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 11 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 13 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.

May 19, 2022 at 12:01am

#1032584

One Great Thing

"Invalid Item"

prompt today.

The biggest plus for living in your town/area

It's not Cleveland.

Seriously, though, I've already covered this: the proliferation of breweries both in and around town. I've mentioned several of the more urban ones, but there are just as many outside town.

But it's not just beer. No, lest anyone start to wonder if I can only think about One Thing, nothing could be further from the truth.

For example, we also have, right here in town, one of the few sake breweries in the US. While it's not the best sake, it's ours.

There are also several cideries in the area, making delicious beverage from apples and pears.

In addition, there's the wineries.

Those aren't technically in town, either, but the surrounding area is home to a large number of vineyards. As I mentioned before, it was Jefferson's idea to plant some French vines in the area, as he figured the climate was somewhat similar to parts of France. The attempt wasn't very successful.

But science advanced over 200 years, and in the 1970s, on a nearby farm dominated by the ruins of a house also designed by Jefferson, an oenologist figured out the trick to producing relatively large quantities of delicious spoiled grape juice. That vineyard is still around. I got married there once. I don't know why, but they shared their secret with other vineyards. If you've ever wondered about the explosion of wineries over the past 40 years when you're driving through unlikely places (like Indiana or New Jersey), well, you have Barboursville to thank for that.

One time, I was in California, a place much better known for its wines. Whilst sampling some good ones, I was talking to the owner (it was a small place) and told him I was from Virginia. He scoffed at the idea of Virginia wine. I of course took it poorly. The next time I was there, I brought with me a bottle of Viogner, one of the grapes that do really well around here, to give to him.

Look, I'm not saying we're as good as France or California, or even a good Argentinian Merlot or an Australian Shiraz, but our vineyards don't suck and, you know... shop local.

We're also home to Trump Vineyards, about which the less said the better. Naturally, it's the one wine I won't drink.

I can't end this without also discussing distilleries.

Our neighbors in Kentucky and Tennessee get all the publicity when it comes to distilled spirits (primarily bourbon and rye). And for the most part, it's well-deserved. But recently, we've become home to several small distilleries, including Vitae in town, and Silverback out in the country. Decent product from both of them.

Now, plenty of locations have a good selection of fine fermented and distilled beverages, but the particular combination here makes the town a great place to live.

So you see, I don't just think about One Thing.

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