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Complex Numbers

by Robert Waltz

Rated: 18+ · Book · Personal · #1196512

Not for the faint of art.

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/

Signature for Honorable Mentions in 2018 Quill Awards

A signature for exclusive use of winners at the 2019 Quill Awards

Signature for those who have won a Quill Award at the 2020 Quill Awards

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10 Most Recent Blog Comments:

Kåre เลียม Enga

a day ago
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

a day ago
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.

Aiva Raine

Nov 5, 2024 at 3:03pm
In response to "It's Hard Not To Scare"

Shaking head. Aside from the inevitable Simpson's spider goat song playing in my head which I will now share with you below, whenever I come across a story that has all these scientists doing ridiculous experiments for no logical reason, I must go- ah- scientists- CANCER? You do remember that problem right? Can we maybe cure that before we make cockroaches magnetic and elephants see pink elephants?

spider pig clip- and if you all now are singing the spider goat song- you're welcome! grin
https://www.youtube.com/watch?v=BARjPuUN36Y

1 Like

Aiva Raine

Nov 5, 2024 at 2:55pm
In response to "Anomalies"

One of your more humorous entries- I especially enjoyed- "Lofoten Vortex can be the name of my Bjork cover band."

Thanks for the laughs. Much appreciated.

2 Like

JayNaNoOhNo

Nov 5, 2024 at 1:16pm
In response to "Anomalies"

*Toronto, Canada submits a sternly-worded petition and disputes the center of the universe claim.*

2 Like

May 9, 2019 at 1:18am

#958596

Far from Home

What is the farthest you've ever been from the place you call "home?"

Ever wonder what's meant by "halfway around the world?"

Like, there's a spot in the Indian Ocean (I looked it up on Google Earth some time ago) which is exactly 180 degrees opposite my house. If I could walk in any direction at a steady pace, it wouldn't matter which direction I went in; they all result in the same time or distance to that spot. It is the south pole to my north pole, or vice-versa.

Should we consider that spot "halfway around the world" for me? Because if I were able to walk to it, and did so, but then kept walking, eventually I'd end up back home - all the way around the world. And yet, every other spot on the Earth's surface is closer, by definition, so every other spot would be less than halfway around the world. After I passed my personal South Pole, I'd be 3/8 of the way around, then 1/4, then 1/8, and all of the infinite points in between. I'd never be 3/4 of the way around the world, even though I'll have walked that far, and more.

There's no paradox here, of course; it's purely a matter of the inconsistency and imprecision of written language. Mathematics would describe the journey with far less ambiguity. To really answer the prompt question, we'd have to define exactly what was meant by "farthest," and probably "home" also.

Even defining "farthest" simply as "greatest distance along a Great Circle route" and "home" as "the place you live most of the time," perception can be thrown off by something as simple as a map. It is impossible to accurately represent every feature of a sphere on a flat map. Something is always distorted. With the most common (Mercator) projection, it's mostly distance and size; for example, a Mercator map makes it look like Greenland is bigger than South America, when it's simply not.

Distance is affected, too. Look at a Mercator map, and it appears that the closest point in the continental US to the continent of Africa is, like, somewhere near Miami, Florida. But no - the closest point (and it's not even much of a competition) is in Maine.

It's a nice old lighthouse called Quoddy Head, not far from the little town of Lubec, which features a few comfy tourist inns, a border crossing into Canada, and the absolute worst place to eat breakfast in the entire Western Hemisphere.

I know that because I've been there. Some years ago, and I think I've mentioned this in here before, I decided to drive all the way across the country. Since it would be my first time doing so, I decided to do something special: travel from the easternmost point of the contiguous US to the westernmost point. And the easternmost point happens to be... Quoddy Head Lighthouse.

From there, the most direct route would cut across Maine, then into Canada, and then pretty much stay in Canada. I have nothing against Canada. I love Canada. But my purpose was to see the US, so I took a route that meandered around south of the Great Lakes, back up through Wisconsin, then across North Dakota and Montana - avoiding interstates the whole way. Took me about two weeks; I wasn't in any hurry. There is no such thing as destination; there are only stops along the journey.

On the off-chance anyone is interested in my chronicles of that journey, well, here's the link to my offsite travel blog. Some of the links therein are broken; time will do that on the internet. First, this

and then starting with this post

and continuing as far as you can stand.

The westernmost point, incidentally, is about a three mile hike west of a map point called Ozette, in Washington State. Ozette is little more than a ranger station; the nearest town is some miles away, and features a bed and breakfast. I stayed there the night before I hiked through the woods to the Pacific. They had a calendar on the wall. The calendar picture at the time featured... Quoddy Head Lighthouse.

Life is weird.

So you ask me what's the farthest I've ever been from home. I could pace it off on a globe; it's Bethlehem. No, not the one in Pennsylvania; the original one. But that's boring. I choose to interpret "farthest" as the most removed psychologically, and for that, I'd have to say: the empty stretch of Washington coastline, a three-mile hike west of Ozette. Hiking six miles, there and back, along a poorly-maintained trail through bear-controlled rainforest, all to fulfill a whim I'd come up with - well, it's not something I'd ever done before, and it's not something I'll ever do again.

But I'm sure glad I did it.

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