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 6 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 8 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.

August 17, 2020 at 12:08am

#990991

The Real You

And who, exactly, do you think you are?

https://www.cracked.com/article_27124_totally-random-things-that-make-you-well-y...

Totally Random Things That Make You, Well, You

Let me get this out of the way: These are not "totally random." Yes, yes, I know, I'm quibbling about the definition of "random" when talking about a dick joke site.

6. Your Eye Color Can Determine Your Alcohol Tolerance ... And Propensity For Alcohol Addiction

When the time comes for drinking, some of us can down eight shots and still beat anyone in the bar at blindfolded darts, while others have to be carted home after sniffing a single beer-soaked napkin.

And some of us don't play drinking games. But I guess I'm a bit older than Cracked's intended demographic.

Blue-eyed people consistently hold their liquor better than those of us with brown-eyes.

Well, that explains everything (I have blue eyes).

Blue-eyed folk's high tolerance seems to put them on track for drinking more and for ultimately being more likely to become alcoholics.

If I'm not one by now, I never will be. If by "alcoholic" you mean "anyone who drinks," then sure.

Whatever the explanation, people now know simply by looking at you how easily they can get you drunk, which is why, personally, we recommend wearing sunglasses at all times.

That also helps to hide the dilated pupils and the red sclera. And mutes the hue of the pink elephants.

5. Having A "Normal" And Easy-To-Pronounce Name Can Make You More Likable And Successful

Students with common names tended to succeed more than their weird-named counterparts, while the ones with the uncommon names were the ones most likely to have psychological problems.

Okay, here's the thing: I'm painfully aware that this sort of thing has racist (or at least cultural) ramifications. It's been shown that identical resumes, where one applicant was named, like, "Steve" and another named, like, "Hamal," led to the former being accepted and the latter rejected, more times than not. At least in the US. So I don't really like the way this section is presented.

But that's no excuse for anyone to name their kid Kaiyleeighh. There is, in fact, no excuse for that at all.

4. Have A Bunch (Or Very Few) Friends And A Huge (Or Tiny) Social Network? It's Probably Your Gut Bacteria

You're on the internet and reading a humor article, so there's a fair chance you call yourself an introvert.

How... how did you know?

What made you that way? Was it your upbringing -- your parents took you to one party when you were three, and it was so loud and sticky that you swore off people forever? Is it in your genes -- your parents were both introverts, and it's frankly amazing that they even socialized enough to ever meet each other? Maybe it's both those, but there could be other factors too. Ten trillion of them in fact, and they're living in your intestines.

I'm starting to get the impression that this is the bacteria's body, and I'm just living in it.

3. The Air Quality In Your Home And School Probably Determined How Intelligent You Are

In 2015, Southern California experienced a gas leak that was absolutely huge -- you can't really compare gas leaks to oil spills, but if you could, this leak would be bigger than 2010's Deepwater Horizon disaster. People around Los Angeles got pretty worked up about the thousands of tons of methane and other gases suddenly hissing around them, and they responded by installing air purifiers in schools, places where sudden emissions of methane are otherwise normally just a source of comedy.

A year later, test scores in schools with the new air filters (and only those schools) went up, measurably.

One data point does not a conclusion make, but the article goes on to point out that, yes, it's happened in other places too.

Every study on a more specific group comes up with the same thing. Baseball umpires? Chess players? Pear pickers? Everyone gets dumber as the air gets dirtier.

Which explains so much.

2. Your Birth Weight Partially Determines Your Respiratory And Cardiovascular Fitness As An Adult

Speaking of respiration, your respiratory prowess might have been set from the moment you were born. And sure, a lot of things are set at that point, that's kind of how genes work, but in this case, your parents were able to view your hidden settings using something clearly observable from the start. We're talking about birth weight, which is approximately the first thing the nurses check when you're born, right after what kind of genitals you're packing and whether you have a tail.

Pretty sure the first thing they check is if you're squirming or not, but whatever.

Obviously, you're still able to change your heart and lung health by exercising, and you should really get on that. Just know that the game is slightly rigged from the start. And when you have a baby of your own, make sure to weigh them immediately and take a commemorative photo with them hanging from a scale at the dock.

Trophies are trophies, after all.

1. Your Facial Features And Tone Of Voice Can Determine How Much You Care About The Environment

One fun thing psychologists like to do is think of some idea that kind of feels like it could be true, put together a study to see if it's true, and then cackle in delight when they find that, whoa, it's true.

And then get mad when another team of psychologists can't replicate the experiment and accuse the first team of p-hacking.

Like what happened last year when a Canadian team wondered, hey, you know how research says that the more masculine a man looks, the more selfish and violent he in fact turns out to be?

The obvious problem with this is that "masculinity" is subjective.

The subjects then answered an environmental survey, and the more noticeably masculine guys gave answers indicating that they didn't care much about the environment at all. The researchers had a pretty clear explanation handy. This all came down to testosterone.

Ummm... what? No, it's more likely due to socialization of gender roles.

So what does that mean? Was this whole study a bunch of bullshit? Maybe, or there could be another factor here: the socialization of gender roles.

Aha! Confirmation!

Anyway, as always, don't take anything from Cracked as Truth. I just think the whole thing's interesting. Or, rather, my gut bacteria think it's interesting and made me type this. Help I'm a prisoner in someone's colon!

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