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:

Wordsmitty ✍️

about 2 hours ago
In response to "Gray Matters"

"who were between age nine and 39," - so the old folks need the rejuvenation formula first to turn back the years.

I wonder if anyone has studied why hair goes gray in certain places and not everywhere. Maybe different kinds of stress?

1 Like

🌖 HuntersMoon

about 5 hours ago
In response to "Gray Matters"

I've always believed "better gray than gone away..."

1 Like

tj-Merry Mischief Maker

a day ago
In response to "A Mammoth Discovery"

This makes think of a book, "Evolution and Genomic Diversity" by Gene Poole.

1 Like

tj-Merry Mischief Maker

a day ago
In response to "Hot Shoulder"

Maybe if we say, "Betelgeuse" three times in a row we can witness it.

1 Like

tj-Merry Mischief Maker

Dec 23, 2024 at 4:55pm
In response to "Trash Pandas"

Being an avid camper, keeping items out of the reach of raccoons can be challenging. We've had them figure out how to open coolers, open chests, and even gnaw through rope used to suspend a cooler out of their reach.

2 Like

🐕GeminiGem🎁

Dec 23, 2024 at 12:56pm
In response to "Trash Pandas"

I wrote one of my Danger Dog stories based on a childhood camping trip that involved a raccoon stealing grapes from our cooler "Danger Dog's Camping Capers"

I had always heard that cockroaches will be the only thing that survives a nuclear war/apocolypse, but maybe it will actually be the raccoons.

4 Like

🌖 HuntersMoon

Dec 23, 2024 at 10:04am
In response to "Trash Pandas"

Just wait until we team up with the cats...

4 Like

Jayngle Bells

Dec 23, 2024 at 10:03am
In response to "Trash Pandas"

The back of my property is sheltered by buildings and cedar trees, so I live on a "racoon highway." Tinfoil works until one takes it for the team and moves it. The only scent that somewhat deters then is ammonia - but they alter their path within the already preferred walkway, not avoid it completely. They generally leave the garbage bin alone, but that's because of the bears, not because of anything we do 😂

I've had more wildlife problems in the city than I ever did in the country. The first time I saw a deer up close? In the middle of my street directly in front of my house. Probably looking for the trash panda party. Lol

4 Like

Annette

Dec 21, 2024 at 11:59am
In response to "Neutral Zone"

It made me happy that I was able to like it. I take the small victories.

1 Like

tj-Merry Mischief Maker

Dec 20, 2024 at 6:06pm
In response to "To Err Is Human"

Right from the beginning I find myself asking, "How many times can anything go extinct?" On a side note, the other day while shopping I saw a neanderthal in the produce section. He was trying to fit in with modern clothes, but it was pretty obvious he was a neanderthal.

1 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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