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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Writer_Mike

about 8 hours ago
In response to "Paris, I Guess"

Elisa: Middle Aged Stik

- I thought it very interesting, to say nothing of extraordinarily coincidental, that your video mentions a lock-adorned bridge in Paris. I just finished reading Spymaster by Brad Thor, in which the Russian mastermind takes strolls from Kaliningrad's mainland to Kant Island. The bridge he uses is also festooned with locks placed by couples declaring their undying love for one another.

1 Like

QPdoll

a day ago
In response to "Landmarks"

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

2 Like

Wordsmitty ✍️

a day 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.

2 Like

Annette in France

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

Bienvenue en France.

1 Like

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

1 Like

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

1 Like

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

2 Like

WakeUpAndLive️~Happiness

Sep 21, 2024 at 6:46am
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/

3 Like

WakeUpAndLive️~Happiness

Sep 21, 2024 at 6:16am
In response to "Paris, I Guess"

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

2 Like

Aiva Raine

Sep 20, 2024 at 10:13pm
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.

March 17, 2023 at 2:19pm

#1046645

Wrong

Heading back to "JAFBG"

[XGC] for a random prompt...

Tell us about a time when you were proven wrong.

Me? Wrong? Unpossible.

Okay, no, that's a joke. I'm wrong a lot. Sometimes even right here in this blog. I'm an adult, though; if you can show me I'm wrong, I'll swallow some corvid and admit it.

There's lots of things I could be wrong on, but until shown otherwise, I'll stick to my opinions. Things like the absence of free will, the arbitrariness of the Gregorian calendar, the reality of time and space, the mistaken definition of a Blue Moon (which is related to that Gregorian calendar thing), the absence of space aliens on or around Earth, the dangers of evolutionary psychology, and much more—all of those things are topics that I've discussed in here, that I have a high level of confidence in. But, say, if a flying saucer landed in my street and disgorged a three-tentacled, seven-eyed mauve creature with a tricorder and universal translator, well, I'd change my mind about aliens real quick. Assuming that I haven't decided to experiment with magic mushrooms, that is.

Worse, though, are the things I'm absolutely certain of. Those are dangerous, because, like anyone else, I get stubborn, and invested in my ignorance.

Let me just come up with an example off the skin of my ass:

Long ago, when I was married, my wife bought a shower curtain. This is not such an unusual thing, I think; I'd been using the same shower curtain for way too long, and I can own that. Fine. This particular one didn't even have any designs; it was just the clear plastic kind that hangs down inside the tub to keep most of the shower water inside.

She also bought a bottle of, I don't recall the brand name, Tilex or something. Used for spraying down the shower tiles after use, supposed to keep mold and mineral scale from showing up. Again, something I'd been neglecting. What can I say? I lived (basically) alone before she moved in. Had housemates, but also had another bathroom for them.

So I proceeded to, every time I took a shower (which, contrary to what you might have inferred from all of the above, was every day), spray down not only the tiles but also the shower curtain, on the (what I thought at the time was a) perfectly sound principle that it would keep us from spending money on another shower curtain for a while.

She found out that I was doing this and we got into an argument. "We can just buy a new shower curtain" was basically her position on it. I disagreed; a shower curtain, like any piece of furniture, should last a lifetime. That was my position.

It was years—years—later that I had an epiphany:

A bottle of Tilex (or whatever) was like five bucks.

A clear plastic shower curtain was like ten bucks, and takes about five minutes to install.

The epiphany was that all the Tilex I had been blasting on the curtain would end up costing more than it would to, as she suggested, simply replace the shower curtain every year or so. So, from a purely financial perspective, I was completely, utterly, incontrovertibly, wrong. Now, I could still argue that, environmentally, I was in the right, what with all the plastic waste and all, but to be perfectly honest, that didn't enter into my calculations.

Thing is, by then, we'd already separated, and I didn't feel the need to confess my wrongness to her at that point. These days, I use two shower curtains: the inside clear plastic one which gets replaced every year or so (environment notwithstanding), and an opaque cloth outer one. I'm not even sure why; I don't share the bathroom with anyone, so it's unlikely that somebody would come in while I'm showering. I just like the design, which has cats.

That epiphany, though, made me wonder: what else am I sure of that just isn't the case? As I'm no longer married or otherwise partnered up, there's no one to set me straight on these things. Not even my cats. The only time they complain is when I'm five minutes late with their dinner.

So, yes, I was wrong. And probably still am, but if so, I wouldn't know it.

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