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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QPdoll

about 5 hours ago
In response to "Landmarks"

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

1 Like

Wordsmitty ✍️

about 8 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 19 hours ago
In response to "Paris, I Guess"

Bienvenue en France.

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WakeUpAndLive️~Happiness

about 23 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 23 hours ago
In response to "Paris, I Guess"

Could not resist sending the link: https://www.atlasobscura.com/things-to-do/paris-france

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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

January 9, 2024 at 8:36am

#1062104

Journey to the New Word

From that ivory tower of knowledge known as Cracked, an article about words:

5 Words That Never Existed Till Oddly Recently

As December draws to a close, it’s time for the world’s dictionaries to name their Words of the Year.

Yes, this article was from last month.

Most words feel like they must have been around forever. But all words are made-up, and some of them were made-up pretty recently.

My goal in life is to make up a word and have it be widely used. I mean, sure, that's already happened: I came up with "rad" (the slang word, not the radiation level, which admittedly inspired me) back in the 70s. But no one knows it was me, and so no one believes me.

In fairness, someone else might have invented it independently. But I'm going with "they stole it from me."

Anyway. The article:

5. Boy

And yet, the word we have now for a male child, “boy,” isn’t that old. It only came about in the 16th century.

I've looked into this before. Maybe even here; I don't remember. Seems to me there's some debate about the actual origin. In this case, though, I didn't make it up.

Before that, for a few centuries, you know what we called male children? We called them girls. We also called female children girls. “Girl” was a unisex word.

I'm also unclear on the meaning of "we" here. 16th century would have been just after the end of Middle English.

Not to mention there were hundreds or thousands of words added to English since "boy."

4. Guy

For starters, the word was a name for around a thousand years longer than it was a common noun. Then came the big guy, Guy Fawkes.

Ah, yes, one of the prototypical religious terrorists.

To complicate things, I've heard this word pronounced both like it rhymes with "die" or rhymes with "key."

It now refers to people of all genders, though that final shift wasn’t so much a gradual expansion of its meaning as an assumption that any word for men can also represent all humankind.

No, that's not the assumption at all. We're moving away from that as a culture. The problem is that English now lacks a second person plural, and y'all won't adopt y'all, instead opting for "you guys."

3. Smoking

We say a fire is smoking up the place when it creates smoke, so it’s pretty weird that we say the same thing when we (or ham) take in smoke.

Not that weird at all. The smoke is happening whether you're inhaling or not, so sure, smoking is still being used in its older meaning, and this section is interesting but nonsensical.

2. Genocide

Gotta admit, the origin of this word surprised me. Specifically, that it's less than a hundred years old.

The word “genocide” was invented by one man: Polish lawyer Raphael Lemkin... The book was published in 1944. By 1948, the U.N. wrote up a treaty criminalizing genocide by name.

1. Forecast

Well, it's not like we needed a word like "aftcast." That would, however, be a fun synonym for "recent history."

By coining the term “forecast,” FitzRoy made the idea sound magical, which wasn’t particularly conducive to convincing skeptical scientists.

Sure, but I bet it got eaten up by the superstitious public.

So, yeah, just some fun with words. Nothing too rad, but I didn't want to just yeet the article into the dustbin.

4 Like • Comment • View 4 comments

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