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 6 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 8 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.

July 13, 2021 at 12:01am

#1013550

Deep Dive

Yet another "JAFBG"

[XGC] prompt.

Tell us about a time you stepped outside of your comfort zone. Did you enjoy it? Would you do it again?

Every time I travel, I step outside my comfort zone (which is about a three-mile radius centered on my house).

Oh, sure, there are times I can step back in, even when traveling. Most of the time, I feel at home in a brewery or brewpub. I speak the language, I mostly fit in (more on this in a later entry), and I can be pretty sure I'll enjoy the experience. Sometimes I don't, but even that is a data point for me.

The most recent time I did something I don't normally do, I wrote about in here. There's a local corner dive bar in a small town in Minnesota, and while I spent a lot of time in dives when I was younger, it's not something I usually do these days. I'm pretty sure I described it as "a return to my roots," but I'm far more comfortable, now, in hotel bars or breweries or other, slightly fancier, establishments.

I mean, the place was a pit. They kept the door open for ventilation, so flies were a thing. A few sad locals sat inside, contemplating their drinks. The pizza on the buffet (uncomfortable enough by itself during a pandemic) had been there for hours, and the salad bar was... well. The less said about the salad bar, the better. I ate both the pizza and the salad anyway; I've had far worse, like a corn dog that had been sitting under a lamp for days in a gas station in northern Nevada. Don't do that, by the way. Not recommended.

Nevertheless, the owner/bartender was friendly and helpful. My vision has deteriorated to the point where I can't read the names on the beer taps or the labels on the bottles behind the bar (in other words, this is a Crisis and Something Needs to Be Done), but he helpfully conveyed their information to me as I sat at the bar, squinting.

A bit of an aside: I usually have a workaround for that. My near-visioin is still pretty clear, so often, what I'll do is take a picture of the taps or the bottles with my phone, and hold the phone up close to my face to read the words. At this point, you may be asking "But Waltz, how can you drive if your vision has gotten that bad?" Answer: prescription sunglasses, and prescription clear glasses for night driving. They make everything clear on the road, but in the dimness of the bar, the sunglasses' darkness keeps me from properly reading the labels. And since it was full daylight, I'd neglected to bring the clear prescription. So, no, vision wasn't an issue when I had the accident; I was wearing the sunglasses and it was daytime.

Anyway, point is, usually I do the trick with the phone (it also serves as a helpful record from my travels), but in this case, I didn't want to be flashing around a brand-new Samsung in what was obviously an economically depressed area. So... bartender.

Speaking of comfort zones, I've lived most of my life with near-perfect vision, with only a slight astigmatism to deal with -- this was only a real impediment to stargazing, and I never even noticed it in my daily life. Later in life, I learned to deal with needing reading glasses for computer work and books. Now, in the space of two years, I've gone from near-perfect distance vision to not being able to read beer taps from across the bar. That's uncomfortable, and I don't like it.

But venturing into a dive bar as someone who Ain't From Around There? That was uncomfortable, but I ended up enjoying it. They even had some of the more widely-distributed craft beers, so I wasn't limited to crap beer or water. (I may descend into dive bars, but I still have standards).

As for "Would you do it again?" -- the thing about doing the same thing again is it becomes part of your comfort zone. Maybe not immediately, but with enough repetition, even the most foreign experience becomes commonplace. So, would I dive into a dive again? Sure. But I think the real question is, "Would you step out of your comfort zone again?" And the answer to that is still yes.

At least, yes, once I buy a car again, which I might just put off until after cataract surgery. Which itself is so far out of my comfort zone that it might as well be on a planet orbiting Proxima Centauri, and an experience that, once I have it, I doubt I'll be in any hurry to repeat.

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