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!

a day 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

a day 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.

June 25, 2019 at 12:45am

#961495

Home

Today, we are visiting one of the most sacred places on the Big Island - Waipio Valley. The valley was home to many early Hawaiian Kings and chiefs because it was well-protected and the land was extremely fertile for growing native taro plants. Today, the switchback road into the valley is so steep, only the most adept drivers with 4-wheel drive vehicles are able to visit the valley’s base and the pristine beach. Native Hawaiian people still farm the land in the same way they did hundreds of years ago. The cultural roots in this place are profound.

Where do your cultural roots run the deepest? Where is your family’s ancestral home? Do you feel most connected to the place you grew up, where you live now, or somewhere else?

Short answer: no fucking idea.

Long answer:

When I was a kid, in rural Virginia, you couldn't kick a clod of dirt without dislodging an arrowhead. I always had this idea that the place had been a Native American village. It was the perfect spot for it: right next to an estuary of the Potomac, also accessible by land.

Later, I found out that it was, indeed, a Native center of population. John freaking Smith himself described the place. The tribe that lived there was part of the Powhatan confederacy - you probably heard of them if you know anything about the non-Disneyfied story of Pocahontas - and the Potomac itself is named after them.

Descendants of this tribe finally got state, but not federal, recognition recently, and my friends and I donated a good number of artifacts to them. Their ancestral home was my childhood home, which is mine now by inheritance, and I couldn't imagine doing anything but honoring their legacy by gifting them these bits of history.

Archaeologists continue to explore the site. Nothing earth-shattering, nothing anyone would have heard of, and I've been pretty careful to keep it off the news. And not much is left; the flat parts have all been farmed since before I was born, and the hilly parts are covered in trees that get timbered every generation or so. There had also been a minor Civil War battle in that location, and in addition to the arrowheads and other artifacts, we found a lot of Civil War relics: buttons, bullets; in one spectacular find, an actual saber in an actual scabbard.

That's my only real connection to history. My genetic legacy is anyone's guess, and I'm not interested in doing one of those intrusive DNA tests that probably aren't very accurate, anyway. My light skin and blue eyes betray, as I've noted before, some sort of Northern European ancestry; I don't really care to know more. It makes no difference to who I am now, and I feel no more connection to anywhere in Europe than I do to, say, Nigeria.

As for my adoptive legacy, well, that doesn't matter much either, but I'm starting to piece together some things that are probably better off unknown. Again, nothing to do with who I am now.

I've mentioned before that I love travel. I've been all around the US, and a few other countries, and there are places that stick with me: the desolation of the Mojave, the cool blue of the Caribbean, the white peaks of the Cascades, England's hill country, the stink of the Niagara River, Maui's Haleakala, the Maya Mountains of Belize with their hints of an ancient civilization.

Virginia is my home, be it Tidewater or Piedmont. More than that, though, I'm more interested in what connects us than in what divides us - and what connects us is the ball of mud we all live on. I want to see more of it.

1 Like • Comment

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