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 2 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 4 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.

December 30, 2021 at 12:03am

#1023825

Risk, Revisited

This one has been hanging around in my queue for a while, and I don't remember the point I was going to make about it. If there ever was one.

It doesn't help that a friend of mine smuggled me a bottle of delicious mead all the way from Utah, and I just drank it.

Our Worst Idea About “Safety”
A concept that took hold in the ’70s has haunted everything from seat belts to masks—and it’s going to keep putting us in danger.

The concept many medical experts can’t seem to loosen their grip on is known as “risk compensation.” It’s an idea that comes from the study of road safety and posits that people adjust their behavior in response to perceived risk: the safer you feel, the more risks you’ll take.

As far as transportation engineering is concerned, yes, that is a thing.

If you’re driving on a precarious cliff-side road without guardrails, you’d probably drive more cautiously. But some proponents of the idea make a stronger claim: that guardrails cause so much reckless driving that any potential safety benefits of guardrails are offset or even reversed. Under this reasoning, a road with guardrails would cause more accidents than a road without guardrails. Guardrails aren’t helpful; they’re counterproductive.

When I was in England several years ago, I was walking along a trail next to a stream when I noticed that something seemed off. After a while, I figured out what it was: even though the trail was, for the most part, significantly higher than the stream, there was no railing, no fence, no barrier of any kind. Here in the US, there would have been something like that -- or else they would have put the trail somewhere else, or, because this is America and we do what we want, moved the stream.

The idea behind the railing is to keep people from accidentally falling into the stream. Without the railing, though, I think we had a greater sense of the risks involved and were therefore more careful.

That's the kind of thing this article is talking about.

But whenever risk compensation has been subjected to empirical scrutiny, the results are usually ambiguous, or the hypothesis fails spectacularly.

Okay. I'm not going to argue with the science, if indeed it's saying what they claim it's saying. But I've thought for a while now that things are just too safe -- at least in the US. There needs to be some risk, some idea of the consequences of, say, falling off a two-meter retaining wall.

But in 1975, University of Chicago economist Sam Peltzman elevated what might have remained armchair speculation to a powerful argument against safety regulations. Writing in the Journal of Political Economy, Peltzman hypothesized that 1960s-era federally mandated vehicle regulations such as seat belts were actually making the roads less safe because they encouraged so much reckless and careless driving. In his thinking, any safety advantage of the new regulations was being offset. He analyzed traffic accident data before and after the regulations and found that not only did the regulations fail to decrease fatal accidents, but traffic-related fatalities increased after regulatory action.

I vaguely remember some discussion about that at the time, but I was nine or ten years old, so I generally ignored it while riding unsecured on a tractor, swimming in questionable water, or learning to drive a truck without seatbelts. Yes, I learned the basics of driving before I hit puberty -- one dubious advantage of living on a farm.

Note how one always "hits" puberty, by the way. One never eases into it.

Decades of traffic data now leave little doubt that, overall, safety regulations have indeed reduced traffic-related fatalities.

The fun thing about science is that sometimes science gets it wrong. And then it self-corrects, but people are so attached to the first conclusion that they ignore the correction. This is not a problem with science. This is a problem with humans.

Anyway, like I said, I don't remember the point I originally had in mind, but it came up today so I'm sharing it. I've been thinking about risk management in general for a while now, and there is, as I think the article points out (after all that mead, I can't be sure), a difference between personal and public risk management.

Tomorrow is the last day of the year, so it's very likely I'll do something other than riffing off of an article from somewhere, opting instead to do a review of the year or possibly a crystal-ball prediction for 2022. Maybe both. I haven't decided yet and probably won't until like 11:45 tonight.

And as the following day is New Year's, don't expect a blog entry at the usual time, because I may be drunk now, but I plan to be completely danchu at that time.

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