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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Elycia Lee ☮

about 7 hours ago
In response to "There Is A Such Thing As Coincidence"

I loveeee the photo! You guys look good!

Coincidences are great! So happy you met!

2 Like

Wordsmitty ✍️

about 20 hours ago
In response to "Two Castles and a Cave"

That's definitely a wonderful picture that shows the whole story. When enlarged, it's a little fuzzy as if after quite a few bottles of wine.

1 Like

Annette in France

about 23 hours ago
In response to "Two Castles and a Cave"

After being in Tours for an afternoon/evening and only, my favorite part of my time there was to see you.
Old Tours with its medieval houses is charming to stroll through, but France has many charming old towns.

The most impressive cave system that I went to this vacation is under the Chateau de Brézé. It is where the Troglodytes lived. Their caves didn't only house people, they even had underground horse stables.

Interestingly, Chateau de Brézé is also a wine castle complete with an attached tasting room. Even the gift shop sells mostly wine along some postcards.

1 Like

Elisa: Middle Aged Stik

a day ago
In response to "Two Castles and a Cave"

Oddly enough, Tours is a sister city to Minneapolis. That still isn't enough of a reason for me to visit France, but it's still a nice bit of trivia.

2 Like

Kåre Enga in Montana

a day ago
In response to "There Is A Such Thing As Coincidence"

That's awesome. Enjoy your visit of the Loire.

TGV = tren à grande vitesse = HST. I only took it once, Paris > Luxembourg. Everything was a blur.

2 Like

Annette in France

a day ago
In response to "There Is A Such Thing As Coincidence"

That was a total coincidence. I didn't even plan to go to Tours at all. I was there because of the way I planned out a travel leg. To run into you on the other side of the Atlantic in a random town was pure luck.
It was fun to catch up with you.

2 Like

Brandiwyn🎶

a day ago
In response to "There Is A Such Thing As Coincidence"

Love it when stars align!

2 Like

Wordsmitty ✍️

Sep 24, 2024 at 5:15pm
In response to "All Out of Louvre"

I agree. You should always check the "Five Must Dos in ..." before you go, so you know what to avoid from crowds.

Smurfs are universal.

1 Like

Elisa: Middle Aged Stik

Sep 24, 2024 at 11:30am
In response to "All Out of Louvre"

That street art is definitely a step up.

1 Like

Annette in France

Sep 24, 2024 at 9:55am
In response to "All Out of Louvre"

I'm already out of Paris.

3 Like

February 5, 2020 at 12:02am

#974890

Maths and Polymaths

Ever feel inadequate?

Well, today's link isn't going to help.

https://www.bbc.com/worklife/article/20191118-what-shapes-a-polymath---and-do-we...

Why some people are impossibly talented
Polymaths excel in multiple fields. But what makes a polymath – and can their cross-discipline expertise help tackle some of society’s most pressing challenges?

If it weren’t for an actress and a pianist, GPS and WiFi might not exist.

In the late 1930s and early 40s, Hedy Lamarr was already the toast of Hollywood, famed for her portrayals of femme fatales. Few of her contemporaries knew that her other great passion was inventing.

Even today, most people don't know how awesome Hedy Lamarr was.

Besides helping to outline the specific traits that allow some people to juggle different fields of expertise so successfully, new research shows that there are many benefits of pursuing multiple interests, including increased life satisfaction, work productivity and creativity.

Despite having once served as an "expert witness," I don't consider myself an expert at anything. I do have a lot of interests, I think, but I can't say I score high in life satisfaction; I don't care about work productivity; and I'm about as creative as a paramecium.

There are, of course, some good reasons why we might be hesitant to pursue multiple interests. One is the fear that we might spread ourselves too thinly if we devote ourselves to more than one avocation. With a divided attention, we would fail to achieve success in any domain – the idea that the “Jack of all trades is the master of none”.

Yep, that's me. Hardly "all" trades, though.

In reality, there is some evidence that developing diverse disciplines can fuel creativity and productivity. So while the pursuit of a second or third interest may seem like a distraction, it can actually boost your success in your primary field.

Now we get to why I bothered to read and link this article. Presumably, the people reading this are writers, and what have we writers always heard? "Write what you know." I've said before that this advice should probably be turned around to "Know what you write," but either way, a lot of writers can benefit greatly from cross-discipline study. Some of the things I learned about personal finance, for example, can be applied to weight loss; both involve budgeting and tracking.

And when we write, we draw from all kinds of different sources. Sometimes it's sufficient to look something up when you need it, but better still if you have some background in the subject in question. While I'd wager none of us have experience in microgravity -- few people do -- some basic knowledge of orbital mechanics and Newtonian physics can help one write more compelling science fiction.

Some people figure they just go to school for some number of years and train in their chosen field. That's one way to get through life. Me? I'll stop learning when my brain turns to mush, through disease, injury, or death. I can't be afraid or unwilling to learn new things, because I never know when some obscure fact will pop up in my writing.

Problem is, the universe of human knowledge is expanding exponentially. I doubt that it's possible for any one person to know everything, as much as some people like to act as if they do. So we all have to be selective. But you can't know ahead of time what will be useful and what won't, so for me, I just assume that all knowledge is potentially useful.

There's no such thing as useless trivia, and yes, all of this will be on the quiz.

2 Like • Comment • View 3 comments

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