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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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.

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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.

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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.

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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.

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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

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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.

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June 11, 2024 at 1:26pm

#1072479

No Cornea Jokes Today

One of the immeasurable joys of getting older is we get to go visit doctors and other medical professionals more often. For instance, me, this morning, so this is going to be a fairly brief personal update rather than deep philosophical musings about whether Chicago pizza is actually pizza (it's not) or scorn at the sorry state of science reporting.

Some long-time readers might remember that back in the dark ages of 2021, I had cataract surgery in both eyes. Today's visit to the eye doctor served as a general check-up and 2.5-year follow-up to that.

Now, I know I like to kvetch, whinge, moan, and complain in here. But this time, I have nothing to complain about. Everything, according to the eye doctor (don't make me spell optha-whatever), is fine. Sure, my vision is slightly worse than it was 30 years ago, but it's a lot better than it was 5 years ago.

What I mostly wanted to share, though, was that the doc, who I go to both because she's very thorough and because she's married to a former colleague of mine, is a massive nerd. I mean, you expect that from science-oriented professions, but she keeps up with the latest ocular-testing gadgets, and this time, she had a new toy: a virtual reality headset programmed to do part of the eye exam.

You put on the headset, and a little blue floating cartoon lady (you can tell the gender from the voice and the ponytail; I have a ponytail sometimes, too, but my voice and goatee give away the game) walks you through an exercise where they measure your visual focus distance. This involves staring at a target while a light show goes on all around you like you're moving through space in a game.

Turns out my focus isn't quite right, but not off enough to cause serious eye problems or require glasses. Which is still weird to me, not needing glasses for anything at my age. Well, sunglasses for protection from the harsh glare of the accursed daystar, sure, but not prescription ones, and I've always needed shades.

And the only reason why?

Science.

I'd be effectively blind right now without science. Well, not really; without science, I'd have died before I could go blind.

So if you ever wonder why I bang on about nerd stuff in here a lot... well, that's one reason why.

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