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:

Brandiwyn🎶

a day ago
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.

Aiva Raine

Nov 5, 2024 at 3:03pm
In response to "It's Hard Not To Scare"

Shaking head. Aside from the inevitable Simpson's spider goat song playing in my head which I will now share with you below, whenever I come across a story that has all these scientists doing ridiculous experiments for no logical reason, I must go- ah- scientists- CANCER? You do remember that problem right? Can we maybe cure that before we make cockroaches magnetic and elephants see pink elephants?

spider pig clip- and if you all now are singing the spider goat song- you're welcome! grin
https://www.youtube.com/watch?v=BARjPuUN36Y

1 Like

Aiva Raine

Nov 5, 2024 at 2:55pm
In response to "Anomalies"

One of your more humorous entries- I especially enjoyed- "Lofoten Vortex can be the name of my Bjork cover band."

Thanks for the laughs. Much appreciated.

2 Like

November 7, 2020 at 12:02am

#997822

Beauty's Snare

I'm sorely tempted to respond to this prompt with "differential calculus" or "matrix algebra."

PROMPT November 7th

It's said that beauty is in the eye of the beholder. Describe something that you think is beautiful or attractive that someone else might consider unattractive or ugly.

But if I talked about those subjects then no one would read it. Hell, I probably lost half the potential readers just by mentioning them.

My first impulse, other than math (hey, where are you going?), was to describe my spirit animal, the turkey vulture. Majestic creatures they are, embodying the ideal of maximum gain from minimum effort, an efficiency that appeals to me as an engineer and a lazy person.

But objectively, those birds are ugly as hell.

Okay, not "objectively." There's no such thing as objective beauty or ugliness, and that's even before you get into the "beauty is only skin deep" cliché. But really, I don't know if even I would hang a photograph of a close-up of a turkey vulture's head in my living room.

There are, of course, certain things that are generally agreed upon to be aesthetically pleasing, such as the Venus de Milo, the Mona Lisa, or, well, pretty much everything else in France with the possible exception of this apartment complex in Paris

, and that's still more attractive than anything I've ever seen in New Jersey except maybe the Statue of Liberty, which... oh, right, that's from France too.

You know what's pretty, though? Well, no, you probably won't agree with me, because that's the whole point of the prompt. But I think spiders are kinda elegant.

Not at first, of course. When I see a big damn spider, my reaction is the same as most peoples': aaaah get it away GET IT AWAY. But then my rational mind takes over (provided, of course, that the arachnid isn't preparing to chomp on me) and I can see their beauty: eight symmetric, articulated legs; multiple jeweled eyes; delicate mandibles that I definitely do not want to get too close to.

And then there's their webs. Not all spiders spin webs, of course, but I marvel at the evolutionary process, whatever it might have been, that led some spiders to be able to instinctively create those works of art. Yeah, I know, I'd think differently about them if I were a fly, and I really, really hate walking into one unexpectedly (because that's a good way to get munched on by a spider), but they captivate the eye just as effectively as they snare insects.

Not that I want to live with one or anything. But when they're outside, where they belong, and they don't sneak up on me, I'm fine with them; they spin their webs and eat the bugs and gleam in the sunlight.

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