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:

tj-Merry Mischief Maker

about 16 hours ago
In response to "Extro Special"

I'll also use the spelling extrovert because if I use extravert my spell checker flags it. In college psychology our professor termed his own words, innervert and outervert because he thought they fit better. I, however, thought they sounded like definitions of belly buttons.

1 Like

Kåre เลียม Enga

about 22 hours ago
In response to "Extro Special"

I probably fit omnivert best. I'm temperamental.

Words change... definitions change... we change.

1 Like

tj-Merry Mischief Maker

a day ago
In response to "Sensational"

They say carrots can improve your eye sight, but it's been proven that drinking alcoholic beverages can make you see double.

After stating this, I find myself wondering why hasn't anyone tried fermenting and distilling carrots and marketing it as a vision tonic?

2 Like

🌖 HuntersMoon

a day ago
In response to "Sensational"

OK, look around you and observe (where possible) the reality of the newly revealed "facts." It's pretty clear that "Smartphones make you a hell of a lot of folks stupid."

2 Like

Elisa: Snowman Stik

a day ago
In response to "Sensational"

Thank you for reminding me to check on Retraction Watch.

https://retractionwatch.com/

3 Like

tj-Merry Mischief Maker

Dec 26, 2024 at 4:20pm
In response to "Gray Matters"

I wonder how many nine year old people have gray hair? I also wonder how much stress most nine year olds have?

3 Like

Wordsmitty ✍️

Dec 26, 2024 at 1:07pm
In response to "Gray Matters"

"who were between age nine and 39," - so the old folks need the rejuvenation formula first to turn back the years.

I wonder if anyone has studied why hair goes gray in certain places and not everywhere. Maybe different kinds of stress?

3 Like

🌖 HuntersMoon

Dec 26, 2024 at 10:00am
In response to "Gray Matters"

I've always believed "better gray than gone away..."

2 Like

tj-Merry Mischief Maker

Dec 25, 2024 at 3:02pm
In response to "A Mammoth Discovery"

This makes think of a book, "Evolution and Genomic Diversity" by Gene Poole.

1 Like

tj-Merry Mischief Maker

Dec 24, 2024 at 5:16pm
In response to "Hot Shoulder"

Maybe if we say, "Betelgeuse" three times in a row we can witness it.

1 Like

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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