Complex Numbers (Book)

Blog Calendar

◄ December ► 2023
S	M	T	W	T	F	S
					1	2
3	4	5	6	7	8	9
10	11	12	13	14	15	16
17	18	19	20	21	22	23
24	25	26	27	28	29	30
31
Archive				RSS

About This Author

Contact Me

Come closer.

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/

Signature for Honorable Mentions in 2018 Quill Awards

A signature for exclusive use of winners at the 2019 Quill Awards

Signature for those who have won a Quill Award at the 2020 Quill Awards

Show Recent Comments

10 Most Recent Blog Comments:

QPdoll

about 7 hours ago
In response to "Landmarks"

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

2 Like

Wordsmitty ✍️

about 10 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 21 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

December 19, 2023 at 9:36am

#1061184

Human (maybe not so) kind

I found this one interesting, but one needs to bear in mind that there might be an ideological agenda involved. Guardian link, so brace yourselves for British spelling:

Where did they all go? How Homo sapiens became the last human species left
At least nine hominin species once roamed the Earth, so what became of our vanished ancestors?

Just 300,000 years ago – a blink in evolutionary time – at least nine species of humans wandered the planet. Today, only our own, Homo sapiens, remains. And this raises one of the biggest questions in the story of human evolution: where did everyone else go?

Clearly, they built ships and left for interstellar space before we could take over.

“It’s not a coincidence that several of them disappeared around the time that Homo sapiens started to spread out of Africa and around the rest of the world,” says Prof Chris Stringer, head of human origins at the Natural History Museum in London. “What we don’t know is if that was a direct connection.”

Maybe professors in England use words differently than I do, but "it's not a coincidence," to me, implies "a direct connection."

There are many theories around the disappearance of our human cousins, and limited evidence to decipher exactly what happened. But recent studies are providing tantalising clues.

Also, these "theories" are what I'd call "guesses." Any extrapolation from one or two "tantalising clues" is almost as much speculation as my spaceship hypothesis up there.

In all seriousness, we can rule out spaceships. Despite recent sensationalist reporting, if there had been an industrial/technological civilization before ours, we would have found evidence, such as a layer of pollution in sedimentary rock.

Palaeoanthropologists continue to argue (quite vociferously) over who the last ancestor of H Sapiens was, but so far there is no conclusive evidence. Also, there is no single origin for H sapiens. There are ancient remains of early H sapiens in Jebel Irhoud in Morocco, Omo Kibish in Ethiopia and Florisbad in South Africa, suggesting that our species arose from multiple sites.

Yeah, okay, I'm not an expert. But this seems misleading. All it does is push back the possible date for last common ancestor.

“Hominin species were likely dying out all the time,” says Prof Eleanor Scerri, head of the human palaeosystems group at the Max Planck Institute of Geoanthropology in Jena, Germany. “It’s probably unusual that we are still around.”

Another instance of words that might mean something different in other countries. It's not "probably unusual that we are still around;" it is, by definition, definitely unusual.

From Neanderthal and Denisovan genomes, researchers have inferred that they lived in small groups and frequently interbred.

Again, I'm not an expert. But the definition of "species" that I learned includes the idea that members of the species can breed together and produce fertile offspring. One classic counterexample is how horses and donkeys can breed, but the offspring is sterile. It's well-known today that most humans carry DNA from Neandertal ancestors; there was interbreeding there, too, and, clearly, their offspring weren't sterile, else we wouldn't be here. (The article mentions this, too.)

Part of the problem is our nearly OCD need to categorize, label, and neatly box everything. But biology is messy and resists being pigeonholed, as illustrated by this quote from the article:

Prof Rebecca Ackermann, co-director of the Human Evolution Research Institute at the University of Cape Town in South Africa, says that it depends on how you define species. This is the source of much debate among palaeoanthropologists: some recognise many species, while others acknowledge only a handful. “My take is that they probably weren’t distinct species,” she says, with the exception of obvious outliers such as small-brained H naledi. “We should really be talking about them as regional variants.”

Later in the article:

But some groups – whether a different species or not – definitely fared better than others, with our own direct forebears surviving. This is in large part because of luck and their behaviour, agree the experts I spoke to – and is something people living today need to recognise to overcome the challenges on the horizon.

And this is what I meant by an ideological agenda being involved. Just because it happens to be one I (mostly) agree with, doesn't mean I'm going to accept all this speculation at face value. Science shouldn't work like that. Make all the guesses you want; collect evidence to rule some of them out.

“Humanity will be faced with either cooperating in the face of those crises or competing. And what we see from Neanderthals and H sapiens is that the groups that cooperated better were the ones that got through.”

I like the conclusion. I agree with it. I've been saying for a while that we probably owe our success to cooperation more than competition; that it's the former that truly drives progress. But that only makes me more wary of the way they reached the conclusion, lest I fall victim to confirmation bias.

5 Like • Comment • View 2 comments

© Copyright 2024 Waltz en France (UN: cathartes02 at Writing.Com). All rights reserved.
Waltz en France has granted InkSpot.Com, its affiliates and its syndicates non-exclusive rights to display this work.