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:

Write_Mikey_Write!

about 16 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 18 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.

March 1, 2020 at 12:39am

#976650

It Doesn't Count

Leap Day was a Good Day. My local cinema/drafthouse (which is conveniently located within stumbling distance) hosted a Cat Video Fest. This was just a compilation of cat videos and gifs. If you're wondering who would go to such a thing when cats occupy approximately 95% of the non-porn internet, well, me. And a lot of other people -- all three showings were sold out, and there's another cinema/drafthouse in town that also had showings, though I don't know how well attended those were.

The hated month of February is finally over. This one wasn't so bad; not as good as the February I spent on Maui, but definitely better than the one in which I spent my birthday in a hospital bed.

I wish good luck to those participating in the 30DBC this month, though I'm not one of them this time. January took a lot out of me, I expect to miss a day or two (because beer), and I still have an immense backlog of random blog fodder to share.

Case in point:

http://nautil.us/blog/-5-languages-that-could-change-the-way-you-see-the-world

5 Languages That Could Change the Way You See the World

I went to my neighbor’s house for something to eat yesterday.

Think about this sentence. It’s pretty simple—English speakers would know precisely what it means. But what does it actually tell you—or, more to the point, what does it not tell you? It doesn’t specify facts like the subject’s gender or the neighbor’s, or what direction the speaker traveled, or the nature of the neighbors’ relationship, or whether the food was just a cookie or a complex curry. English doesn’t require speakers to give any of that information, but if the sentence were in French, say, the gender of every person involved would be specified.

Behold, those words are ones I can probably mangle in French. Je suis allé à la maison de mon voisin pour quelque chose à manger hier. Maybe. Something like that. If I were female, I think "I went" would be "Je suis allée," so that's how my gender is specified. If the neighbor were female, "my neighbor" would be "ma voisine" instead of "mon voisin." And of course, "la maison" itself is gendered feminine. Anyway, my translation is probably bad, and I can't be arsed to plug it into Google Translate to check it; besides, those translations are often suspect. This is apropos of nothing ("apropos" itself having been stolen shamelessly from French), except that I guess six months on Duolingo has taught me a little bit.

Having learned a bit of Hebrew and Latin a very long time ago, the concept of gendered language isn't new to me. But some of the other things in this article were.

English speakers and others are highly egocentric when it comes to orienting themselves in the world. Objects and people exist to the left, right, in front, and to the back of you. You move forward and backward in relation to the direction you are facing. For an aboriginal tribe in north Queensland, Australia, called the Guugu Ymithirr, such a “me me me” approach to spatial information makes no sense. Instead, they use cardinal directions to express spatial information. So rather than “Can you move to my left?” they would say “Can you move to the west?”

Now that's interesting.

Other studies have shown that speakers of languages that use cardinal directions to express locations have fantastic spatial memory and navigation skills—perhaps because their experience of an event is so well-defined by the directions it took place in.

My personal experience is largely limited to the English-speaking world (which, to be fair, is a good chunk of Earth), and while I think I have a pretty good sense of direction, this took years of dedicated practice as well as knowledge of astronomy -- how to find Polaris at night, and how the sun's apparent path changes through the day and year.

In a series of experiments, the linguists had Kuuk Thaayorre speakers put a sequential series of cards in order—one which showed a man aging, another of a crocodile growing, and of a person eating a banana. The speakers were sat at tables during the experiment, once facing south, and another time facing north. Regardless of which direction they were facing, all speakers arranged the cards in order from east to west—the same direction the sun’s path takes through the sky as the day passes. By contrast, English speakers doing the same experiment always arranged the cards from left to right—the direction in which we read.

This kind of relates to something I wrote in here a few days ago; I wonder if such a study on readers of Hebrew, Arabic, or Chinese (for example) would produce a different result.

Rossel Islanders speak Yélî Dnye, which is quite dissimilar to other neighboring language groups. It has little specific color terminology—indeed, there is no word for “color.” Instead, speakers talk about color as part of a metaphorical phrase, with color terms derived from words for objects in the islander’s environment.

Color is weird in how it relates to language. Our current understanding of color owes something of a debt to Newton, as I understand it, who identified seven colors on the spectrum. I suspect he did this for metaphysical reasons; in truth, the spectrum is, well, a spectrum, a continuum. It seems odd to our 21st century post-Enlightenment sensibilities that Sir Isaac, who is known for his scientific achievements, was also a mystic, but he was; his investigations sprung from an interest in alchemy. Anyway, point is, it could have been three colors (the primaries), or seventeen. Color isn't an intrinsic "thing" the way a cat or a chair is.

In Nuevo San Juan, Peru, the Matses people speak with what seems to be great care, making sure that every single piece of information they communicate is true as far as they know at the time of speaking. Each uttered sentence follows a different verb form depending on how you know the information you are imparting, and when you last knew it to be true.

Take a moment to think about how this approach would improve politics.

And the last example in the article involves a culture that doesn't count. No, I don't mean that they're marginal (though they are pretty small); I mean that they literally don't have words for numbers. This makes a kind of sense based on my limited understanding of anthropology and the history of number theory, though; until we developed agriculture, there probably wasn't a great need for counting.

I'm guessing Sesame Street doesn't play well in that culture.

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