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:

tj-Merry Mischief Maker

about 20 hours ago
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🎁

a day ago
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

a day ago
In response to "Trash Pandas"

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

4 Like

Jayngle Bells

a day ago
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

Annette

Dec 20, 2024 at 1:00pm
In response to "To Err Is Human"

Comments work.

"Like" button sends me this message:

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2 Like

Annette

Dec 20, 2024 at 12:58pm
In response to "To Err Is Human"

Can I leave this comment?

Wordsmitty ✍️

Dec 19, 2024 at 9:03pm
In response to "Five Seconds"

Way, way back when I was little, it was a three second rule that I remember. Sometime since it was upped to five seconds because of all the chemical preservatives that are pumped into everything now.

1 Like

tj-Merry Mischief Maker

Dec 19, 2024 at 3:43pm
In response to "Five Seconds"

I also ate carrots right out of the garden growing up, I also picked and ate fruits and berries growing in the wild without washing. I do remember food dropped on the floor was either rinsed off and then eaten, or if it was something not easily rinsed, like buttered toast buttered side down, we gave it to the dog.

In our house, there's not a five second rule; unless it's something dogs can't eat, it's whichever one gets it first.

2 Like

November 30, 2024 at 8:50am

#1080660

Tongue-Tied

Something interesting I found at BBC Future:

The weird way language affects our sense of time and space

It's also a book ad:

In our new book, we explore the many internal and external factors that influence and manipulate the way we think – from genetics to digital technology and advertising. And it appears that language can have a fascinating effect on the way we think about time and space.

And I'll do my usual pointing out that this isn't settled science.

But a growing number of experts believe language can influence how we think just as our thoughts and culture can shape how language develops.

Honestly, I'd kind of figured that was the case. I think in words, myself. Usually English ones. But I've asked other people, and some say they don't.

I imagine kids these days think in emoji.

For example, we know that people remember things they pay more attention to. And different languages force us to pay attention to an array of different things, be it gender, movement or colour.

I'm still just focused on getting French pronunciation close to correct.

Linguists, neuroscientists, psychologists and others have spent decades trying to uncover the ways in which language influences our thoughts, often focusing on abstract concepts such as space and time which are open to interpretation.

I hope they took into account my anecdotal evidence about not everyone thinking in words.

There follows a good bit of examples of the science people did to investigate this. I could probably nitpick some of the methods, but I don't feel like it today.

This is the part that most interested me, as someone trying to be bilingual:

Things start to get really strange, however, when looking at what happens in the minds of people who speak more than one language fluently. "With bilinguals, you are literally looking at two different languages in the same mind," explains Panos Athanasopoulos, a linguist at Lancaster University in the UK. "This means that you can establish a causal role of language on cognition, if you find that the same individual changes their behaviour when the language context changes."

While it is true that I complained about how bad this year's Beaujolais Nouveau was (it was really bad), and joked about how that means I'm turning French, that doesn't mean I'm turning French. When you learn a new language at my advanced age, things go a lot slower.

We aren't necessarily prisoners to thinking a certain way, though. Intriguingly, Casasanto has shown that you can quickly reverse people's mental time representation by training them to read mirror-reversed text, which goes in the opposite direction to what they're used to.

This refers to a part I didn't quote, which stated that people generally conceive of time moving in the same direction as their language writing. In contrast to language learning, I taught myself how to read mirror-reversed text (and upside-down, and upside-down mirror-reversed) when I was a kid, and I still conceive of time's arrow as moving left to right.

Maybe that's a case of just because I can do it doesn't mean I've internalized the connection to conception of time.

There's a whole lot more to the article, but I'll skip to near the end:

As this body of research grows, it is becoming increasingly clear that language is influencing how we think about the world around us and our passage through it.

Again, though, I wouldn't take it as settled science but more of a working hypothesis.

And while being multilingual won't necessarily make you a genius, we all can gain a fresh perspective and a more flexible understanding of the world by learning a new language.

A reasonable assertion, I think. One reason to actually learn a language rather than relying entirely on smartphone apps for translation.

2 Like • Comment

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