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.

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.

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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.

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').

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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
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

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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.

2 Like

June 18, 2024 at 11:44am

#1072853

Luna Ticks

It's refreshing to find a science article that seems to get the facts straight (though necessarily simplified). Surprisingly, it's from CNN.

Why scientists say we need to send clocks to the moon — soon

Before I get into it, though: no, time is still not an illusion. It's just tricky. This is analogous (but not exactly so) to answering a question like "How long is this iron rod?" when the answer depends on temperature and other factors, and probably can't be answered past a certain degree of precision.

On the lunar surface, a single Earth day would be roughly 56 microseconds shorter than on our home planet — a tiny number that can lead to significant inconsistencies over time.

Sure, that doesn't sound like a lot. Over one (earth) year, it only adds up to about 2 hundredths of a second. But they're talking about long-term effects, and highly precise instruments.

Never mind that the length of a year (and a lunar month) change slowly over time, too.

Scientists aren’t just looking to create a new “time zone” on the moon, as some headlines have suggested, said Cheryl Gramling, the lunar position, navigation, and timing and standards lead at NASA’s Goddard Space Flight Center in Maryland. Rather, the space agency and its partners are looking to create an entirely new “time scale,” or system of measurement that accounts for that fact that seconds tick by faster on the moon, Gramling noted.

CNN calling out other sources' headlines for being misleading probably falls under "irony."

Astronauts on the moon, for example, are going to leave their habitats to explore the surface and carry out science investigations, she said. They’re also going to be communicating with one another or driving their moon buggies while on the lunar surface.

It should be obvious that they can't exactly use Google Maps to navigate. I have no idea if it would even be technically possible to set up an equivalent GPS array for the moon. Even if it is (our natural satellite's gravity field has been described as "lumpy"), I imagine the cost / benefit analysis might not work out.

GPS comes with its own time issues, too, having to do with relativistic effects, which the article notes later. And it requires a great deal of precision.

General relativity is complicated...

Ya THINK?

...but in broad terms, it’s a framework that explains how gravity affects space and time.

Eh, fair enough.

The article goes on to explain quite a bit more, fortunately mostly in plain language.

Then it gets into the bit I really wanted to address:

Accurate clockwork is one matter. But how future astronauts living and working on the lunar surface will experience time is a different question entirely.

On Earth, our sense of one day is governed by the fact that the planet completes one rotation every 24 hours, giving most locations a consistent cycle of daylight and darkened nights. On the moon, however, the equator receives roughly 14 days of sunlight followed by 14 days of darkness.

If we don't doom ourselves or give up entirely, this won't be the last time such adjustments have to be made. While there are robots on Mars now, there's always been talk about sending humans there. As I'm sure we all know, a Martian "day," (which they call a "sol" to distinguish it from our familiar 24-hour cycle) is longer than 24 hours, but not by much. As a reader (and sometimes writer) of science fiction, I've often wondered whether we'd keep our circadian rhythms there, or try to adjust them to the Martian day/night cycle.

Considering that any Mars colonies will be entirely indoors, and probably underground, I could see it going either way. In a way, Mars is harder than the moon to figure out this sort of thing, because it's much closer to the cycle we're used to.

Not to mention we'd have to answer these questions for any other planet we choose to visit. (Please note that this isn't an entry about whether we could or should. Mostly, I very much needed to make the pun in the title.)

I mean, sure, the article mentions this sort of thing, too, but without details. Guess I'll have to do what other writers do and just make stuff up.

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