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 10, 2020 at 12:01am

#999970

Resolve

"30-Day Blogging Challenge ON HIATUS" [13+]:
10. Human Rights Day
Write anything you wish about this.

"JAFBG" [XGC]:
How do you feel about New Year's resolutions? What are you definitely not going to work on in 2021?

You know, I almost -- almost -- wish I were more sober to do this particular combination.

Yesterday, I made latkes. This may not seem like a big deal to anyone else, but to get me to actually do the work to create a food thing is a major undertaking. My motivation was simple: I couldn't visit my friends in California this year, and when I do visit them, there's a tradition of eating latkes (among other things) and going wine-tasting. So I approximated this at home by making my own latkes and drinking wine. Not California wine this time, though, no, I had to go all the way and drink a Vouvray.

Allow me to expand upon the wonder that is Vouvray.

All along the Loire Valley in France, there are vineyards. This should not be shocking; it is, after all, France. But there's something about that particular region. Vouvray is created from the chenin blanc grape, and anywhere that chenin blanc grows one can make wine out of it. But most chenin blanc is terrible. Not so with the Loire Valley. There's something about the soil there that imparts an earthy character to the wine. This is called -- and I knew this word before I even started learning French seriously -- terroir.

One of these days, I want to spend like two weeks (or maybe more) in France, mostly touring that area, if they'll put up with an ugly American for that long. Oh, sure, you hear about Bordeaux and Champagne and Burgundy or whatever, but of course I have to seek out the lesser-known regions because I'm secretly a hipster. Shh, don't tell anyone.

Anyway, the reason I'm not entirely sober right now is I drank most of a bottle of Vouvray while eating the latkes that I made (with an applesauce topping created by my housemate -- said applesauce being improved by my idea of adding pears in addition to the apples).

I guess maybe I have some creativity after all.

But regarding the prompts.

I think I've made my opinion on New Years resolutions more than clear over the many years I've been here. My go-to joke in the Comedy newsletter has been how we make resolutions and they're mostly broken by the end of the month. I've even encouraged people to go ahead and break them on New Year's Day just so you can get that shit right out of your system. I always avoided going to the gym in January, because it takes that long for people to give up. I call them tourists, because they invade the gym in January and are mostly gone by February. This was in the Before Time, of course, when going to the gym was a viable option.

I hate the entire idea of resolutions.

So allow me to expand upon what, exactly, I am NOT going to work on in 2021:

I am not going to stop smoking cigars.

I am not going to go on a diet (I mean, sure, I've been careful about what I eat, but I'm not making any further changes just because a calendar flips over).

I am not going to commit to writing, reading, reviewing, or really anything else.

I am absolutely, definitely, not going to stop, or even slow down, my drinking.

I think everyone can agree that 2020 sucked major camel ass. Right? Anyone think this was a great year? No? I didn't think so. A lot of people seem to be looking forward to 2021, but I have to reiterate: it's just going to be more of the same suckage.

New Years are strictly arbitrary anyway, and the older I get, the more I resent the Gregorian calendar and its utter disconnection from anything real. There are certain astronomical events that define the passage of time much better than the fake-ass calendar ever can, and again, I've harped on these things to the point where I'm sure people have quit listening to me. But I'll say it again anyway: the only markings of the passage of time that are meaningful to me are sun and moon milestones. Solstices. Equinoxes. Full and new moons. The dance of the sun, moon and planets means something to me, while the arbitrary counting of days, months, and years is just a thing imposed upon us, like the idea of a number system or a language.

Which

SEGUÉ

leads me to point out just how arbitrary "human rights" are.

Now look, I'm not arguing against human rights, here. Nor am I nearly sober enough to enumerate what I think human rights should be.

But.

Rights are what we, collectively, decide that they are.

Now, I could get into things like your "right" not to wear a mask as opposed to my "right" not to collect your germs. But I'm not going to.

Instead.

I want to point out that instead of talking about "rights," we should be talking about "responsibilities."

Like. For instance. If we declare, collectively, that there is, as enumerated in the USA's founding documents, the "right to life," that implies that we all have the responsibility not to kill each other. (And don't get me started on how that applies to abortion. All I'll say about that right now is that life begins at birth, period, end of discussion, I have spoken.)

You have the right to free speech? The associated responsibility is that I do not have the power to silence you.

I have the right to free exercise of religion (or lack thereof)? The attendant responsibility is that you may not impose the tenets of your religion (or lack thereof) upon me.

I'm always arguing with my conservative friend over health care. "Health care is not a right," he says. "It should be," I say. This is a legitimate area of discussion, and while I have my opinions on the subject, I understand that there is room for debate and compromise. The point being that we can only decide what "human rights" are through careful consideration, philosophy, and science -- rights are, in the end, abstract ideas.

Perhaps if I hadn't been drinking I could delve more deeply into the abstractions involved. But right now, I just can't be arsed. I will say that, drunk or sober, I think we all need to take the responsibility to end discrimination based on race, gender, religion, etc. That's something we can all work on, and it shouldn't take a purely arbitrary calendar shift to decide this. "Oh, it's the New Year, I think I'll quit being racist this year." No, fuck that. Just quit being racist, all of y'all.

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