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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Writer_Mike

about 6 hours ago
In response to "Paris, I Guess"

Elisa: Middle Aged Stik

- I thought it very interesting, to say nothing of extraordinarily coincidental, that your video mentions a lock-adorned bridge in Paris. I just finished reading Spymaster by Brad Thor, in which the Russian mastermind takes strolls from Kaliningrad's mainland to Kant Island. The bridge he uses is also festooned with locks placed by couples declaring their undying love for one another.

QPdoll

about 23 hours ago
In response to "Landmarks"

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

2 Like

Wordsmitty ✍️

a day 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

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

Bienvenue en France.

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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

Sep 20, 2024 at 10:13pm
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.

April 11, 2023 at 10:16am

#1048020

Word to the Wise

Speaking of words...

Why people hate or love the sound of certain words

We’ve all experienced how certain sounds can grate on our nerves, such as the noise made by dragging your fingernails across a blackboard or the cry of a baby...

I suspect most people don't know what their own fingernails sound like dragged across a blackboard. Partly because who uses blackboards anymore, since like the 90s, and also partly because most people aren't evil enough to do it themselves. I, however, know exactly what my fingernails sounds like dragged across a blackboard.

Certainly I was a baby, but I don't remember how annoying my cries were. Knowing me, "very."

...but it turns out that the sounds of some words (like “virus”) can also affect how we feel and even give us a clue to what they mean (something to avoid).

I'd expect it to be the other way around: the name for something you want to avoid takes on a bad connotation. But that's why I'm reading the article, isn't it?

This phenomenon, where the sound of a word triggers an emotion or a meaning, is referred to as “sound symbolism”. Yet the idea that there might be a link between the sound of words and their meaning flies against accepted linguistic thinking going back more than a century.

And? Sometimes paradigms get reversed.

In our book...

Of course it's another book ad. What else is free on the internet anymore? Apart from this blog, I mean.

...we outline a radically new perspective on how we, as humans, got language in the first place, how children can learn and use it so effortlessly, and how sound symbolism figures into this.

Okay. Well, that first part sounds like speculation, but okay.

For example, if you pick a language at random that has the concept of “red”, the corresponding word is more likely than not to have an “r” sound in it — such as “rød” in Danish, “rouge” in French, and “krasnyy” (красный) in Russian.

Um.

The way an R is pronounced can be way different in different languages, such as the trilled R of Spanish or the nearly-German-CH sound of the letter in French.

Made-up words can be sound symbolic too.

All words are made up. Some just longer ago than others. And I'd be more surprised if newly minted words didn't have sound symbolism. Like when people go "whoosh" when a joke flies over someone's head.

In a classic study from 1929, the German psychologist Wolfgang Köhler observed that when Spanish speakers were shown a rounded shape and a spiky one and asked which one they thought were called “baluba” and which “takete”, most associated baluba with roundedness and takete with spikiness.

So much for flying against "accepted linguistic thinking going back more than a century."

Computer modelling of how children learn language has revealed that, as a child’s vocabulary grows, it becomes harder and harder to have unique sounds to signal different aspects of meaning (such as that all words relating to water should start with a “w”). Indeed, in a study of English sound-meaning mappings, we found that words that tend to be acquired earlier in development were more sound symbolic than words that are acquired later.

Computer modeling is a powerful tool, but it's only part of science. Also, I'm quite surprised these authors don't go into the "m" sound widely present in words associated with motherhood. Maybe they do, in the book.

I don't have much else to say, really (except "moist"). The article is less substantive than most book ads (which, again, I don't have a problem with here on a writing site), but the little bit of speculation I saw in it doesn't make me want to delve deeper by buying the book... and I'm predisposed to appreciating books on linguistics.

Still, I feel like it's something to keep in mind while writing, especially (but not limited to) writing that's going to be spoken, like a speech or screenplay: that the sound of words matters as well as their literal meaning.

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