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:

Turkey DrumStik

about an hour ago
In response to "Barley There"

And approximately 15 seconds later, someone's going to screech about genetically engineered crops.

That would be me...screeching for more genetically engineered crops.

Write_Mikey_Write!

a day 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

a day 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

February 8, 2019 at 1:47am

#951470

Anonymous

https://getpocket.com/explore/item/the-irrationality-of-alcoholics-anonymous

The Irrationality of Alcoholics Anonymous

Its faith-based 12-step program dominates treatment in the United States. But researchers have debunked central tenets of AA doctrine and found dozens of other treatments more effective.

Now... as much as I talk about drinking, I don't have a drinking problem. But see, that statement right there is enough to convince AA that I actually do and am in denial.

They also list in their "signs you're an alcoholic" thing the following:

-You can't control your drinking
-You have to control your drinking

I mean, what? Also, "You drink alone." I'm a goddamn introvert - I do everything alone. Piss off.

But I digress.

I do understand that some people have a drinking problem, in the same way that I understand that some people have a peanut allergy. Okay, that's not a very good analogy, because while I'm not allergic to the little fuckers, I don't consider them to be food unless they're processed into butter, whereas I like pretty much anything that's been distilled or fermented.

By the time he was a practicing defense attorney, J.G. (who asked to be identified only by his initials) sometimes drank almost a liter of Jameson in a day.

And that is what I call a drinking problem. I mean, Jameson? Really?

Okay, that's insensitive.

In the spring of 2012, J.G. decided to seek help. He lived in Minnesota—the Land of 10,000 Rehabs, people there like to say—and he knew what to do: check himself into a facility.

I'm using that nickname from now on.

He felt utterly defeated. And according to AA doctrine, the failure was his alone. When the 12 steps don’t work for someone like J.G., Alcoholics Anonymous says that person must be deeply flawed.

I've seen this sort of thinking in other contexts. "If you give yourself over to the Faith, good things will happen to you. Therefore, if bad things happen to you, it means you have not truly given yourself over to the Faith." It's not only terrible logic, but it's also dehumanizing.

Nowhere in the field of medicine is treatment less grounded in modern science.

And now we get to the part that pisses me off.

Whereas AA teaches that alcoholism is a progressive disease that follows an inevitable trajectory, data from a federally funded survey called the National Epidemiological Survey on Alcohol and Related Conditions show that nearly one-fifth of those who have had alcohol dependence go on to drink at low-risk levels with no symptoms of abuse. And a recent survey of nearly 140,000 adults by the Centers for Disease Control and Prevention found that nine out of 10 heavy drinkers are not dependent on alcohol and, with the help of a medical professional’s brief intervention, can change unhealthy habits.

While science is always in the process of improving itself, at some point, you have to pay attention to statistics.

As I researched this article, I wondered what it would be like to try naltrexone, which the U.S. Food and Drug Administration approved for alcohol-abuse treatment in 1994. I asked my doctor whether he would write me a prescription. Not surprisingly, he shook his head no. I don’t have a drinking problem, and he said he couldn’t offer medication for an “experiment.” So that left the Internet, which was easy enough. I ordered some naltrexone online and received a foil-wrapped package of 10 pills about a week later.

And then you stopped doing science. You know, the mad-scientist trope is a fun one in fiction, but this is the reason people don't trust science reporters. At least she admits later that it's not science.

Claudia Christian, an actress who lives in Los Angeles (she’s best known for appearing in the 1990s science-fiction TV show Babylon 5), discovered naltrexone when she came across a flier for Vivitrol, an injectable form of the drug, at a detox center in California in 2009. She had tried Alcoholics Anonymous and traditional rehab without success. She researched the medication online, got a doctor to prescribe it, and began taking a dose about an hour before she planned to drink, as Sinclair recommends.

As a big fan of Babylon 5 and especially of the character she portrayed, I found this ironic, as one of the other characters was named Sinclair.

Religious fervor, aided by the introduction of public water-filtration systems, helped galvanize the temperance movement, which culminated in 1920 with Prohibition. That experiment ended after 14 years, but the drinking culture it fostered—secrecy and frenzied bingeing—persists.

Here we have every problem with America encapsulated. We tend to think in binary terms. When enormous McMansions became popular, we didn't go back to modest 2000sf houses; we started yearning for trailers (though the polite term now is "tiny house" or something). Netflix couldn't abide the nuances of a 5-star rating system, so it switched to thumbs up or down. Everything either is awesome or it sucks; there's no middle ground. And with booze, you're either a teetotaler, or you have a problem - in peoples' minds. It's an all-or-nothing mentality, and it doesn't leave room for the much-vaunted "moderation" I keep hearing about.

That was the original meaning of "temperance," you know: moderation. People would "temper" wine with water to get a weaker drink. It didn't mean the same thing as abstinence.

In 1934, just after Prohibition’s repeal, a failed stockbroker named Bill Wilson staggered into a Manhattan hospital. Wilson was known to drink two quarts of whiskey a day, a habit he’d attempted to kick many times.

Hm... I wonder what else was going on in that time period that might drive a stockbroker to become a drunk...

In any case, not everyone who drinks is an alcoholic. I'm certainly no professional in this regard, but it seems to me that if your drinking gets in the way of something else like work or relationships (or video games), or if it negatively affects other people - I mean, really affects them, not just earns their disdain because they've chosen not to drink themselves - then, maybe, you should consider that you have a problem.

Just remember, the solution probably isn't AA.

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