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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Show Recent Comments

10 Most Recent Blog Comments:

Write_Mikey_Write!

about 17 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 19 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.

March 20, 2020 at 12:04am

#978582

The Friendly Skies

This seemed to matter more in a world where people still used airplanes, but I'm posting it anyway.

https://www.vox.com/2016/5/17/11687014/tsa-against-airport-security

The TSA is a waste of money that doesn't save lives and might actually cost them

The TSA's inefficiency isn't just aggravating and unnecessary; by pushing people to drive instead of fly, it's actively dangerous and costing lives.

People make fun of the Trolley Problem. But as I've said before, it has real-world applications.

The TSA is hard to evaluate largely because it's attempting to solve a non-problem. Despite some very notable cases, airplane hijackings and bombings are quite rare.

I've been saying this for years, possibly even before the 9/11 singularity: people have no sense of risk management. As a group, they're scared shitless of flying, but think nothing of speeding to the airport if they think they might be late. Which of those is more risky, in terms of mortality? Go ahead, take a guess.

"No terrorist is going to base his plot on getting a gun through airport security if there's a decent chance of getting caught, because the consequences of getting caught are too great."

Yeah, except that if you have a 95% failure rate in your screenings, that implies that 19 times out of 20, your bomb (or whatever) is going to make it through security. Send two bombers through security, and your success rate climbs to 99.75%. If I'm doing my drunk math right. Someone check me. Regardless, it's less than 5%.

Still, they had no problem practically sending me to Gitmo for having a cigar cutter I'd forgotten about in my carry-on.

Also, stop being sexist. Chicks can be terrorists too.

What about the most loathed TSA rules: the shoe removal requirement, and the ban on all but the tiniest containers of liquids? There's never been any evidence that these are effective. Remember: We caught the people who tried to attack with their shoes and with liquid explosives, without these rules in place. Europe is gradually phasing out the liquid ban.

The shoe thing is bullshit, has always been bullshit, and by making people take off their shoes, they're basically saying, "our nudity scanners aren't good enough to detect shit in shoes."

The TSA doesn't save lives, but it probably ends them. One paper by economists Garrick Blalock, Vrinda Kadiyali, and Daniel Simon found that, controlling for other factors like weather and traffic, 9/11 provoked such a large decrease in air traffic and increase in driving that 327 more people died every month from road accidents.

And this is where the trolley problem becomes relevant.

I'm too drunk to quote any more of the article, so go read it for yourself, but here, I'll relate that situation to our current state of abject fear.

People are dying from COVID-19. There's no denying that (unless you're a hardcore conspiracy theorist, of the kind who believes that "crisis actors" or whatever staged Sandy Hook to promote gun regulations, in which case, buddy, you are reading the wrong blog).

But consider the measures we're taking to mitigate that mortality: social isolation, closing both optional service industries like hairdressers and essential ones like bars, making it socially unacceptable (and in some cases illegal) to go out.

Now, look, I'm an introvert. This doesn't bother me much. Hell, I'm home most of the time anyway, and I have an enormous stash of booze here at home, subscriptions to four different streaming services, and more video games than I can play in what's left of my life. But I recognize that most people are extroverts, just like most people are right-handed or most people watch sports on TV, another thing we don't have anymore.

Here's the trolley problem: given that social isolation is poison to extroverts, how many people will die from suicide or other causes, or go literally crazy, because of the isolation -- compared to the death rate from getting sick?

That's the social calculus we need to be using. No, it doesn't take into account one's particular situation, or the idea that different people will die in each scenario. I'm just talking cold, hard numbers. And no, I don't have these numbers. If I were more sober, I could probably get a handle on it, but as it is, I can't be arsed. I'm just saying this: just because this virus thing is new and different doesn't mean we should fear it more than other causes of death.

It's the driving vs. flying thing all over again. Now, look, I prefer to drive, myself, but it's not out of fear; I'm well aware that my chances of getting killed in a car wreck on a trip to Vegas (24 hours by interstate) are much higher than my chances of getting killed in an airplane incident on the same journey (4 to 5 hours each way). But that's not my primary concern; I simply like to drive, and the TSA has made flying, which should be a fucking miracle, into a nervous chore. Not because of any fear of terrorism, but because I could get arrested for having goddamn nail clippers in my carry-on.

Not that it matters now. Vegas is closed. Hooker out front shoulda toldja.

It's the unknown that we, as a group, fear. I'd just like to see some more rationality, is all I'm saying. And if I can be rational after downing a fine Belgian beer, everyone else can be rational while sober. In theory, anyway.

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