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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QPdoll

about 11 hours ago
In response to "Landmarks"

I hope you're enjoying every minute in France! What an opportunity.

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Wordsmitty ✍️

about 14 hours ago
In response to "Landmarks"

Hey! Why are the signs at that Mac's in English if you're in France?

With all that walking, I hope you stayed hydrated.

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Annette in France

a day ago
In response to "Paris, I Guess"

Bienvenue en France.

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WakeUpAndLive️~Happiness

a day ago
In response to "Paris, I Guess"

last one today:https://parisjetaime.com/eng/event/olivier-giraud-how-to-become-a-parisian-in-one-ho ur-e047

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WakeUpAndLive️~Happiness

a day ago
In response to "Paris, I Guess"

Could not resist sending the link: https://www.atlasobscura.com/things-to-do/paris-france

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Elisa: Middle Aged Stik

a day ago
In response to "Paris, I Guess"

Your remarks on Paris looking like many cities reminds me of this video (which I promise is short). It's a sassy look at Paris, and I'm here for it.

https://youtu.be/ByUPFYxy5vw?si=lT57RpTDntAhFdBs

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WakeUpAndLive️~Happiness

a day ago
In response to "Paris, I Guess"

if you have two hours to spare tomorrow? A beer bike tour in Paris. Check it out, still available. https://parisbeerbike.com/

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WakeUpAndLive️~Happiness

a day ago
In response to "Paris, I Guess"

Je suis sure, vous avez aimez la ville!!!!

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Aiva Raine

a day ago
In response to "Get Baked"

FYI- basting them with olive oil before placing them directly on the oven rack- NO tinfoil- will allow the skin to crisp up a little as well and add a nice flavor to the skin. Recommend Idaho baking potatoes as well for best results.

Sum1's In Kalispell Montana

Sep 20, 2024 at 9:08am
In response to "Get Baked"

I have long advocated using only the oven for baked potatoes, and am guilty of punching holes in the tops of one. I don't use Aluminum Foil either. But my other half likes to nuke them first, or only nuke them. It's the speed thing for her. I prefer to take my time when I cook, and do things the way I do them, not her way. But, sometimes, ,ya gotta keep the peace in the house. She won't buy Russet potatoes, but those are the only ones to use for this. I may have to exert a little effort an buty Russets just for Baked Potatoes. She loves the English Chips (really enjoyed those in England when I was there, so maybe she'll like the link I sent her (from you!))

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October 29, 2023 at 11:11am

#1058242

Revisited: "Man"

Today's callback is a difficult one, and I'm not sure I have the mental energy for it this morning. The original entry was from about two years ago: "Man"

The article I referenced there was pretty new when I wrote that, and it's still up. If you don't want to read the earlier entry (understandable, though I promise the entry, if not the article, is short), here it is

again.

My intro then: "I can admit when I just don't understand something. This is one of those times."

And I can't say I've developed a full, nuanced understanding now, two years later. The comments and later blog entry on the topic from Elisa: Middle Aged Stik

did help me to comprehend some things, but it's just not a subject I'm attuned to.

For one thing, I thought, for the vast majority of my life, that being a man simply meant that I was in possession of a todger. Presumably, when I was born, the doctor or nurse checked me out, saw a tallywhacker, and said something like "It's a boy!" I learned of this practice at a fairly young age, and also saw it used on farm and domestic animals, so my mind went "If you see a wangdoodle, that means it's male. Therefore, 'man' is simply defined as an adult human with a pecker."

All that other superfluity, such as "boys have shorter hair" (I didn't always, and don't now) or "girls wear dresses" (they don't always, and historically, boys wore dresses too) just seemed like a decision we collectively made as a society, one which could be reversed. Now I see that this stuff is what people mean by "social construct." And it still feels optional to me.

After all, my cats don't have a gender identity. Their gender is simply their biological sex. The queens don't wear makeup or hair bows, and the toms don't deliberately grow bushy beards or smoke pipes. Even being surgically altered doesn't change what pronouns you use for the animal.

More recently, I realized that this kind of thinking gets you labeled a transphobe, and no amount of denial of that on your part can ever change that label. It's far easier to change your gender than to get people to stop calling you a transphobe. It's kind of like if you say "I'm not an alcoholic," everyone will conclude that you are, in fact, an alcoholic. Especially if you're drunk and slur the words.

Anyway, that kind of thinking is why I always considered those outward markings of "manliness" to be superficial and optional.

Then, later in life, I realized that I was wrong. I can admit that, too; I'm man enough.

So I won't bother offering any support for my assertion that I in no way fear (or hate) trans people. Think what you will of me; I don't care.

And that brings me to the actual crux of the matter, at least for me, and for now: people seem to care a great deal about what others think of them, even, in some cases, when they claim not to. In another grand cosmic coincidence (see yesterday's entry for the last time this happened), one of my favorite webcomics touched on this very subject today. {xlink:https://www.smbc-comics.com/comic/escape-2}This link should be as permanent as anything is on the internet.

In the comic, a man (you can tell from his clothes, hairstyle, and facial features) goes to visit the archetypal Hermit on the Mountain (you can tell from the mountain, his robe/dress, and his beard). His question begins: "Wise Master, how do I escape? I'm a social ape. I'm obsessed with status. All my actions, even private ones, can be perfectly explained if you assume I'm seeking the esteem of other apes."

I'll tell you what, though: once I found a way to live that does not require the approval, or escaping the disapproval, of other social apes, I found a freedom that most people can't even conceive of. Once I stopped trying to peacock or scrabble for this mysterious "status" crap, life became a whole lot easier.

In short, I'm becoming the Hermit on the Mountain, and I'm okay with that.

One other unrelated thing about my earlier entry: Yes, I used to do Merit Badge Mini-Contests. I stopped because readership dwindled (or at least the number of commenters; being not obsessed with status, I don't often check my readership statistics) and because I quit doing entries at midnight in favor of being free to drink in the evenings. The schedule is much more flexible, these days, so it's harder to do deadlines.

But I'm also inclined to be generous, so who knows? I may start that up again in some form.

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