything wrong with that. Point is, if I lose, I switch to a different Hawaiian shirt. It doesn't "work" or "not work;" it's just my thing.

There's no logical reason for this, or for most superstitions for that matter. Hell, I live with several cats, including a black one. Her name is Robin, and she's a good kitty. She crosses my path several times a day, and yet, I don't feel especially unlucky.

What's important is that we acknowledge and try to be understanding of the superstitions of cultures that are not our own. Not because bad things will happen, but just out of courtesy. So yeah, don't take lava rocks from Hawai'i. That culture has suffered enough, what with all the haole invading their islands, despite all the care they took to avoid pissing off Pele. We can't say we're entirely rational; how can we judge another culture's irrationality?

I guess what I'm saying is:

Shaka, brah.

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