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 9 hours ago
In response to "Landmarks"

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

2 Like

Wordsmitty ✍️

about 12 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.

2 Like

Annette in France

about 23 hours 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

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/

3 Like

WakeUpAndLive️~Happiness

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

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

2 Like

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!))

2 Like

November 23, 2023 at 10:04am

#1059989

Who?

Inspired again by "Invalid Item"

, because, here in the US, whatever else is commemorated today gets overshadowed by a national holiday, I wanted to highlight a few other observances that fall on November 23 every year, or happen to fall on November 23 this year.

First, let's get the turkey in the room out of the way:

Unthanksgiving Day

and Turkey-Free Thanksgiving.

The first link, there, provides a good bit of detail about Unthanksgiving. In brief, it's also known as The Indigenous Peoples Sunrise Ceremony, and it takes place on Alcatraz on the fourth Thursday in November. Now, I can't claim Native American heritage, and it's against my principles to do anything but sleep at sunrise (especially in that time zone, which, if my luck holds out, I'll enter in a couple of weeks), but I appreciate that it happens. The history of it is absolutely worth reading.

In contrast, no information is provided for the second link, except for its name and that it also coincides with US Thanksgiving. There's another link to follow in there, but I can't be arsed. My only comment on it is that if you're thinking "tofurkey," please stop that.

Some of these "holidays" for the year seem to be made up by kids. In the case of Wolfenoot

, it actually was made up by a kid. It's always on November 23, so it won't always coincide with US Thanksgiving; nor was it, as you might infer from the name, created in the US. According to that link, "Wolfnoot[sic] is a 'celebration of canines, kindness, and humans who embrace both.' Showing kindness to animals, especially to wolves and dogs, is an important part of the day."

I mean, technically, wolves are lupines, but even I tend to cut 7-year-olds some slack when it comes to technicalities. I'm more of a cat person, as you know, but I still like the idea of this holiday. Except for the social media aspects, as detailed at the link.

Another one always observed on November 23 is Eat a Cranberry Day.

I'd love to, but sadly, Dolores O'Riordan

is no longer with us. Nootch!

Confession time: I'd originally thought to make this entire entry about Fibonacci Day.

Like Pi Day (3/14), it's a mathematical pun, and we all know I can't resist those; it's right there in the name of the blog. Instead, I'll keep it brief in the interest of keeping people reading: The Fibonacci Sequence (or Series), which is important in the branch of mathematics known as number theory, begins with 1,1,2,3... And since today is 11/23, well... you know. You get the next number in the series by adding together the previous two; hence, the next number in the series is 5. So maybe I'll raise a glass in celebration at 5pm. Or maybe not; more likely, I'll be napping then.

There are other observances this day, but the last one I want to highlight is of great personal relevance to Me, which makes it the Most Important: Doctor Who Day.

It, too, always takes place on November 23, because that's the Gregorian calendar anniversary of the airing of the first episode, back in 1963. Math-inclined readers may notice that this year's is significant because it's the 60th anniversary. Even more astute readers will note that the first episode aired the day after JFK was assassinated; somewhat famously, that first programme (look, it's British, so I'll use their spelling) was interrupted by ongoing coverage of that event, a big deal even in England.

The personal relevance, though, is only that I was introduced to the show, in syndication here in the US, when I was a kid, and immediately loved it. And still do, in all its regenerations. Not as much as Star Trek, maybe, but one must remember that, at the time, the only Star Trek was reruns of the Original Series, and a few episodes of the possibly non-canonical Animated Series (which is severely underrated). Apart from the movies, new Trek wouldn't come to be until 1987, so I got my SF fix from Doctor Who in the meantime. (The BBC missed a grand opportunity by not making "Doctor Who: The Next Regeneration." Don't look at me like that; they pun more than I do.) Like many people around my age, to me, Tom Baker will always be The Doctor—though I appreciate what other actors brought to the role, especially Tennant.

So of all the things to celebrate today, for me, I think it's going to be that one.

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