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:

Turkey DrumStik

about 5 hours ago
In response to "Barley There"

And approximately 15 seconds later, someone's going to screech about genetically engineered crops.

That would be me...screeching for more genetically engineered crops.

Write_Mikey_Write!

Nov 14, 2024 at 2:04pm
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

Nov 14, 2024 at 12:10pm
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

December 4, 2018 at 12:43am

#946804

Tea, Totally

I got hit by anxiety yesterday.

Historically, this isn't something that happens to me. I mean, sure, maybe if there's a specific thing generating it, like public speaking or having to go to a party where I don't know many people, I'll feel a bit nervous, or whatever you call it. But this is the first time I can remember just feeling it with no obvious reason.

I don't like it.

The only thing I can figure is after a long time, I'm finally feeling like being social again. I feel something other than a generalized sense of malaise and the certainty of impending doom. Those things, though, I've gotten used to. They comfort me. If this is what actual emotion feels like, then... no, thanks. I'd rather stay depressed.

Fortunately, there is beer.

This evening, BJ's (a chain brewery, but a decent one) had a Belgian Grand Cru style beer. It is delicious. I almost didn't go home.

But even I can't drink beer all the time. Mostly I imbibe Coke Zero; I want the caffeine but not the sugar/HFCS of regular sodas. And don't give me that "aspartame will kill you" crap; there are worse things out there.

Also, there's tea.

https://www.nbcnews.com/better/health/best-teas-energy-digestion-sleep-more-ncna...

Tea is the second most widely consumed beverage in the world (next to water), according to the Food and Agriculture Organization of the United Nations. While coffee drinking overshadows tea consumption in the United States, the Tea Association of the U.S.A. reports that 80 percent of American households have some form of tea in the cupboard, and more than 159 million Americans drink it on a daily basis.

I utterly despise coffee. Never got a taste for it. This, apparently, sets me apart from civilization in general. As an American, I'm supposed to be addicted to coffee. I think it's written into the Constitution or the Declaration of Independence: "We hold these truths to be self-evident, that coffee is the national beverage..."

But no, apparently I never got the memo. Can't abide the stuff. The smell gives me a headache, and being around people with coffee breath is probably the source of my antisocial tendencies. But that's okay, because once I tell people I don't drink coffee, they look at me like I'm a reptilian alien from Antares 7. It's like:

"I don't have kids."

"That's cool."

"I'm an atheist."

"Hey, I understand."

"I don't drink coffee."

"WHAT THE STEAMING PILE OF FUCK IS WRONG WITH YOU?"

So like I said, there's tea. "So why don't you just drink tea instead of that nasty cola?" Well, because for me, even the most highly caffeinated teas put me to sleep. I can drink a pot of black tea just before bedtime and sleep like a... well, not like a baby, because babies wake up every hour and a half which is why I never wanted one, so let's say sleep like a computer that's been powered down and unplugged.

If I drink it with breakfast, all I want to do is go back to sleep. It's nice when I want to relax, though. I can drink non-caffeinated herbal teas, but there's no freakin' point in drinking decaffeinated tea.

My favorite tea isn't on the list I linked above. It's harder to find in the US, but I've ordered it from Amazon. Its name is going to trigger your inner Beavis and Butthead, so let's just get that out of the way right now:

https://en.wikipedia.org/wiki/Pu%27er_tea

When you're done snickering like a 10 year old, I'll continue. Meanwhile, I'm going to go make me a pot of the stuff.

Ready?

Okay.

So yeah, Pu'er is strong, dark, earthy and bold. I drink it straight, or sometimes with just a bit of lemon because steeping it too long sometimes brings out a hint of bitterness. With a fairly high level of caffeine, it does what no other drink can do for me: make me relaxed and alert at the same time. It's a weird feeling.

And it sure beats the hell out of anxiety.

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