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:

Robert Waltz

7 minutes ago
In response to "Dis Dress Signal"

tj wanderlust-words-in-motion

- Ah, but that's what they want you to think.

tj wanderlust-words-in-motion

44 minutes ago
In response to "Dis Dress Signal"

Are you sure it's the lighting? I've heard the "dress" was a government scheme to divide the people and distract us from what's really important. It's all conspiracies, you know!

1 Like

tj wanderlust-words-in-motion

about 21 hours ago
In response to "Cerebral Musings"

I like to think of it as right brained and wrong brained, and many these days are wrong brained.

1 Like

tj wanderlust-words-in-motion

a day ago
In response to "Solitary"

Having five brothers and one sister, I often seen less family stress in friends that were an only child.

1 Like

tj wanderlust-words-in-motion

Feb 24, 2025 at 6:17pm
In response to "Mystery"

#tiktokgirl: octothorpe, number, and as you indicated poundtiktokgirls. That's the problem with "the style these days", they distort all the coolness of the past styles.

1 Like

tj wanderlust-words-in-motion

Feb 23, 2025 at 4:32pm
In response to "Revisited: "Forgive Me, For I Have Zinned""

My oldest son, while in college, would buy it and mix it with Kool-Aid when he had friends over.

2 Like

Allan Charles 🐾

Feb 23, 2025 at 12:03pm
In response to "Revisited: "Forgive Me, For I Have Zinned""

I find White Zinfandel to be disgusting. Defining it as the wine equivalent to American cheese is a great comparison.

At my job, we only sold a cheap house wine for white zin, and when people ordered it I had to try to restrain my facial muscles from giving stank face.

2 Like

tj wanderlust-words-in-motion

Feb 22, 2025 at 7:36pm
In response to "Apparatus"

And then we have "cocktail" which can, if one takes the time to look it up, waste a good hour or more of time and provide various meanings and origins, including shoving ginger up a horses arse!

tj wanderlust-words-in-motion

Feb 21, 2025 at 3:02pm
In response to "Thwip"

Put it on ATVs so when they get stuck they can shoot out a web and winch themselves out.

2 Like

tj wanderlust-words-in-motion

Feb 20, 2025 at 6:10pm
In response to "One Fine Thing"

Well, it was a fine idea.

February 27, 2025 at 9:32am

#1084443

Dis Dress Signal

I'm jumping the queue with this article from Slate. I feel like it touches on some things that might help us better frame national and world political discourse, and I didn't want to wait for blind chance to select it at random some indeterminate amount of time from now and probably in a different blog.

Oddly enough, it all starts with a woman's dress from ten years ago.

It’s Been 10 Years Since “The Dress”
The viral image holds a lesson in why people disagree—and how we can learn to better understand each other.

You know the one, unless you were living under a rock, have a memory even worse than my own, or are way younger than my usual readership demographic: a single photo of a dress, illustrated at the link, which some people saw as white and gold and others saw as blue and black.

It was not a tranquil time. People argued with their friends about the very basics of reality. Spouses vehemently disagreed. Each and every person was on one side or the other side. It could be hard to imagine how anyone in their right mind could hold an opinion different from your own.

That sounds like hyperbole now, but I'm pretty sure relationships ended over this thing.

To recap: A cellphone picture of a wedding guest’s dress, uploaded to the internet, sharply divided people into those who saw it as white and gold and those who saw it in black and blue—even if they were viewing it together, on the very same computer or phone screen.

And this wasn't your usual optical illusion, either. Normally, you can trick your eyes and/or brain into seeing through the illusion. The dancer moves clockwise, until you decide she should be moving counterclockwise. The plates are all upside-down until you stare long enough and they flip right-side-up. This square is darker than this other square, until you directly compare the colors used with each other, ignoring the other inputs. That sort of thing. This one involves color, too, but, as far as I know, once you saw the dress one way, no amount of convincing or trickery could make your brain flip to the other color scheme. I know that was true for me: how I saw it was how I saw it, and no trickery or convincing made me see it the other way.

No, I'm not telling you what colors I saw. It's irrelevant. What is relevant is that I believed other people when they said they saw different colors (well, except for the trolls who insisted on, say, mauve and pink just to mess with the rest of us) and, being the curious type, I always wanted an explanation.

I don't recall any from back then; this article, however, almost satisfies my curiosity on that front.

The notorious dress, under natural lighting conditions, is unambiguously black and blue, for (almost) everyone who saw it in person, or in other photographs. It was just the one image, snapped by a mother of a bride and uploaded to Tumblr by one of her daughter’s friends, that caused so much disagreement. How can it be that there is such strong consensus about the colors of the actual dress, but such striking disagreements about its colors in this particular image?

And no, it's not because everyone who sees it a different way is brain-damaged... which is a preview of the point made soon after in the article:

While the colors of a piece of clothing might be a trivial thing to disagree about, we can all learn a thing or two from the dress about how to navigate high-stakes disagreements.

And no, it's not just about how to argue or debate effectively.

Why did people disagree about the dress? It’s all in the lighting.

There's more of that explanation at the article; I don't see the need for, or wisdom in, reproducing all of it here. There are also some other examples of color perception differences.

One thing that you might notice about all of these examples: Your brain never tells you “We really can’t tell what the color is because we don’t have all necessary information available.” There’s no flag that goes up saying “Just FYI, your assumptions did much of the heavy lifting here.” The brain prioritizes decisive perception (giving you the ability to take decisive action) over being paralyzed by uncertainty and doubt.

A lesser author might have made up some evolutionary guess for why that is. Like "This is because our ancestors needed to act quickly when they thought they saw a tiger, instead of standing there wondering if it's a predator." I just made that up. It might be true. It likely is not. Sure, there's an evolutionary reason for it; what's guesswork is what that reason is and how far back it goes along the evolutionary tree. In any case, I appreciate the lack of made-up evo-psych "explanations."

This might be all fun and games when applied to internet memes, but similar convictions—sincerely held and self-evidently true to the individual—in domains like religion or politics will also be determined in large part by differential priors.

And that, simply put, is the metaphor that makes this article both useful and timely.

The phrase "differential priors" isn't strictly defined here, but it can be inferred by the examples used: as I understand it, it refers to unconscious assumptions based on one's unique experience. Kind of like how someone growing up poor will have a much different relationship to money than someone who grew up rich.

Rather than thinking people must just be plain wrong, or stupid, a better way might be to take the disagreement seriously and try to actively elicit and discern the differential priors that led to the diverging conclusions.

I get the impression that this is easier said than done. But it may be necessary in a world where we're calling anyone who disagrees with us things like "stupid," "woke," "Nazi," or "evil."

Between sincere and well-meaning parties, the very fact that the disagreement exists in the first place must be due to a difference in the priors that informed the formation of the conviction.

The caveat there is "sincere and well-meaning parties." That must be discerned, too. And that's hard to do when you don't see your political opponents as sincere or well-meaning. I think internet trolling contributes to muddying the waters here, but trolling existed long before the internet.

All that remains is to determine what those [differential priors] might be.

And like I just said, that's work.

I know I'm lazy. I've built my entire life around being lazy. But this is too important, the stakes are too high, to be intellectually lazy. So, next time someone says something I vehemently disagree with, I'm going to try to get a better handle on their point of view before dismissing them as an idiot.

They might still be an idiot, but as idiots have the right to vote in my country, it may be wise to see things from an idiot's perspective. And it might turn out that they're just coming at the topic from a different angle, or in different lighting.

For what it's worth, I never assumed the people who saw the dress differently were stupid and trying to destroy the country and/or world. And I should probably apply that to political disagreements as well—at least until I'm sure they're trying to destroy the country and/or world.

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