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 13 hours ago
In response to "Landmarks"

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

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Wordsmitty ✍️

about 16 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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September 7, 2023 at 8:33am

#1055294

Less Than Super

Today's link, from Atlas Obscura, is now outdated, having languished in my queue for several weeks. It's just as well, because if it'd come up anytime before August 30, it would have led to a rant, and I ranted about that particular subject enough for this year.

Just How Super Is The Coming Supermoon?
July’s full moon will be thousands of miles closer to us than usual. Learn how to measure it with your hand.

I'm linking it despite its lack of timeliness, because it's got general information of value to skywatchers. And some dubious information that I, of course, am going to question.

The Moon is growing now, more and more of it visible every evening. Though it is technically full for just one night, it will look mostly full for a night or so before and after that.

Again, not current information. And yet, I'm going to pick on it. It's technically "full" for just one moment, and half the time, that moment occurs during daylight hours wherever you happen to be. Traditionally, in most cultures that observe this sort of thing, a full moon spans three nights.

The term “supermoon” is a relatively recent introduction to the astrological lexicon, but these events are nothing new; they occur several times a year.

Okay, fine, I'm going to rant anyway. "Astrological?"

I'm not even going to legitimize the sentence that follows that quoted one by pasting it here. Suffice it to say it made veins pop out in my forehead with its utter wrongness.

A supermoon takes place when the Moon is nearest to Earth in its elliptical orbit around our planet.

And that happens once per orbit: at perigee. It doesn't always coincide with a full moon. When it does, that's what they've taken to calling a supermoon. I don't object to the terminology here; what I object to is when people draw the wrong conclusions from imprecise language.

The Moon can look about 14 percent bigger during perigee, its closest approach to Earth, than during its apogee, its furthest point. It can appear 30 percent brighter, too. If you pay attention throughout the year, you might notice how much smaller or bigger the Moon can seem from month to month.

Fair enough, though the article starts out by comparing perigee to average, then compares perigee to apogee. Also not mentioned: the horizon moon illusion, where the rising or setting moon might appear larger to the eye. There's a very reasonable explanation for this illusion, and it has nothing to do with earth-moon distance.

Incidentally, the article doesn't do a very good follow-up on its promise in the subheadline, the "measure it with your hand" thing. There might be a way to do this, very roughly, but I suspect that individual differences in things like arm length and hand size overshadow the supermoon/micromoon difference.

Which, admittedly, is one I've never seen. I wouldn't know a particular full moon was a supermoon but for the websites I frequent, which all gush about it every time it happens. I know some people claim to be able to tell the difference, and I don't doubt them, but my perception just isn't that fine-tuned.

I make up for it by insisting upon pedantry.

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