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.

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').

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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 21, 2024 at 9:17am

#1069306

Revisited: "Hack This"

I'll be on what passes for vacation for me this week—I have nothing to vacate from, but I do like to go elsewhere from time to time—so entries will happen whenever I get a chance and probably short and even more pointless than usual.

For today, though, I'll do my usual Sunday thing and look back at an old entry. This one's from 2020: "Hack This"

Worth reading if you want a decent takedown of "lifehacks." Or even if you don't. Especially if you don't.

I still see "lifehacks" from time to time, but I see more parodies of them, which warms my heart. Things like: "Life hack: Don't have a mental breakdown at home. Have it at work so at least you're getting paid for it."

The article I linked

(Medium, 2017) is still up as of today. And my opinions haven't changed much, but maybe a little, and maybe some points need clarification. But first, I'll address the end of the entry, where I discover, too late, that this advice article that is a takedown of advice books and articles is actually an ad for the author's advice books.

Also, I hate reading this far along a halfway decent article only to find that it's a commercial in disguise. Bah.

Yes, I have said numerous times that I don't mind taking a look at book ads here, on a site that caters to readers and writers, so long as the content is worth commenting on. This one, however, did manage to catch me by surprise. Usually the book-flogging is near the beginning, or in a sidebar, or otherwise obvious when you start reading the article. This one was, I felt, deceptive—moreso because, as I wrote then: "I feel like he has good points, but those are somewhat muted by the fact that he's doing exactly what he's railing against."

Now... one might say, "But Waltz, here you are doing it too." Yes. I am. But I'm not trying to sell anything.

Besides, if I really objected to it, I'd have scrapped the entry and done a different one.

Rereading this entry, I realize it might read as if I'm against self-help books and articles in general. I'm not, necessarily. It's just that most of them only "self-help" the author to make money. There's nothing wrong with making money; most of us want to do that. But doing it by misleading others into doing something that doesn't help, and may even actually harm, is generally called "fraud," and is frowned upon.

So, just one more comment on my previous comment:

I keep seeing that the true enemy is "processed" or "overprocessed" foods, but I haven't found a good definition for those, yet. I mean, technically, cooking is a process, and - raw-food-diet bullshit aside - cooking is what makes a lot of food more nutritious and digestible. Potatoes, for example. It's probably what allowed our ancestors to evolve these great big brains that most of us don't use.

Obviously, it's been four years, so a) "processing" has been better-defined, while at the same time I haven't seen much about it lately; I wonder what the next anti-fad will be. And b) I shouldn't have typed that last sentence; it's misleading and does what I'm railing against here: assertion without evidence or experience. Not that I'm above doing that, but in this particular case, it's evolutionary guesswork, which I have issues with, and it's also what's commonly known as a "chicken/egg" scenario: did we evolve big brains because our ancestors cooked their food, or did they cook their food because they had big brains from some other adaptation? Or was it a synergy of some kind? My only point should have been that cooking food is good.

In reality, the "chicken/egg" scenario is easily resolved: eggs existed long before what we call chickens, and the first chickens hatched from eggs laid by some dinosaur descendant that was almost a chicken. That's still evolutionary speculation, but at least in this case, it fits with what we know of evolution.

Therefore, it was the egg that came first.

Fortunately for chickens and those of us who enjoy eating them, it wasn't cooked.

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