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 2 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 4 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.

December 7, 2021 at 12:02am

#1022904

Meh. Whatever.

Yesterday, I managed to score a vaccine booster on a walk-in. Consequently, my arm is sore and I don't much feel like typing. So of course this one comes up at random. Now, I could cheat and just link to something fluffy instead, but no, I'll just present the link and try to keep my comments short.

It’s Time to Stop Talking About “Generations”

Oh, I don't know, I thought the chemistry between Kirk and Picard in that movie was... oh, you mean like the demographic concept. Never mind.

The discovery that you can make money marketing merchandise to teen-agers dates from the early nineteen-forties, which is also when the term “youth culture” first appeared in print. There was a reason that those things happened when they did: high school. Back in 1910, most young people worked; only fourteen per cent of fourteen- to seventeen-year-olds were still in school. In 1940, though, that proportion was seventy-three per cent. A social space had opened up between dependency and adulthood, and a new demographic was born: “youth.”

Contrary to The New Yorker's usual meandering style, the lede is actually useful and relevant. I think some people might be under the impression that high school has always been a big thing, that the way things are now is generally the way they've always been, at least since the Industrial Revolution.

Why is it relevant? Because the concept of naming generational cohorts is primarily a marketing tool.

The rate of high-school attendance kept growing. By 1955, eighty-four per cent of high-school-age Americans were in school. (The figure for Western Europe was sixteen per cent.) Then, between 1956 and 1969, college enrollment in the United States more than doubled, and “youth” grew from a four-year demographic to an eight-year one.

Incidentally, a part of me rebels against the categorization of teenagers as "young adults" (again, for the purpose of marketing). I see "young adult," and I think very late teens, early 20s. But apparently they mean like 13 year olds. This only leads to more confusion. But I digress.

To keep this market churning, and to give the consulting industry something to sell to firms trying to understand (i.e., increase the productivity of) their younger workers, we have invented a concept that allows “youth culture” to be redefined periodically. This is the concept of the generation.

Like I said. And I've been trying to say stuff like this for years, to not much effect, mostly because I don't have the bones to express it properly. That's why TNY writers get paid and I don't, I guess.

The new idea was that people born within a given period, usually thirty years, belong to a single generation. There is no sound basis in biology or anything else for this claim, but it gave European scientists and intellectuals a way to make sense of something they were obsessed with, social and cultural change. What causes change? Can we predict it? Can we prevent it? Maybe the reason societies change is that people change, every thirty years.

Change, as I've noted, is generally a continuum. Though of course that continuum is punctuated by defining events.

The theory also seems to require that a person born in 1965, the first year of Generation X, must have different values, tastes, and life experiences from a person born in 1964, the last year of the baby-boom generation (1946-64).

Which I've been freakin' saying.

Anyway, the article is long but worth a read if you care about this sort of thing. I try not to, but it's hard when all I see around me are assumptions made based on highly questionable generational stereotypes.

Fortunately, as a Gen-Xer, I can turn to the rallying cry of my cohort:

"Meh. Whatever."

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