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🎶

about 22 hours 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

July 26, 2024 at 11:39am

#1074411

Unbelievable?

I'll give Cracked a pass for using the phrase "scientific facts" like this, because they're a juvenile humor site, not a scholarly publication.

4 Scientific Facts Rational People Should Refuse to Believe
If we told you someone’s hiding edible microchips in your cheese, you’d think we were nuts

Mmmmmm... cheese and nuts...

Ahem. Sorry. Where were we? Oh, yeah.

We love telling you unbelievable facts. When we call a fact “unbelievable,” though, we still do expect you to believe it. You’re supposed to say, “I didn’t know this was true before, and it’s surprising, but I believe it now because I trust you.”

I trust you to make jokes. Sometimes those jokes are even funny. The idea that I'd trust this source for anything else is funny.

Note: We’re not telling you the following facts are untrue. Some of them definitely are true, and they’ll remain true whether you believe them or not.

As usual, skepticism serves well here.

4 Baby Animals Evolved to Look Cute So We’ll Protect Them

These cute baby animals share a specific set of characteristics, which we’ve dubbed kindchenschema. They have large eyes and small mouths.

Animators know this, which is why your favorite cartoon character protagonists have big eyes and small mouths. That much, I don't dispute. What I have a problem with is the reversal of cause and effect.

Also, the scientist who first described kindchenschema was a full-on Nazi.

Yeah, well, so was Heisenberg. That doesn't mean he was right or wrong about the science.

What if so many animals are born with big eyes and small mouths just because they’re still growing into their parts, rather than as a specifically evolved survival trait?

Okay, I can accept that hypothesis, but it doesn't address the cuteness factor: why do we find those traits adorable?

Look, Occam's Razor demands the simplest hypothesis, which is not that a diverse number of animal species began producing "cute" offspring to please their bipedal, mostly hairless overlords, but that said overlords themselves evolved to find those attributes "cute" because if people didn't find their own babies cute, there would be less reason to hang on to the whiny, poopy bastards.

I'm not saying that's correct, either. But it makes more sense.

3 Muscles Grow More When You Concentrate on the Mind-Muscle Connection

The fitness industry is full of new and exciting science explaining the right way for you to build muscle.

And most of it is bought and paid for by companies wanting to sell their latest torture devices, much like nutrition science is often bought and paid for by companies wanting you to consume their products.

One of the strangest factors here is something called mind-muscle connection. You will see more muscle growth if you think very hard about the muscle you’re exercising.

Okay, but that's not totally out there. The idea that mind and body are separate entities is obsolete. I'd just want to see real evidence, which I imagine is hard to come by.

What if this is a conspiracy, whose sole goal is to occupy your full attention when you exercise, so you’re oblivious to all else around you? This way, the conspirators can take sips out of your water bottle when you’re distracted, saving them from having to walk all the way to their own bottles, which are on the other side of the room.

See, now, that's funny.

2 They’re Putting Microchips in Parmesan Cheese

If you buy some parmesan cheese in the United States, the word “parmesan” on the label means basically nothing.

Well... it means it resembles cheese, but you can be pretty sure it doesn't resemble steak or an energy drink.

Europe is a different story. In Europe, that parmesan cheese you buy had better be the real thing, made in northern Italy.

One of these days, I'll figure out why this works for, say, champagne vs. sparkling wine, or tequila vs. mezcal / agave spirits, but not parmesan. But today is not that day. Tomorrow doesn't look good, either.

The blessed cheesemakers from Parma and Emilia-Romagna have a new tool to let you distinguish their wares from fakes: microchips.

Well, now, that's a reference joke that's growing increasingly obscure. It's from Monty Python's Life of Brian. Buncha peasants are standing around trying to hear the Sermon on the Mount, but they're pretty far away and microphones haven't been invented yet. So there are lines like "What'd he say?" "Blessed are the cheesemakers."

We’re going to find it hard to invent a conspiracy to explain this story, since our whole issue with it is it already sounds like a conspiracy. If we try, we’ll probably end up concluding that every other piece of food we eat also contains microchips, and the Parmigiano Reggiano ones only got so much attention because they got caught.

How about: we only think it's a conspiracy because the microchips we've absorbed from other sources control our minds to think so as a way to deflect blame?

1 Balls Hang Outside the Body to Keep Cool

Oh, hey, after the cheese entry, we're back to nuts. Yeah, not going to quote this one. I'll just point out that exposed balls are certain proof that men weren't intelligently designed. Jury's still out on women.

Which, of course, is the exact opposite of the author's joke at the end.

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