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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Write_Mikey_Write!

about 9 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.

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Adherennium Plotting something

about 11 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.

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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.

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.

3 Like

Turkey DrumStik

Nov 6, 2024 at 10:22am
In response to "Lo!"

Not gonna lie I need the holiday season chatter.

February 10, 2024 at 9:16am

#1063912

The Abyss Stares Also

Today's article, from Aeon, is about holes.

I'm going to pause here for a moment so you can get the obvious juvenile jokes out of your system.

Ready?

Okay.

In pursuit of the hole
Dig into the voids, pin-pricks and cut-outs of art and history, and those absences speak volumes about what’s been missed

Now, the article itself is about art history, written by an art history teacher. This is not a subject I touch on much, but it's not like I've never presented subjects here that I knew very little about.

Besides, while the article is interesting, that's not the primary reason I'm sharing it. So there's the link if you want to read it. I'm only going to quote a couple of passages from the not-very-short text, and besides, as befits an art history piece, there are illustrations there.

Holes are full of potential.

Okay, but this is your absolute last chance to get your mind out of the gutter. Seriously, I'm not talking about those holes.

Archaeologists use soil analysis to identify postholes, the marks of ancient settlements. They excavate sewers at the Colosseum to find things the Romans thought not worth writing about. Holes leave space for projecting both forwards and backwards in time.

That last bit is a little too philosophical, even for me. I'd say rather, in the context of the article, holes can inspire imagination and creativity; I don't know about this timey-wimey stuff.

Museum and library collections are full of holes. Some are the product of hungry bookworms and moths. Others are deliberate, made by humans. These holes are, arguably, the most crucial bits, even though they are missing. This is the hole’s paradox. A hole points, through absence, to importance. The object wasn’t just used; it was used up. Philosophers struggle mightily over this question: is the hole something or nothing at all?

And that's all I'm going to quote from the article, because it provides me a jumping-off point for my own thoughts, which go way beyond just one academic subject.

The concept of a "hole" (look, seriously, stop snickering now) has intrigued me for a while. Leaving aside those astronomical objects called black holes, which are, in some sense, not holes at all but rather locations where space and time switch places, a hole can only be defined by what it's a hole in.

That is to say, they occupy a place on the reality spectrum that's neither completely real nor completely unreal.

Still doesn't help? Okay, well, try this:

Some nouns are concrete, and others are abstract. Concrete nouns describe everyday objects: a bed, a table, a tree, the sun, raindrops, my cat. They are what I consider real. I know some philosophers love to deny the reality of these things, and their arguments are worth thinking about, but these are, for all practical purposes, real things. Abstract nouns point to concepts that we humans have come up with, such as justice, mercy, liberty, or Santa Claus (a name is a proper noun).

There are also, and there's probably a word for this already but I don't know it so I'll call them "Platonic nouns." These are concepts that have some bearing on reality, such as a circle, a number, food, writing, or a game. They're not real in the same sense that the Sun is real; they're on the abstraction spectrum, being more conceptual than concrete. And yet, they're not entirely abstract, either. You can point to examples of circles, for instance, but you can't really point at a specific, everyday object, and say "that is liberty." You can count multiples of any object, like "I have two cats" or "I have two sleeping bags." Cats and sleeping bags are (arguably) very different objects, but the same number can apply to both. Likewise, the concept of "sphere" can apply (approximately) to any number of objects: a basketball, the Sun, a crystal ball.

The point of all this philosophizing is that, while a hole is a concept as well, it has no physical presence. No one points to some place in the air around them and says "This is a hole." No, a hole is, and can only be, defined by what's around it. A bagel, say. The handle of a teacup. The ground. Your memory.

This reminds me of the concept of zero, for which I have another article in my queue. Would zero have meaning if we didn't conceptualize the number line? Hell, the way we represent it is telling: 0. An oval or circle, which has a hole. That doesn't mean much, though, as many of our numeral representations have holes: 4, 9, 8, 6, and even that polar opposite of zero, ∞.

Holes are real, as anyone who's twisted their ankle by stepping into one can attest. And yet, they're also... not real, because they're voids. I mean, sure, here on Earth, they're almost always filled with air or water, but they're not defined by air or water.

Topologists also draw a distinction between an indentation (which is what a hole in the ground or the place where you put your morning coffee before you drink it actually is) and a true hole (like the handle of the coffee cup, or the inside of a donut). But for my purposes today, I'm lumping them together. Insofar as you can lump several examples of nothingness together, anyway.

Imagine a hole in the ground, then, viewed from the side. From this perspective, the Platonic ideal of "hole" is something like a cylinder with no top: straight sides, flat bottom. Now imagine the sides getting dug out, like in a strip mining operation. It becomes like a trapezoid, as viewed from the side. Now, at what point does it cease being a hole and start being a crater? And at what point in this mining operation does it become an indentation? Taken to the logical extreme, you can further excavate until it's absolutely flat, its edges intersecting the curvature of the planet. Keep excavating, and you no longer have anything even resembling a hole, but what used to be the bottom becomes the peak of a hill. Or mountain.

I keep running into classification problems, don't I?

Anyway, those are my thoughts, completely tangential (another abstract concept) to the article's topic, and yet something I wish they'd touched on more.

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