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.

May 28, 2021 at 12:02am

#1010894

Fail

PROMPT May 28th

What would you do if you knew you could not fail?

Fly.

No, I don't mean go over to the airfield and take piloting lessons. I could have done that before, given enough time and money. I just couldn't be arsed. I'm talking about whooshing through the atmosphere, like Superman or whatever.

I mean, if I can't fail, that mans I'd be able to soar under my own power, right?

There are levels of failing. Trying to fly and failing could, depending on the circumstances, be fatal. "Oh, I'll just jump off this cliff and trust that I'll miss the ground below entirely." Trying to get a pilot's license, on the other hand, like I said, just takes time and money... along with a visual acuity that I no longer have, so that's right out at the moment. But even absent the vision problems, the thing stopping me has not been fear of failing (or crashing), or lack of funds (it's really not much more of an investment than many other lessons, unless you actually buy an airplane), or even lack of time. No, it's that I have no purpose for it other than being able to say "I can fly an airplane."

And that's not enough for me. I need at least two reasons to do something before I do it, preferably more. Funny enough, that doesn't apply to sitting on my ass and doing nothing. I don't need any reason to do that.

But I digress. Levels of failing. Something could be fatal, but so what? You could trip in your bathtub. Or you could fail by getting bored and giving up. That happens to me a lot, because squirrel. Or it could turn out that you're just no good at it, as is the case with me and musical instruments. None of these possibilities ever stopped me from trying something, though. One thing that does stop me is the amount of work involved for a certain level of reward. Brewing is like that. I have no interest in brewing my own beer, because a cursory examination of what is required told me that it's 1% enjoying the fruits of your labor, 2% actual brewing, and 98% cleaning. And yes, I'm aware that adds up to more than 100%.

Not worth it, not with thousands of breweries operating in the US alone, the products of which I could spend the rest of my life sampling with almost no effort on my part.

In other words, I have to believe that I'm going to enjoy the process, not just the end result.

Now, if I were assured that I couldn't fail at brewing -- meaning not only producing drinkable beer, but not getting bored with the process or finding out I have no talent for it after all -- I'd do it.

It's not flying, but it's close enough.

As I've mentioned before, I'm trying to see at least one movie a week at the cinema. Yesterday was the last chance for me to see Raya and the Last Dragon, so I opted for it over newer releases such as A Quiet Place Part II and Cruella, both of which I hope to see eventually but should be in the theater for the next few weeks. So hopefully I can continue to do brief reviews, because condensing a movie review into one sentence is as much a writing exercise for me as anything.

One-sentence movie review: Raya and the Last Dragon

This solidly-plotted, excellently-animated, and beautifully-voiced fantasy action movie from Disney succeeds in telling a compelling story; however, I question the appropriateness of the movie's central lesson (which seems to boil down to "trust your enemies;" a hell of a thing to teach a kid), and besides, I couldn't stop thinking that the movie's multi-pastel-colored dragons closely resemble My Little Ponies.

Rating: 4/5

6 Like • Comment • View 8 comments

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