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 11 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 13 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.

June 9, 2020 at 12:21am

#985316

Blame it on the Rain

This will be entry #3 for "Journalistic Intentions"

[18+]

9. You have to look through the rain to see the oncoming bus, dump truck, and motorcycle.
Solace.Bring

I'd probably have gone with "clouds." You have to look through the rain to see the clouds.

I don't really know the original quote here, but I suspect I would be disdainful of it. As it is...

Well. I go through optimistic phases, but I'm not what you'd call an optimistic person. There's a scene I vaguely remember from Doctor Who, in the Matt Smith incarnation. I don't really remember the situation or the exact wording, and I can't seem to find it anywhere because most people associate the Doctor with optimism, but in this case he says something like, "It's not like things could get any worse." Things proceed to get worse, and then he says, "There are some sentences I should just stay away from."

In other words, it's always darkest just before things fade entirely to black.

I mean, obviously, I know it's not true. Simple statistics show that if something unpleasant happens, there's a good chance that something nice will happen afterwards, in addition to the possibility that things will get worse or that they'll stay about the same.

But, see, that's not helpful. I like surprises, but only pleasant ones. If I find an unexpected creature in my home, it would be a nice surprise if it were a unicorn, and a lousy surprise if it's a rabid coyote. So if I expect the rabid coyote and then I get the rabid coyote, I can be like, "Yep, I was right," and I'd feel good about being right all the way up until the time I'd have to get rabies shots. But if I expect the coyote and get the unicorn, well, that's a pleasant surprise. On the other hand, if I expect the unicorn and get the coyote, that's all kinds of disappointing unpleasantness.

So, yes, I'm going to expect the worst, because that way, I can feel good about being right, or I can feel good about the pleasant surprise. One thing I can never be is disappointed.

Well, that's the theory, anyway. In practice, I get disappointed quite a bit, so I just keep becoming more negative until I'm no longer unpleasantly surprised.

So to get back to the remixed quote above, I find rain to be generally pretty neutral. I know people who love it or hate it, or maybe it's situational, but to me it's like, well, I don't live in a desert (which makes me wonder just why all of these rabid coyotes are running around) and so I have to expect it to rain sometimes. It does no good to love it or hate it; it just is.

But it does tend to put a crimp in one's plans, and when that happens, I can only expect things to get worse, like I get caught in a three-way weather-induced accident between a bus, a dump truck, and a motorcycle.

And when that doesn't happen, I feel pretty good. When it does happen, I'll be dead.

Yesterday's entry included a mini-contest. The challenge was to describe the smell of coffee, because some stuffy German philosopher once said it couldn't be done.

And there were some great responses. All of them, actually. You guys can't make this easy on me. But as per the above, I expected that.

Based on the responses, I'm starting to think Wittgenstein might have been onto something after all. You can compare the aroma of coffee to some other scent, which is useful but also kind of kicks the can down the road; how would you explain these other scents? You can use metaphor, as in "the liquid essence of a black sun," which I thought was particularly poetic, but none of us have experience with black suns. And then there's the neurological route, as in "triggers the pleasure centre" (which I disagree with, but as I said, I'm not going to dismiss an attempt just because I don't like the aroma myself), but while that's accurate (for people who aren't me), it doesn't really convey the sensation.

But maybe there's really no way to convey it on a practical level. I used the example yesterday of describing "blue" to a person blind since birth, but I suspect that trying to describe the scent of coffee to someone who has never had a sense of smell would be a problem just as intractable.

As I said, I thought everyone provided a great response, but I'm going to go with the one that made me laugh.

How would a non-coffee drinker describe the aroma of coffee? Well, like this. Coffee's aroma is the closest we'll get to someone trying to set moist soil or peat on fire using a non-petroleum based ignition fluid. You could imagine something like vodka, 151 rum, or a very small amount of C-4 in this scenario. These ignition fluids can account for any trace scents that may appear against the soil-based backdrop. If it's sweet, blame the rum. If it's clean, blame the vodka. If there's a flinty or mineral note, well, C-4 could account at least for part of that.
Turkey DrumStik

, I owe you a Merit Badge.

This is fun. I'll keep doing it from time to time as long as I keep getting enough responses to make it worthwhile, so everyone will have another chance soon.

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