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:

Write_Mikey_Write!

about an hour 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 3 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.

November 16, 2023 at 9:51am

#1059568

Beaujolais Nouveau Day

Once again prompted by "Invalid Item"

, I present yet another wine-related celebration, one that I look forward to every year.

Beaujolais Nouveau Day

Unlike some of the other observances I've noted, which are international or US in nature, this one's French. Or, at least, it's French the way Oktoberfest is Bavarian: no one's stopping us ugly Americans from celebrating.

There's a bit about Beaujolais Nouveau at the link above, but, naturally, I'm going to say my piece (actually pieces) about this utterly delightful wine.

The "Nouveau" part of the name means, it doesn't take four years of Duolingo French lessons to know, "new." But it's not like it's a new product; no, I think it's "new" in the sense of "fresh." The "Beaujolais" part is a place name, like Bordeaux or Champagne. Apparently, according to that link, the capital of the region is Beaujeu, which might translate to "beautiful game," which honestly would be really cool if so.

But I digress. This is a "new" wine, meaning the time from harvest, through fermentation, to the important part (drinking it) is very short. This results in a clean, crisp, fruity wine, without many of the more complex flavors or profound textures of the slower reds.

It is also, unusually for a red wine, best served chilled, like a white.

Another digression: it's well-known, even outside of wine circles, that most reds aren't served chilled. But this does not mean they should be served at room temperature (around 20C). Same goes for many dark beers. No, these heavier fermented beverages are at their best at cellar temperature, which is around 13C, though there's some leeway there. Owing to its status as a young wine, Beaujolais Nouveau is, instead, served cold like a white, or that better-known French product, champagne.

As delicious as the wine is on its own, there's a special, serendipitous connection for those of us in the US: Beaujolais Nouveau Day is, as is also noted at the link, celebrated on the third Thursday of November.

If this sounds vaguely familiar, it's because Thanksgiving is celebrated over here on the fourth Thursday. And yes, Beaujolais Nouveau is, normally, available in the US. Today is the day I usually make a pilgrimage to the grocery store (as opposed to ordering a delivery therefrom) to pick out a bottle. Might have to be tomorrow, though; I have a beer dinner to attend tonight and I need to make sure I'm well-rested for that.

And I can tell you this from experience: there is, quite simply, no better wine to accompany a traditional Thanksgiving feast.

Like I said, it's not a complicated wine. But usually, one of the distinctive fruity flavors it fronts is that of cranberry. And what better accompaniment for turkey than cranberry?

So there's that linking the two countries, as if the Statue of Liberty, the design of our capital city, and, you know, the entire concept of representative democracy weren't enough.

"But, Waltz, the Pilgrims didn't have French wine to drink." Okay, fair. And Beaujolais Nouveau hadn't even been invented yet, not to mention the refrigeration that would keep it at its ideal temperature. But if you think your Thanksgiving feast is a perfect reconstruction of what the Pilgrims ate, or, more appropriately, what the settlers at Jamestown managed to scrounge up, think again. I don't know; maybe you try to do that, keeping it all to truly native American harvest foods. If so, great. You do your thing.

As for me, while I have no intention or desire to be part of a crowd at Thanksgiving, I may give the holiday a nod by warming up a frozen turkey pot pie.

It will be accompanied by a nice, cold bottle of Beaujolais Nouveau.

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