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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Show Recent Comments

10 Most Recent Blog Comments:

Write_Mikey_Write!

about 6 hours ago
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

about 8 hours ago
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

Turkey DrumStik

Nov 6, 2024 at 10:22am
In response to "Lo!"

Not gonna lie I need the holiday season chatter.

July 2, 2021 at 9:50am

#1012904

Walking in Minneapolis

Yesterday, I managed to visit four breweries.

How, you might ask, can anyone survive four breweries in one afternoon?

Fair question, even though I totally just made it up.

I might have mentioned this before in here, but most breweries serve some variation on a tasting flight. This can take several forms depending on the brewery and whatever convoluted, Byzantine state laws govern the manufacture, serving, and drinking of fine fermented beverages. Whatever the details, the point is to be able to try small samples of a wide variety of craft beer selections. There's certainly a time for a pint or two (or three), but when I visit a brewery or brewpub for the first time, I like to get an idea of their diversity of offerings.

Now, it's also true that most breweries will let you taste a sip of a brew for free; I know I've noted before that taking advantage of this by drinking a bunch of samples and then not buying anything (or at the very least, tipping the bartender) is a dick move. While I'm not above dick moves from time to time, beer is too important to me to pull one at a brewery. At least not on purpose.

The point is, though, that it's not like I drink a whole lot of beer at any given brewery -- usually. I'll take a couple of sips of a 3 or 4 ounce sample. If I really like it, I'll finish the sample. Where this gets complicated, though, is at the rare place of libation that doesn't do some variation of the sampler flight. This lack has become more common, unfortunately, due to the pandemic -- which doesn't make a lot of sense to me as it's not like it increases the risk of exposure over the baseline for a brewery; perhaps it's because it's a bit more labor-intensive than just pouring pints.

So of the four breweries I tried yesterday, one of them didn't do flights. It was probably just as well, because half their menu was hard seltzer anyway. And that's fine; a lot of people like that, but I don't classify it as "beer." I'm old enough to remember Zima, as much as I'd like to have forgotten that crap.

Incidentally, walking around Minneapolis is a fairly pleasant experience. It's relatively flat, and as long as it's not winter or riot season, it's a nice place. And there's a river to look at from some vantage points.

After the whirlwind brewery tour, I got a chance to meet Turkey DrumStik

for dinner at a delicious pizza place. Well, I assume the pizza is delicious; I had spaghetti and a "small" Caesar salad that was about the size of the Colosseum. I just ordered Coke to drink there, though; I'd had about enough beer prior to that, and didn't want to be so drunk as to become the subject of one of her blog entries.

Sorry, no pictures this time.

Today perhaps I'll visit two or three more breweries, because that is, after all, one of my main reasons for traveling.

5 Like • Comment • View 3 comments

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