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 12 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 14 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 29, 2020 at 12:08am

#984523

Dream Home

Gah! It's almost the end of the month already. Again.

PROMPT May 29th

When I was little, I dreamed of living in a treehouse surrounded by a pool with a spiral staircase going up the middle of the trunk.

In your second to last entry of the month, write about your dream home. Describe the rooms in your fantasy house and any unique characteristics. Be creative!

"Be creative." It's hard to come up with a two-word sentence that strikes more fear into my stony heart, except maybe "I do."

There's been pushback in recent years against the trend of ever-growing housing. "McMansions," they call them, cookie cutter castles in usually gated subdivisions with tyrannical homeowner's associations. Well, there is no way I'd ever live in a place with a homeowner's association, not if I have any choice in the matter at all. But I've always wanted a gargantuan house.

I know I'm supposed to want a "tiny house," which is the modern euphemism for "trailer." That's the form the abovementioned pushback takes these days. Nope. I need room. Not just for my stuff, but I'm home almost all the time so I want to look at different walls sometimes. In typical American binary fashion, these closet-sized domiciles are touted as the one and only antidote for McMansions. Apparently, I'm the only one who thinks that a middle ground is not only possible, but desirable: having a modest-sized home with a few rooms, rather than being owned by a modern incarnation of a king's palace, or living in a space smaller than my friend's shoe closet (yes, she has an entire room devoted to her footwear collection). Or maybe the tiny house manufacturers have an enormous advertising budget, which leads me to believe that such trailers are way overpriced.

But the mid-sized house is my reality, not my fantasy. In reality, I wouldn't want anything much bigger or smaller than what I've got right now, because I'm not wealthy enough to have staff.

No, my dream home is a supervillain lair.

Cut deep into the bones of a mountain, this would be a massive underground complex with hundreds of rooms, a spaceport, a helipad, and a missile silo. I'd have guards and flying drones patrolling the perimeter at all times (on a randomly rotating schedule), and a household staff consisting of scantily-clad supermodels. Most importantly, the air ducts would be crisscrossed by thick titanium bars at regular intervals, so the protagonist can't use the tired old "sneak in through the air vents" trope.

I'd need a dungeon, of course, to imprison my enemies, and an interrogation room (use your imagination) to extract information from them. There would have to be a movie theater with Dolby surround sound and 3D capability, an Olympic-sized swimming pool (as well as an infinity pool elsewhere), a bowling alley, and a skating rink. I don't skate, but I'd watch my staff do it, assuming I decide they can have some time off.

As for the other features, well, in addition to the obvious like a kitchen, several bathrooms with gold fixtures, and a gargantuan master bedroom (in the broadest sense of the word "master"), a well-stocked library would be a requirement; I may be evil, but I read. And a bar. Not just any bar, though, but a drafthouse with 100 taps and a large selection of wine and spirits. I might even let some of the staff use it.

I'd be tempted to put in a Starbuck's, but I really hate coffee.

Don't forget the broadband internet. Can't take over the world without it.

Oh, and I'll need a Planning Room. Maybe two Planning Rooms. You can't come up with dastardly plans if you don't have a room in which to concoct them. The reason I'd need two is that one would be like a conference room where I can brainstorm with my most trusted minions, before they attempt to betray me and I have them sent to the dungeon. The other would be dark and foreboding, with just one black chair in the middle on which I could sit and brood.

Yes, a supervillain lair would suit me quite nicely. But I'll settle for my modest split-foyer on the edge of town. There's a marginally lower chance of it being nuked from orbit.

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