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 5 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 7 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.

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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.

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').

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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.

3 Like

Turkey DrumStik

Nov 6, 2024 at 10:22am
In response to "Lo!"

Not gonna lie I need the holiday season chatter.

April 23, 2024 at 11:38am

#1069487

Going to the Dogs

One thing that always strikes me about New York City is the scale of the dog infestation.

I'm pretty sure that having a dog is a requirement of living here. I wouldn't be surprised if people are issued one when moving in: "Welcome to New York. Here's your dachshund." At the very least, even though the internet is telling me that having a dog in NYC requires a license, I'd bet they charge you a fee if you don't have a canine.

And the furry barking tail-waggers are everywhere. At any given moment on a typical block (on the Upper West Side, because I can't be arsed to take a subway ride just to support a blog entry), there are at least three dog-walkers on the sidewalk with an average of two dogs on leashes. Doing some quick math leads me to believe that not a single resident of the city is dog-free. Maybe they occasionally just appear in your expensive, cramped apartment via wormhole, complete with a regulation 6-foot-or-shorter leash and a roll of plastic doggie doo bags.

At least New Yorkers tend to be good about using both of these items. Whether that's out of a heightened sense of civic duty, or because they have a healthy fear of the NYPD, I don't know, and it doesn't matter, because the result is the same.

This isn't a case of "stop liking what I don't like." It's not that I dislike dogs, any more than I dislike kids just because I don't want to own one. Although I can't be arsed to do all the work a dog requires (and I absolutely will not pick up dog shit, with or without gloves), I understand that a lot of people want to hang out with the mopey, needy quadrupeds. It's just that I know what the housing situation tends to be around here, and, even with daily walks, having a retriever in an apartment the size of my spare bathroom can't be good for the dog.

Not only that, but right now, it's pretty close to the middle of astronomical spring. This, right now, is one of two weeks of the year when it's actually pleasant to be in NYC, the other being the week of the fall equinox. Between these two events, in one direction, it's either pouring down rain or oppressively hot, or both; in the other direction, it's either pouring down rain or snow, or oppressively cold, or both. And yet, every day, you gotta walk those dogs. Twice, at least. While juggling the three jobs and two side gigs you need to afford to live in NYC.

I get that some people want to live with dogs. I get that some people want to live in NYC. I'm just left shaking my head in confusion over the intersection of the two.

5 Like • Comment • View 4 comments

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