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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Brandiwyn🎶

a day ago
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.

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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.

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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.

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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.

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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').

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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.

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Turkey DrumStik

Nov 6, 2024 at 10:22am
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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

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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.

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January 3, 2021 at 12:05am

#1001313

Portal

PROMPT January 3rd

If you had a portal that would instantly transport you to one specific place on Earth whenever you wanted, where would you want to go? You can use the portal as often as you want, but it will only transport you to the one location you choose and then back again to where you came from.

Oh, this one's easy: Belize.

Look, it's really very simple: I despise cold weather, but I like Virginia. Unfortunately, Virginia is cold for 3-4 months out of the year, and kinda iffy for another 1-2 months.

On the other hand, Belize is in Central America and in the potential path of hurricanes... but generally, only in the months where it's not cold in Virginia.

I could also say "Hawai'i," but having been to Hawai'i, it's expensive as hell unless you can get kama'aina, but only residents get that and with the portal I still wouldn't be a resident.

But really, any place that is warm when it's cold here will do. Oz, maybe, though I'm not a huge fan of 40C weather either, or having 90% of the flora and fauna actively trying to kill me. Or New Zealand, which I've never been to either but has the benefit of it being summer there when it's winter here, and also doesn't have drop bears.

Obviously, just about anywhere in the Southern Hemisphere or tropics would meet my requirements, but Belize was the first thing to come into my head and now I'm not sure I can even justify it. I mean, it's got its problems, but I did enjoy my visit there lo these many years ago and would like to go back, portal or no.

In the hypothetical situation we're discussing, though, I'd want the portal to take me someplace I'm familiar with already, and when it comes to the tropics, for me that's limited to Hawai'i, Belize, Antigua, Montserrat, and St. Thomas. Of all of those, only Belize is not a tiny island.

And I've never been south of the Equator, a situation I would like to remedy before I kick it.

Now. I'm just drunk enough to point out a potential issue with such a portal. Okay, more than one issue.

Let's start with air pressure.

The average air pressure at the surface of the Earth is, by definition, one atmosphere. There are myriad and confusing ways to report said pressure, in both imperial and metric units, but let's go with the metric standard of 101,325 Pa.

That is, as I mentioned, an average. An arithmetic mean. It varies according to weather conditions and elevation and, notably, when it's warm the pressure tends to be significantly lower than when it's cold. As this portal specifically exists to take me from someplace cold (defined by me as anything less than about 12C) to someplace warm (defined by me as anything above around 20C), and I currently live at 500' above mean sea level, I don't know how such a hypothetical portal could handle pressure differentials.

Air pressure gradients in adjacent areas of Earth's surface can generate hurricane-force winds. Hell... that's usually what wind is: an attempt to balance out pressure differences. Change that to non-adjacent areas connected by a magical or scientific portal, and the problem gets even worse.

That's always bugged me in science fiction, by the way. Bad enough when it's a portal from one place on Earth to another, but when it's between worlds, like the Stargate or something, how do you justify being able to send people through the portal when there's almost certainly an air pressure differential on both sides? How does it send people but not air molecules?

I'm sure a clever writer can hand-wave the science, but most fiction never even tries.

And then there's angular momentum.

You may feel like you're standing still, but you are not. If you're on the equator, you're moving at 460 meters per second with the Earth's rotation, relative to the planet's axis. You're also being flung around the sun and the solar system itself is moving through the galaxy, but let's just think about rotational velocity right now.

I live at about 38 degrees north latitude, which (math) is moving at about 355 meters per second relative to the axis.

Those numbers are probably outside most peoples' experience, so for the benefit of other Americans: 1000 mph and 800 mph.

That might sound "close enough" but imagine jumping off a train moving at 1000-800=200 mph and that's the kind of velocity change that a portal from my house to somewhere on the Equator would have to deal with. It would be even worse, obviously, the further from the Equator you go.

And, finally, such a portal would need to take into account that the planet is round (yes, really, it is, and if you believe otherwise, boy are you in the wrong place right now), so how do you change a person's angle as you change their latitude? If that makes sense. I might still be drunk from Zoomies, so I'm probably not explaining everything as clearly as I could.

None of which, of course, even begins to answer the question of how such a portal could be built in the first place, but hey, let's not get too technical here.

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