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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Show Recent Comments

10 Most Recent Blog Comments:

Write_Mikey_Write!

about 8 hours 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

about 10 hours 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.

March 3, 2024 at 9:37am

#1065466

Revisited: "Sea What I Did There"

Almost two years ago, in May of 2022, I wrote some commentary on an Atlas Obscura article about the Great Lakes: "Sea What I Did There"

Being relatively recent, the article I referenced is still there.

Among other things, it's a good read for learning more about the Great Lakes. So, today, I'll just critique my own entry.

And now, today's burning (or really, drowning) question

I thought that intro was clever at the time. In retrospect, it was just silly. Partly because parts of the Great Lakes indeed have a history of burning, due to the chemicals that apes like to dump into them (I kind of chuckle every time they're described as "fresh water," though I know that's meant to describe lack of sea salt). And also partly because it was a bit insensitive.

(Responding to "Are the Great Lakes Really Inland Seas" headline): That's right up there with "Is a hot dog a sandwich?" in terms of categorization questions. (It's not, by the way. It's a taco.)

I stand by my categorization of hot dog as not-a-sandwich, but since then, I've been wondering about where gyros fit in. They're kind of like tacos, too. A taco is clearly not a gyro, nor is a hot dog (even if the hot dog has been spinning around on one of those convenience store heat roller machines). But is it fair to lump several different cultures' foods into a category based on that of only one of the cultures? Hot dogs are basically derived from Germany. Gyros are, famously, Greek, though I'd extend that to nearby Mediterranean regions. And tacos are, of course, Mexican. Other cultures have the similar idea of folding some sort of carbs around a filling, but not fully (that would be a "wrap" or a burrito. or perhaps one of those meat pies you can get from street vendors in England, though those are probably closer to calzones). So what we need is an overarching category for foods that are not technically sandwiches, because they're not fillings between two hunks of bread, but also not technically wraps or calzones, but something in between the two. A sandwich sandwich, if you like.

It is entirely possible that I spend too much time considering these important philosophical questions.

I think we're all aware of how angry the Lakes can become.

Frankly (see what I did there?), I have no memory of why I made such an idiotic blanket statement. Maybe I assumed everyone's heard Gordon Lightfoot's most famous song, the one about a ship sinking in the Great Lakes. Another fun categorization question: Ship or boat? The Edmund Fitzgerald was a freighter, which is most definitely a ship, because, well, it shipped stuff between ports, or, in its case, between a port and the bottom of Lake Superior.

In any case, I should always know better than stating "we're all aware" of anything. There are always people who don't know, be it from age or geographical distance or lack of exposure to cultural references.

Is Pluto a planet? Depends on definition of "planet." Under current internationally accepted definition, no.

As of this writing, I'm still obsessed with playing Starfield, a video game that features lots of star systems with planets and moons to explore. One of the star systems is our own solar system. Amusingly, they list Pluto as a planet and Charon as its moon, though here in reality, Charon and Pluto orbit a center of mass which doesn't lie within either body, which should make it a binary system. Further, in reality, this double-not-a-planet has other, smaller satellites.

This is, of course, not the only, or even the worst, technical issue in Starfield. But it's a game, so I just take it for what it is.

Virginia and three other US states are technically Commonwealths. We still refer to them as states.

That was not as apt an analogy as I apparently thought it was at the time. There's a difference between the arbitrary boundaries and technical name of a state or country, and the border between bodies of water and land. Except, of course, when there is no difference; Hawai'i, e.g.

The Dead Sea is famously salty as hell, sure, but so is the Great Salt Lake, which is about 7 times bigger than the Dead Sea.

Since then, I've learned that the GSL is shrinking pretty fast, so now, two years later, I'm not sure about that size comparison. The Dead Sea is also shrinking, of course. In any case, my real point was that they're both endorrheic bodies of water, and we call the bigger one a lake and the smaller one a sea.

In the spirit of what I said above about assuming awareness, "endorrheic" describes a watershed that's self-contained and doesn't allow for runoff into an ocean. They're really common in the American West. Hell, the vast majority of Nevada is endorrheic. Why they're usually found in areas we call deserts, I really do think should be obvious.

At one time, though, around the time the dinosaurs bit it, the Rockies were at the bottom of an inland sea, and the Appalachians were much, much higher (and originally extended into Scotland).

You know, when I make a claim like the bit about Scotland, I really should include a reference. At this time, however, I can't find where I learned that little tidbit, though I do remember that it involved the really stupendous age of the mountain range, combined with continental drift. Since I can't locate a reference, even on Wikipedia, you can probably safely ignore that bit of trivia, because it might well be the result of a misunderstanding on my part.

The relative youth of the Rockies and former height of the Appalachians, though, those are well-documented parts of geological history. here,

for one.

All continents can be considered big islands, and there is really only one world-spanning ocean surrounding all of them.

I object to my use of the word "really" in that sentence. Sure, it's another way to look at things, but there are real differences

between the areas we label as "oceans," even if the boundaries between those bodies are fuzzy.

I'll just end with this little tidbit, relevant to this entire discussion, which takes the categorization problem to its logical extreme: The Earth sandwich.

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