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 13 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 14 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.

May 3, 2020 at 12:27am

#982609

Relax

Does anyone else go through life feeling like an outsider everywhere, or is that just me?

PROMPT May 3rd

What do you do you relax and unwind in the evening? Paint us a picture of your ideal relaxation

I am fortunate in that I have nothing to unwind from anymore.

That wasn't always the case, of course. Used to be, I'd come home from work and play video games. Now I just play video games. Or do whatever: write, learn stuff, browse the internet (outside of these 30DBC months, I have to find blog fodder somewhere), read, watch shows or movies, that sort of thing.

I've heard people say that they wouldn't know what to do with their time if they retired. I have never had that problem, nor have I wrapped my identity up in my work (my profession, sure; once an engineer, always an engineer, working or not). I'm never bored, though I sometimes look for some new experience or knowledge -- everything is potential material for writing, but that's not why I do it; it's mostly for the experience, and the writing is secondary. And I still don't have time to do all the things I want to do.

It's like when I went with friends to see the 2017 total eclipse. I've mentioned that in here before, I think. It was my first eclipse experience. I know a lot of people who prepared with cameras and whatnot. While it was tempting to try to get photos -- I used to be a photographer, though I gave up the professional side of that when everything switched to digital and everybody became a photographer -- I figured there were 30 million other people who would be standing, sitting, or reclining in the path of totality, and at least 5 million of them would be able to get a better photo than I could (I was never all that great at it, to be honest). Rather than trying to compete with that, I decided that instead of fiddling with equipment, I'd just absorb the experience.

And I'm glad I did. But everyone else around me seemed like they had to be doing something else, mostly involving cameras. Again: outsider.

Mobile phone cameras suck for recording eclipses, by the way. As most of those people found out firsthand.

I don't know. If I'm still around in four years, there's another eclipse then, also here in the US. Maybe then I'll try my hand at photographing it, since I already had the experience.

Yeah, I know I've gotten off track; I do that sometimes. The point is, I don't do things just because "everyone else" is doing them, nor do I refrain from them just because "everyone else" is doing them. I want my own experiences. Hence the "outsider" comment above.

There is one place where I feel less like an outsider, and this brings us back to the actual prompt for the day. That place is the quintessential location for relaxation: a bar. While some are more welcoming than others -- there's not much I hate more than a crowded sports bar with pitchers of Bud Light and 35 wide-screen TVs, all showing different sportsball games (or, worse, inane commentary on said sportsball games or, even worse if that's possible, ads) -- in general, if I can sit down at the bar, have a drink, and pass judgment on their liquor and/or draft beer selection, I'm happy.

Best of all are the craft breweries, where I can chat with the bartender or fellow beer drinkers about the brews. Now, don't get me wrong; I also very much enjoy other adult beverages, but the beer culture is where I fit in the most. It's usually an exception to me feeling like an outsider. That is, until someone inevitably asks me if I brew my own beer. No, I do not. There were, before the pandemic, nearly 10,000 craft breweries in the US alone, and there will always be someone who brews better beer than I could. Besides, I know enough about the process to understand that brewing is 90% cleaning, 9% waiting, 0.9% mixing ingredients, and 0.1% enjoying the fruits of one's labor. This doesn't strike me as a worthwhile use of my time, not when I can go to a brewery and enjoy the fruits of someone else's labor. Besides, I hate cleaning.

We'll see how many breweries remain after the pandemic. I'm preparing myself for tragedy, though I'm doing my part to keep my favorite local one in business by ordering delivery from them once a week. And then, if and when I can drive around the country again, I'll resume visiting whichever of these establishments are still... you know... established.

Meanwhile, I have my delivered craft beer and I have my cigars; and I have a deck behind my house with a patio table and umbrella, and a laptop to watch movies on. What could be more relaxing?

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