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:

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.

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').

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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.

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

July 29, 2023 at 9:46am

#1053258

Getting the Drift

Today's article, from Nautilus, is a few years old, but it's not like plate tectonics has stopped since then.

What Happens to Google Maps When Tectonic Plates Move?
Earth’s tremors can tweak your GPS coordinates.

It's not just Google Maps, of course; this applies to any GPS receiver.

This is a thing I've often idly wondered about, but never enough to go and seek out the answer.

As a writer on physics, I’m always seeking new metaphors for understanding Einstein’s general theory of relativity, and while working on my last book, Spooky Action at a Distance, I thought I’d compare the warping of space and time to the motion of Earth’s tectonic plates.

Oh look, a book plug. What a surprise.

Regardless, this is the kind of thing I mean when I say there's no such thing as useless knowledge: you can more easily find similarities between disparate things. The irony (or whatever) here is that GPS wouldn't work worth a damn without relativistic corrections for the satellites' speed and distance.

I discovered a sizable infrastructure of geographers, geologists, and geodesists dedicated to ensuring that maps are accurate. But they are always a step behind the restless landscape. Geologic activity can create significant errors in the maps on your screens.

As if paper maps are more accurate.

Several factors produce these errors. Consumer GPS units have a position uncertainty of several meters or more (represented by a circle in Google Maps). Less well known is that maps and satellite images are typically misaligned by a comparable amount.

All things considered, an uncertainty of a few meters, compared to the size of the planet, may as well be sorcery. I'm pretty sure military hardware has less inherent uncertainty, but it would also be affected by tectonic shifts.

For the most part, misalignments don’t represent real geologic changes, but occur because it’s tricky to plop an aerial or orbital image onto the latitude and longitude grid.

And this makes sense when you think about it. Not only do you have the Earth's curvature (it is, in fact, roughly spherical) to consider, but also elevation—another parameter that can change, albeit usually slowly.

NGS and other agencies recheck survey marks only very infrequently, so what a stroke of luck that a community of hobbyists—geocachers—does so for fun.

I had no idea geocaching was going on. I was a guest at a large gathering of theirs many years ago, and they seemed... obsessed. But I suppose we all have our obsessions.

Confusingly, the U.S. uses two separate datums. Most maps are based on NAD 83, developed by NGS. Google Maps and GPS rely instead on WGS 84, maintained by a parallel military agency, which has a considerably larger budget.

It should be kept in mind that GPS, like the internet itself, started out as a military project, and only later became accessible to civilians.

When NGS introduced NAD 83, replacing an older datum that dated to 1927, it was the geographic version of the shift from the Julian to the Gregorian calendar. If you’d been paying attention, you would have woken up on Dec. 6, 1988, to find that your house wasn’t at the same latitude and longitude anymore. The shift, as large as 100 meters, reflected a more accurate model of Earth’s shape. Vestiges of the old datum linger. You still see maps based on NAD 27.

This change was relevant to my work, as well. As I recall, the datum (the zero reference) for the two systems was different by nearly a foot; I wasn't quite as concerned with the horizontal references, because the parcels we worked with were relatively small, a couple hundred acres at most, and the important thing was relative position, not absolute. Yes, we used inches and acres.

“Most surveyors and mapmakers would be happy to live in a world where the plates don’t move,” Smith explained.

Considering that plate tectonics is probably linked to the processes that generate Earth's magnetic field, which helps protect us from effects such as solar radiation and the atmosphere floating away into space, no, no, they wouldn't be happy to live in that world, not for very long.

Anyway, the article does get a bit technical, but I find the discussion of how this works entirely fascinating... if irrelevant to my quest to find the nearest brewpub from wherever I happen to be at the moment.

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