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 19 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 21 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.

February 10, 2020 at 12:21am

#975188

Well, Actually...

I'm starting to think Cracked has improved again. It was funny, then it kinda sucked, and now I think it's often funny once more. But it's also educational.

https://www.cracked.com/article_26746_5-famous-psychological-studies-that-were-h...

5 Famous Psychological Studies (That Were Hugely Overhyped)

Since this has to do with science and science reporting, I have things to say about it.

We cling to certain psychological studies like they're the only mental life preservers in this great big raging ocean of crazy.

Of course, they are not. There is also booze.

5. Your Answer To The Trolley Problem Has No Relation To What You'd Do In Reality

The infamous Trolley Problem is a popular thought experiment in Intro to Philosophy, as well as a beloved pop culture trope.

It has certainly been mined for Comedy Gold by writers for Cracked and other sites.

The result found almost no correlation to what participants said they would do and what they actually did.

Stop the presses! People say one thing and do another? THIS IS HUGE!

Okay, but seriously, though, there are practical applications to the Trolley Problem, but they don't necessarily involve individuals.

Consider an outbreak of a completely hypothetical contagious virus. This completely hypothetical virus has a fatality rate of 2%, meaning 1 person in 50 who contracts it dies. Further assume that it infects 1/10th of the world population, so it kills one person in 500 overall. These are, of course, all numbers I'm extracting from my nethers, but so are the Trolley Problem numbers.

Now, some scientists invent a vaccine for Hypothetical Contagious Virus, HCV. But owing to genetic differences among individuals, the HCV vaccine is itself deadly to 1 in 50,000 people. Administering this vaccine is, consequently, a Trolley Problem - assuming you could spread it to the entire human population, you're killing 150,000 people with it. But without it, 15 million people could contract HCV and die.

There might be overlap between the two groups, but it's likely that some vaccinated individuals will die who wouldn't have if they'd just contracted the virus. Overall, though, you're saving 14,850,000 lives.

The Trolley Problem is, at least in part, about the ethics of action vs. inaction. The above numbers really are completely hypothetical (and in no way reflective of how pandemics or vaccines actually work), but you can apply a similar calculation to, say, the invention of self-driving cars; if 30,000 people a year die in autonomous vehicle accidents, but 40,000 a year would die in human-driven vehicle accidents, is it not better, overall, to go with robot cars? Or are we just going to focus on the lives lost and the fear of the unknown, rather than the reduction in fatalities?

4. The "Bystander Effect" Isn't The Whole Story About Kitty Genovese

I've known for a long time that the Bystander Effect is completely overblown (although it did create an excellent origin story for Rorschach in the Watchmen graphic novel).

So at this point, "Genovese Syndrome" should be the name for when people parrot old myths to look cynical instead of bothering to check basic facts.

I can be cynical with the actual truth; I don't need myths for that.

3. Power Posing Doesn't Change Anything In Your Body

Your body language says a lot about how you're feeling, which is why you don't give a big presentation while slouched over like you're four hours into a Gears Of War marathon. And supposedly, a "powerful" posture can make you powerful.

Yeah... the only surprise here is that someone took this shit seriously in the first place.

But while the original study found a positive correlation with power posing, later experiments didn't. In 2017 alone, 11 different experiments tried to replicate the original's results, and all of them found that power posing apparently does nothing for your body.

This also speaks to that other cognitive bias we have that I can never remember the name of: we tend to believe the first thing we hear on a subject, and subsequently ignore contradictory information. This is why it's so damn hard to convince people of anything, even though it's one thing the scientific method was designed to overcome.

2. No, Your Ego Doesn't Get "Depleted"

Unfortunately for fans of thinking less, psychologists tried to replicate Baumeister's results and very much could not. A previous meta-analysis was also reviewed, and the results weren't pretty there, either. The theory appears to only apply if you personally have been convinced that your willpower is a finite resource.

Again, a finding that fails upon attempts at replication - and again, one that wasn't nearly as widely publicized as the original hypothesis. This is my problem with science reporting, in a nutshell.

1. Maslow's Hierarchy Of Needs Needed More Real Science

It's the idea that there's a logical order to human need. First you fulfill your physiological needs, like sleep, water, and pizza. Safety, like not being beaten up for your pizza, comes next. Once you're safe, you can look to fulfill your need for social belonging (getting invited to pizza parties), esteem (being lauded for bringing good pizza), and finally self-actualization (building a pizza oven and inviting people over to witness the depth of your pizza skills).

Dammit, Cracked! Now I'm hungry!

When research into Maslow's structure came along, it showed that humans can fulfill their needs in any order, and can feel self-actualized even when their basic needs aren't being met.

Yep. I'll happily eat pizza even when sleep-deprived.

Being connected to others, knowing what the hell you're doing, and feeling like you have some control over your life is the best way to happiness. Well, unless someone tries to replicate the latest data and determines that it's all bunk.

Drinking beer, listening to music, and being snarky on the internet is my best way to happiness.

Your mileage, as always, may vary.

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