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:

Brandiwyn🎶

a day ago
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.

Aiva Raine

Nov 5, 2024 at 3:03pm
In response to "It's Hard Not To Scare"

Shaking head. Aside from the inevitable Simpson's spider goat song playing in my head which I will now share with you below, whenever I come across a story that has all these scientists doing ridiculous experiments for no logical reason, I must go- ah- scientists- CANCER? You do remember that problem right? Can we maybe cure that before we make cockroaches magnetic and elephants see pink elephants?

spider pig clip- and if you all now are singing the spider goat song- you're welcome! grin
https://www.youtube.com/watch?v=BARjPuUN36Y

1 Like

Aiva Raine

Nov 5, 2024 at 2:55pm
In response to "Anomalies"

One of your more humorous entries- I especially enjoyed- "Lofoten Vortex can be the name of my Bjork cover band."

Thanks for the laughs. Much appreciated.

2 Like

November 9, 2022 at 12:01am

#1040431

A Little Pick Me Up

Whenever Cracked claims to be "scientific," I always take it with a huge chunk of pink Himalayan sea salt.

Still funny, though.

15 Scientific Factors That Secretly Influence Who You Have Sex With

Of all the things you want control over in your life, who you have sex with probably ranks pretty high.

Right, because not controlling it means someone's committed a felony.

Fortunately, considering that we have not, in fact, descended into a Handmaid’s Tale dystopian nightmare (yet), it probably feels like you do.

We're getting there.

Well, we’re sorry to tell you that free will is an illusion and you’re as beholden to the tyranny of biology as the grubbiest little worm when it comes to who you rub your genitals on.

Well, duh.

Some of these are not so "secretly." I'm not going to go through all 15 here, just some highlights (lowlights?)

14. Whether You’re Ovulating

Ovulation (that is, the phase of a woman’s menstrual cycle when she’s fertile) lowers women’s standards...

Anything that lowers women's standards can only work in my favor.

12. How Much They Smile

This one depends largely on who you are and who you’re trying to get with: Men prefer women who smile more, while women like men who smile less (or even look vaguely ashamed).

This one didn't sit right with me—it is physically impossible for me to smile in the standard bare-your-teeth fashion, and yet I'm somehow not swimming in sex—so I went to the link

they helpfully provided, and oh boy.

A note to single dudes: If you're looking to pick up a woman at a bar, whatever you do -- don't smile at her.

I have a hard and fast firm (dammit, there's not an adjective here that can't be misconstrued, is there) rule about not picking up women at bars. "But Waltz, didn't you just say that you want women with lower standards? What lowers a person's standards more than alcohol? Don't say 'ovulation.'" Yeah, that was what's known in the rarefied circles of advanced comedy as a "joke."

The obvious difficulty here is that if I'm out in public, I'm probably at a bar.

Full disclosure: I did it a couple of times when I was younger; why else do you think I developed that rule?

Researchers asked more than 1,000 volunteers to rate the sexual attractiveness of hundreds of images of the opposite sex.

Images don't cut it. Smiles, and other expressions of emotion, are generally fleeting, unless you work for a retailer and thus have to have one plastered on your face at all times. No, maybe an image can give someone a good or bad first impression, but I suspect I'm not alone in wanting to see more body language—even if I'm terrible at reading it.

(All were heterosexual, ages 17 to 49 years, with a median age of 21. Fifty-two percent of participants were Asian, and 48 percent were Caucasian.)

I think a few demographic groups are missing here. While it's irrelevant to me what gays, for example, prefer to see in such a study, I'm sure a lot of people do want to know that. What is relevant to me is if that still holds true at age fifty-something. I suspect that, like most studies of this nature, the majority of the guinea pigs were university students (or possibly teachers/researchers in the case of the older ones) who got enough bread for a couple of pints out of the deal.

They found that women ranked the smiling guys as less attractive -- but they were into the prideful and ashamed men. But the male participants were most attracted to the smiling women, and least attracted to the ones who seemed proud.

Missing some instances of "most" here. I seriously doubt everyone had the exact same reaction. It's like asking people what their favorite candy is. Most people say "Reese's cups." I despise Reese's cups.

Anyway, enough of that. Suffice it to say this is one instance of me needing that huge chunk of pink Himalayan sea salt.

(How sea salt got up into the Himalayas, I leave as an exercise for the reader.)

11. Whether They Touch Your Arm

Lightly touching a woman on the arm makes her more likely to agree to dance or give out her phone number because touchers are considered more attractive and (sigh) more dominant.

Touching a woman on the arm (or anywhere else) if you don't know her that well is a good way to get mace in the face. Or so I'm told. Maybe that's just me.

7. Genetics

You have a secret superpower, and it’s sensing immune profiles. (We never said it was cool.) Women prefer the smells of men whose genetic immune profiles are the most different from their own, which is helpful for your future offspring but also to everyone hoping to avoid a distant cousin.

Or, sometimes, people just stink.

You can't avoid a distant cousin, by the way. Close relatives, maybe, but everyone who isn't a close relative is your distant cousin. And I've seen other "studies" that imply that people too distantly related won't be attractive to a given individual.

Case in point:

4. What Your Parents Look Like

The Freudians weren’t right about much, but we do gravitate toward people who look like our preferred-gender parent. This would seem to counteract the whole “genetic diversity” thing...

I have no idea what my genetic parents looked like (other than that they probably resemble me, if they're still around). Who I find attractive has historically been all over the place, though, and I can't think of anyone I looked at and said, "oh, wow, she's short with black hair just like my mom!"

I mean, ew.

3. Your Parents’ Ages

Similarly, people raised by older parents tend to have older partners...

No idea if this is genetic or environmental. My adoptive parents were a lot older than me, too old at the time I was adopted to have given birth to anyone. And while my first wife was two years my senior, my second was nine years younger (fun fact: they were both 27 when I married them).

1. Who You’ve Had Sex With Before

Think you don’t have a type? Wrong. You might not have started out with one, but one of the biggest factors of our perception of beauty is familiarity. We prefer faces similar to those of our friends, loved ones, and yes, exes, because we associate those people with good times.

Yeah, no, not in my case. Women I've dated tend to be all over the spectrum in terms of height, hair color, body type, etc. The one thing they all have in common, the one characteristic that could be considered "my type" is that they were all batshit crazy.

Which I'm aware says more about me than about them (me being the other thing they all have in common). Which in turn is one reason I'm single.

That and my refusal to pick someone up at a bar.

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