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 14 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 15 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.

April 12, 2022 at 12:01am

#1030583

False

It's important to question the things we think we know from time to time.

15 Psychology "Facts" We've Been Getting Wrong Our Whole Lives

This is one of those times, courtesy of that bastion of fact-checking, Cracked.

Brains are weird; we’ve known that since the first caveman tried to impress a girl with the Stone Age version of an acoustic rendition of “Wonderwall.”

Or as I call it, Wonderbra.

We’ve spent a lot of time trying to figure out those little balls of electric meat, but we have to do that with our own stupid balls of electric meat, so sometimes we get it way, way wrong.

Especially when we fall victim to anchoring bias. I've been guilty of that myself, and I have to wonder what else I think is true that has been proven to be not exactly the case.

Since these cases deal with psychology, there's also a lot of confirmation bias involved. That is to say, if we're inclined to believe the worst of people, we'll see examples of it everywhere, and lock onto any purported information that seems to show how shitty we can be.

Now, there are 15 of these (as a countdown, of course), and I won't be repeating all of them here.

15. Stanford Prison Experiment

If you're unfamiliar with this, Google it. Apparently they gave people roles to play: prisoner or guard. The "guards" ended up being shitty to the "prisoners."

Its authors claim that the Stanford prison experiment basically proves that power corrupts, but there’s evidence that its subjects knew what they wanted and intentionally played up their roles of prisoner or guard or even were coached to do so.

This is an early example of the shit they apparently pull on "reality" shows, which also make people look worse than they really are.

14. Kitty Genovese and the Bystander Effect

This one made it into popular culture at least once, in the original Watchmen graphic novel. Turns out nearly everything we knew about it was sensationalized by a media outlet (go figure).

The murder of Kitty Genovese, purportedly witnessed by 38 apathetic neighbors, inspired the theory that people won’t help someone if they assume someone else will do it, but in reality, only about six people witnessed any part of the attack, and no one saw it in its entirety, so it wasn’t clear how serious it was.

While the murder itself was of course a tragedy, it wasn't compounded by witnesses failing to do anything about it.

11. Birth Order

It’s long been believed that oldest siblings are the most mature and responsible, middle children are needy attention seekers, and babies are, well, babies, but a recent study of more than 1,000 people revealed no correlation between birth order and personality.

Though it wouldn't surprise me if, as with astrology and generational labeling, if you tell someone they've got such-and-such a personality, they might actually incorporate it into their personality. This thus becomes a self-fulfilling prophecy.

So why not just tell people that we know they'll do the right thing? I know it won't work all the time, but can it hurt?

7. The 10,000-Hour Rule

Confession: this is one I chose to believe. In my defense, it's not about how shitty people are, but how they can improve themselves.

It turns out that practice only accounts for about 12% of skill mastery and there are plenty of people who will just always suck at stuff.

Like me with music. I know I've spent more than 10,000 hours practicing various instruments, and I still can't carry a tune in a paper bag.

4. Maslow’s Hierarchy

I see this one referenced everywhere, and it honestly surprised me that it's in dispute.

...but more recent research shows little support for the order of the hierarchy or any hierarchy at all.

I can't say I took it at face value, but a lot of people seem to.

2. We Only Use 10% of Our Brains

Perfidious nonsense, and the only thing that surprises me here is that people believed it wholeheartedly.

There are no extraneous brain parts just bumming around in there, but the myth is so pervasive that one Brazilian survey from this century found that half of people still believe it. There was even that terrible Scarlett Johansson movie.

Aw, it couldn't have been that bad if it had Scarlett Johansson in it. Like how Catwoman wasn't so bad because it had Halle Berry in it.

1. The Milgram Experiment

I've seen this one bandied about, too. The idea being that, given the opportunity, it's in all of us to be unquestioning authoritarian assholes.

If anything, it just proved that a lot of people are trolls.

Which is still being an asshole, but at least not one of the fascist variety.

A while back, someone designed a hitchhiking robot.

As described in that link:

It gained international attention for successfully hitchhiking across Canada, Germany and the Netherlands, but in 2015 its attempt to hitchhike across the United States ended when it was stripped and decapitated in Philadelphia, Pennsylvania.

And lots of people used that story to vilify Philadelphia (which, to be fair, is easy to vilify), the US as opposed to other countries, or even humanity in general -- ignoring that the vast majority of people who interacted with HitchBOT, even in the US and part of Philadelphia, either left it alone or actively helped it.

The point being that this supports my Lone Asshole Theory, because eventually it was going to come across someone with a baseball bat (I'm assuming baseball bat because it was Philly) or a crowbar (also Philly), but until then it traveled unmolested. Yeah, some people suck. But all the dubious studies, shitty kids-on-a-deserted-island books, and made-up statistics just don't provide enough evidence to show that all of us do.

3 Like • Comment • View 2 comments

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