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

December 16, 2020 at 12:15am

#1000304

Chocolate-Covered Crap Remains Crap

"30-Day Blogging Challenge ON HIATUS" [13+]:
16. National Chocolate Covered Anything Day
Sounds like a great to 'pig-out' and enjoy!
Have you ever eaten anything 'unique' covered in chocolate?

"JAFBG" [XGC]:
Congratulations! You get to kill one holiday song and never have to hear it again. Which will you choose?

One? Just ONE? Hardly worth it when there are so many that send me into paroxysms of grumpiness.

You know what my favorite thing to have covered with chocolate is?

Chocolate.

And I don't mean milk chocolate, or the abomination that is deceitfully called "white chocolate." There is absolutely no reason to sully the purity of chocolate with any other ingredients, except for a small bit of sweetener.

Which is not to say I don't sometimes enjoy chocolate-covered something. I'm fine with truffles, and it's okay to coat almonds with chocolate. Drizzle some on a dessert, fine. But to me, nothing beats the sublime goodness of pure, unadulterated dark chocolate.

Every year or so, someone will come up with a list of peoples' favorite commercial candy bars, and every damn time, Reese's makes it to the top of the list. As I noted in a previous entry, I utterly despise peanuts, although I'm okay with peanut butter. But to me, that crumbly, sugary, dried-out crap in the middle of a Reese's cup bears little resemblance to actual peanut butter. Okay, I can accept - have accepted - that my tastes are weird, as I've noted in a recent entry. But I simply cannot comprehend how popular Reese's cups are, with their fake, sugary "peanut butter" and waxy milk "chocolate."

I do have weaknesses, of course. Many of them. One of them is, or rather was, Oreos. I think I've mentioned this before, but it's been a while, so I'm going to relate this again:

Used to be I'd go to the grocery store and, in the course of perusing the aisles, the Oreos display would draw me like a moth to a flame. I'd get the standard Oreos, and go about my business, and then at home occasionally eat some of the Oreos.

Then Nabisco started doing different flavors of Oreos. The chocolate-filled ones were good, I thought. And then they came up with fudge brownie cream filled Oreos and OMG those were delicious.

Then they stopped making them. I suppose Nabisco thinks that if they keep coming up with new flavors, people will buy more Oreos. Not me. Once I tried the fudge brownie kind and then couldn't have them anymore, instead of occasionally buying Oreos, now I *never* buy the little fuckers.

Good for me. Bad for Nabisco. Doesn't seem to bother them much, though, as they continue to produce ever-weirder concoctions, the most egregious of which were -- not joking here -- Swedish Fish Oreos. Look, I have nothing against Swedish Fish, and I'm fine with original Oreos, but that combination just strikes me as being particularly heretical, like mixing Skittles with M&Ms.

Anyway, the point is, when it comes to chocolate, I'm happy to try different things; we have a chocolatier here in town that I can say with some certainty is one of the best in the world, and I've enjoyed nearly everything I've had from them. They are expensive, though, and my go-to gift for people who don't drink booze is a box of those chocolates. But of all of them, my favorite is still the pure chocolate ones, sans nuts, sans cream, sans goddamn cherries.

All of this is to point out that yeah, my tastes are obviously different from most peoples'. Another thing I shun with every fiber of my being is commercials, as I have also noted here before. I despise them with an all-consuming fiery passion. And yet... and yet... on Monday, I went to my optometrist. I had to Uber over there because they were going to dilate my pupils and I wouldn't be able to drive back, so in an abundance of caution, I ended up getting there half an hour early. Sitting in the empty waiting room (they've got anti-contagion protocols in place), I was subjected to a holiday music radio station.

I could feel my blood boiling, my lymph roiling, my brain recoiling. As with chocolate, I get that other people have different likes/dislikes than I do. That's fine, I get that, you do you, whatever. But that day, listening to one sappy, gooey holiday song after another, all I could get through my mind was: "Please go to commercial. Please go to commercial. Please go to-" And then finally they did. So yeah, it's official: as much as I hate commercials, I hate standard holiday music even more. Though I could argue that holiday music *is* a commercial.

And yet, in defiance of my wishes, it's still produced and broadcast. Like Swedish Fish Oreos. The only difference is, if SFOs are sitting out on a table somewhere, I can -- and will -- choose not to eat one. Or if there are M&Ms in a bowl and I can tell that they're the bloated, hated peanut kind rather than the inoffensive milk chocolate or delicious dark chocolate kind, again, I can simply pass them up. More for the people who like them, right? Not so with holiday music. It's out there and if I go out in December, November, and sometimes even cocksucking October, I'm subjected to it.

Obviously, a lot of people like it. Fine. And I'll admit that there are some holiday songs that I like. The Grinch theme comes to mind, as you might imagine. And I do appreciate good music, so if it's good music, I don't care if it's holiday or even religious themed; I will appreciate it. The problem is that most of the crap that gets drilled into our collective brains every season is the musical equivalent, to me, of white "chocolate" covered cherries and peanuts.

And I can't think of a bigger abomination than that, except -- maybe -- for Swedish Fish Oreos.

Clearly, I'm going to have to rearrange my schedule so that I get my medical crap taken care of in January, so that I can hermit every December. That way we all win: I don't have to be subjected to smarmy "White Christmas" bullshit, and you don't have to hear me bitch about it.

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