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:

Annette in France

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

Bienvenue en France.

1 Like

WakeUpAndLive️~Happiness

about 12 hours 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

about 12 hours 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

about 17 hours 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

about 18 hours 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

about 19 hours 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

a day ago
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

Cubboo!

a day ago
In response to "Get Baked"

Yes! I discovered this a few years back (slitting the potato about 1/4 - 1/2" lengthwise, then one or two slits across the longer slit.) It came out wonderful! It seems like this was called an English jacket potato, or something like that.

2 Like

Annette in France

a day ago
In response to "Get Baked"

I'll try your oven baked potato in winter when it's sensible to run the oven for 90 minutes.

2 Like

June 3, 2024 at 8:51am

#1072084

One for Sorrow, Two for Joy...

It's fairly well known that corvids are hardly birdbrains, but this recent article from SciAm delves into further evidence, while also punning on one of my favorite bands.

Crows Rival Human Toddlers in Counting Skills
Counting crows proclaim “caw, caw, caw, caw” when staring at the number four

The rock group Counting Crows were onto something when they chose their band name. Crows can indeed count, according to research published this week in Science.

That's not why they picked the name, but by Waltz's Law, never let facts get in the way of a good joke. Or a bad one. Especially a bad one.

But, you know, that's just, like, a pinion, man.

Okay, okay, puns out of the way for now, there's some interesting stuff here.

The results show that crows have counting capacities near those of human toddlers who are beginning to develop a knack for numbers, says lead study author Diana Liao, a postdoctoral researcher in neurobiology at the University of Tübingen in Germany.

So, no, they're not about to take over the world or build rockets to Mars.

“We think this is the first time this has been shown for any animal species,” she adds.

I know there's been research done in this area, but I can't be arsed to search for examples countering what the researcher says here. Yes, there have been claims that horses, e.g., can count, but those usually turned out to be hoaxes or misunderstandings.

They presented the birds with randomly ordered cues, four of which were visual—colored Arabic numbers that appeared on a touch screen—and four of which were auditory, including a short guitar chord and a drumroll.

Please, please tell me they sampled the latter from Counting Crows albums. Well, in my headcanon, they absolutely did.

Onur Güntürkün, a biopsychologist at Ruhr University Bochum in Germany, who was not involved in the research, says the new paper is “excellent”—even if the findings are “not unexpected” given all that scientists already know about crows and many other species’ intelligence.

Like I said: we know they're smart. They also apparently have a long memory. There are plenty of stories about people who pissed crows off, only to find themselves harassed by black-winged birds for the rest of their lives, indicating that not only does the pissed-off crow take revenge, but gets all their corvid buddies and offspring to punish the offender. People associate elephants with never forgetting, but elephants ain't got nothin' on crows.

On the flip side, do something nice for a crow, and they could become your army of darkness. I've thought about doing this by leaving shiny things out for them to collect, but I'm entirely too lazy, and what would I do with bird minions, anyway? Get them to attack surveillance drones? No, they might get hurt. Besides, my cats might get jealous.

Obviously, I can't leave this entry without an appropriate video:

If you think you need to go
If you wanted to be free
There's one thing you need to know
And that's that you can't count on me

2 Like • Comment • View 2 comments

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