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 28, 2019 at 12:19am

#961661

Infinity, Or Close Enough

We’ve spent a lot of time in the stereotypical Hawaiian climate these last few days, but today, as a grand finale to our final Summer Sprint, I want to bring you to a place a little colder: the summit of Mauna Kea. When measured from the ocean floor where the mountain began as a baby volcano, Mauna Kea is the tallest mountain in the world. The peak is regularly white with snow and the observatories on the summit enjoy clear views of the sky since they are above the cloud line. The mountain is a hub of science and spirituality, and a constant beacon of strength and protection over the island.

Today, write about an experience that wowed you. When was the last time your jaw fell open? Where were you the last time you felt awe and admiration? As best you can, share your experience in a way that your blog reader can feel the same wonder you felt.

I have trouble with elevation.

This is, I think, related to my problems dealing with pressure. Not, like, work pressure - I dealt with that just fine, thanks to my old buddy Al Cohol - but actual atmospheric pressure. Like I said, I can't dive because I'm unable to equalize pressure, and water pressure increases very quickly with depth.

I spent my early life at or near sea level, and most of my adult life at a paltry 500 feet. Occasionally, I'd take a trip to the mountains around here, but even those top out at less than 1500 feet.

Driving across the Rockies or the Sierra Nevada, though, there are elevations that make me gasp for air, even while just sitting in the car. After some experimentation, I found that this usually happens at or around 9000 feet or more.

Nevertheless, I've exceeded that elevation multiple times, on purpose. Last summer, Nerd Camp was around 10,000 feet in the mountains of Colorado. I haven't been to Mauna Kea, but I saw it from the summit of Haleakala, also about 10,000 feet.

But the time that sticks out in my mind was one New Year's Day, while crossing from Nevada into California.

When I travel, I generally try to avoid interstates. While they're a triumph of civil engineering, they're not conducive to the kind of travel experience I crave, one that focuses on the journey rather than the destination. When I tell people I once took two weeks to cross the continental US, they seem shocked - using interstates, it's no more than 30 hours from sea to polluted sea.

There are basically two crossings via interstate from Nevada to California. The northern route starts around Reno, and crosses at Donner Pass, the infamous location of arguably the US's most famous cannibalism. Donner Pass tops out at about 7,000 feet - easy, even for me. (The other route goes from Vegas to L.A. and isn't important to my narrative.)

But there are other ways to get across the Sierras, and on this particular New Year's, I was heading for one of them.

I don't know the elevation of it, but it's definitely higher than Donner Pass. I'm not even sure of the route; it wasn't US 50, but something else. I have it recorded somewhere, but can't be arsed to look it up. I'd left Vegas in the early morning, and spent the short, near-solstice daylight hours driving northwest through Nevada. I started feeling like crap - turns out that what happens in Vegas doesn't always stay in Vegas, especially if it's a virus. No, not that kind of virus; get your mind out of the gutter. This was like a cold, or the flu, but somewhere in the middle - not easily shaken like a rhinovirus, but not severely debilitating like influenza. This got worse as the day progressed, and the sun descended toward the Sierra Nevada.

Stopping for dinner in Hawthorne, I paused to watch a rocket launch just as the western sky turned to rosy gold. It was about then that I knew I was ill.

Still needing to make it to central CA before midnight, I just kept going. I got stopped by a Nevada cop somewhere near Carson City, and the cop read me the riot act about one of my headlights being out. I hadn't even noticed; I hadn't been driving at night for most of this particular trip. He stopped short of giving me a ticket or searching my vehicle for contraband that he wouldn't find, so I almost feel bad that he certainly came down with whatever it was within a couple of days. Almost.

This delayed me further, so it was full dark by the time I crested the Sierra Nevada. Full dark, that is, except for the stars. I could see them even through the windshield, against the loom of my one working headlight (yeah, officer, I can see the road just fine, see?)

So I stopped. Reeling from exhaustion - nine hours or more on the road, ill, and down off the adrenaline you feel when a cop pulls you over, even when you know damn well you don't have anything illegal and weren't speeding - I gasped for air, a scarce commodity at whatever elevation that was. Over 9,000 feet, almost certainly.

Alone on the road, chilled, shaking, I turned off the car's lights, leaned back on the hood, and beheld a diamond-studded infinity.

That's the thing about those high elevations - there's less air to muddle your view. Also, it's colder, so there's less water vapor, which is responsible for much of the atmospheric haze. It's like there's nothing between you and the stars but the frigid vacuum of space, and that's very, very close to being the truth.

I was in a hurry, and I probably shouldn't have stopped at all. But how could I not? For probably 20 minutes, until I just couldn't deal with the lack of temperature anymore, I looked at the sky, often cupping my hands against my temples so as to block out any silhouettes of trees in my peripheral vision.

After a while, seat warmer cranked up to full, I started coasting down the western slopes of the mountains. Ended up infecting half of California on that trip. So if you live in California and got laid up by some nasty cough thing for a couple of weeks in early 2014, well, that was my fault. Sorry.

Worth it.

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