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 11 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

September 22, 2018 at 2:30am

#941825

Beer Hunter: The Quest for Fest

It's Oktoberfest season.

Yes, I know it's not October yet. Don't blame me. Blame the Germans - they're the ones that start Oktoberfest in September.

Every year, I look forward to this time because that is when all the Oktoberfest beer comes out. It's kind of the man version of pumpkin spice latte everything (though I'm sure someone, somewhere, has come up with a pumpkin spice latte beer). It's not my all-time favorite style, but since it's seasonal, I like to drink it when it's around. That pic of me in my bio? That's an Oktoberfest, from a visit to New York City a couple of years ago.

But.

I went to my favorite (read: closest and within stumbling distance) bar the other day, and the only fest they had was Sam Adams. Now, I'm not going to rag on Sam Adams - they've done a lot to promote craft beer in general, though they're big and corporate and their beer tends to be inoffensive and therefore uninteresting.

We live in a golden age of beer, with over 6000 breweries in the US alone at the moment. Just about any style you can think of, and more, is represented somewhere in the world of craft beer. Moreover, in my local area, there's probably about a dozen breweries. So there's plenty of diversity in beer selection around here. And yet, the only fest this taphouse could come up with was Sam?

Okay, so there's Alamo, the drafthouse/movie theater I mentioned a few entries ago. I got to the Wonka party early so I could have a couple of beers. I was hoping for an Oktoberfest. But no - I mean, they have a good selection, but dammit, it's nearly the end of September so where's my fest?

So I spent my time drinking a nice stout and trying to use my phone to find out who around me has the holiday spirit.

There's a BJ's in town now, and I looked at their menu and behold! An Oktoberfest. The accompanying text said, "lagered for up to three months." Okay, well, I like beer but I'm not an expert - I thought fests were supposed to lager for six months. Whatever. Their beer is okay for a chain, so perhaps I'll pay them a visit soon.

But really, it shouldn't be this hard to find an Oktoberfest lager in September in a place with that many breweries around. Maybe if the taphouses weren't so fast to jump on board the whole "sour beer" fad, there would be more room for seasonal offerings. And yes, for the uninitiated, sour beers are actually a thing. I don't like them. I prefer my beer to taste different going down than it does coming back up.

1 Like • Comment • View 3 comments

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